{"id":64,"date":"2017-10-08T19:32:36","date_gmt":"2017-10-08T19:32:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/?post_type=chapter&#038;p=64"},"modified":"2017-10-08T19:34:34","modified_gmt":"2017-10-08T19:34:34","slug":"the-ideal-gas-law","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/chapter\/the-ideal-gas-law\/","title":{"raw":"The Ideal Gas Law","rendered":"The Ideal Gas Law"},"content":{"raw":"<h1 class=\"entry-title\">The Ideal Gas Law<\/h1>\r\n<div class=\"difficulty\"><\/div>\r\n<div id=\"post-2650\" class=\"standard post-2650 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>State the ideal gas law in terms of molecules and in terms of moles.<\/li>\r\n \t<li>Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.<\/li>\r\n \t<li>Use Avogadro\u2019s number to convert between number of molecules and number of moles.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"wp-caption alignright\">\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20104241\/Figure_14_03_00.jpg\" alt=\"Figure (Figure_14_03_00.jpg)\" width=\"200\" height=\"553\" \/>\r\n<p class=\"wp-caption-text\">Figure 1. The air inside this hot air balloon flying over Putrajaya, Malaysia, is hotter than the ambient air. As a result, the balloon experiences a buoyant force pushing it upward. (credit: Kevin Poh, Flickr)<\/p>\r\n\r\n<\/div>\r\nIn this section, we continue to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (Most gases, for example nitrogen, N<sub>2<\/sub>, and oxygen, O<sub>2<\/sub>, are composed of two or more atoms. We will primarily use the term \u201cmolecule\u201d in discussing a gas because the term can also be applied to monatomic gases, such as helium.)\r\n\r\nGases are easily compressed. We can see evidence of this in <a href=\"https:\/\/courses.lumenlearning.com\/physics\/chapter\/13-3-the-ideal-gas-law\/chapter\/13-2-thermal-expansion-of-solids-and-liquids\/\" target=\"_blank\" rel=\"noopener\">Table 1 in\u00a0Thermal Expansion of Solids and Liquids<\/a>, where you will note that gases have the <em>largest<\/em> coefficients of volume expansion. The large coefficients mean that gases expand and contract very rapidly with temperature changes. In addition, you will note that most gases expand at the <em>same<\/em> rate, or have the same <em>\u03b2<\/em>. This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates.\r\n\r\nThe answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in Figure 2. Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.\r\n<div class=\"wp-caption aligncenter\">\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20104246\/Figure_14_03_01.jpg\" alt=\"Spheres representing atoms and molecules; the spheres are relatively far apart and are distributed randomly.\" width=\"599\" height=\"279\" \/>\r\n<p class=\"wp-caption-text\">Figure 2. Atoms and molecules in a gas are typically widely separated, as shown. Because the forces between them are quite weak at these distances, the properties of a gas depend more on the number of atoms per unit volume and on temperature than on the type of atom.<\/p>\r\n\r\n<\/div>\r\nTo get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into an initially deflated tire. The tire\u2019s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If we continue to pump air into it, the pressure increases. The pressure will further increase when the car is driven and the tires move. Most manufacturers specify optimal tire pressure for cold tires. (See Figure 3.)\r\n<div class=\"wp-caption aligncenter\">\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20104251\/Figure_14_03_02.jpg\" alt=\"The figure has three parts, each part showing a pair of tires, and each tire connected to a pressure gauge. Each pair of tires represents the before and after images of a single tire, along with a change in pressure in that tire. In part a, the tire pressure is initially zero. After some air is added, represented by an arrow labeled Add air, the pressure rises to slightly above zero. In part b, the tire pressure is initially at the half-way mark. After some air is added, represented by an arrow labeled Add air, the pressure rises to the three-fourths mark. In part c, the tire pressure is initially at the three-fourths mark. After the temperature is raised, represented by an arrow labeled Increase temperature, the pressure rises to nearly the full mark.\" width=\"750\" height=\"189\" \/>\r\n<p class=\"wp-caption-text\">Figure 3. (a) When air is pumped into a deflated tire, its volume first increases without much increase in pressure. (b) When the tire is filled to a certain point, the tire walls resist further expansion and the pressure increases with more air. (c) Once the tire is inflated, its pressure increases with temperature.<\/p>\r\n\r\n<\/div>\r\nAt room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.\r\n<div class=\"textbox shaded\">\r\n<h3>Ideal Gas Law<\/h3>\r\nThe <em>ideal gas law<\/em> states that\u00a0<em>PV<\/em> =\u00a0<em>NkT<\/em>,\u00a0where <em>P<\/em> is the absolute pressure of a gas, <em>V<\/em> is the volume it occupies, <em>N<\/em> is the number of atoms and molecules in the gas, and <em>T<\/em> is its absolute temperature. The constant <em>k<\/em> is called the <em>Boltzmann constant<\/em> in honor of Austrian physicist Ludwig Boltzmann (1844\u20131906) and has the value\u00a0<em>k<\/em> = 1.38\u00a0\u00d7 10<sup>\u221223<\/sup> J\/K.\r\n\r\n<\/div>\r\nThe ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles\u2019 law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle\u2019s law (that for a fixed temperature, the product <em>PV<\/em>\u00a0is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of <em>V<\/em>. The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that <em>N<\/em> is the total number of atoms and molecules, independent of the type of gas.)\r\n\r\nLet us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure <em>P<\/em> is essentially equal to atmospheric pressure, and the volume <em>V<\/em> increases in direct proportion to the number of atoms and molecules <em>N<\/em> put into the tire. Once the volume of the tire is constant, the equation <em>PV<\/em> =\u00a0<em>NkT<\/em>\u00a0predicts that the pressure should increase in proportion to <em>the number N of atoms and molecules<\/em>.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1. Calculating Pressure Changes Due to Temperature Changes: Tire Pressure<\/h3>\r\nSuppose your bicycle tire is fully inflated, with an absolute pressure of 7.00 \u00d7 10<sup>5<\/sup> Pa\u00a0(a gauge pressure of just under 90.0 lb\/in<sup>2<\/sup>) at a temperature of 18.0\u00baC. What is the pressure after its temperature has risen to 35.0\u00baC? Assume that there are no appreciable leaks or changes in volume.\r\n<h4>Strategy<\/h4>\r\nThe pressure in the tire is changing only because of changes in temperature. First we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown.\r\n\r\nWe know the initial pressure <em>P<\/em><sub>0<\/sub> = 7.00\u00a0\u00d7 10<sup>5<\/sup> Pa, the initial temperature <em>T<\/em><sub>0<\/sub>\u00a0= 18.0\u00baC, and the final temperature <em>T<\/em><sub>f<\/sub>\u00a0= 35.0\u00baC. We must find the final pressure <em>P<\/em><sub>f<\/sub>. How can we use the equation <em>PV<\/em> = <em>NkT<\/em>? At first, it may seem that not enough information is given, because the volume <em>V<\/em> and number of atoms <em>N<\/em> are not specified. What we can do is use the equation twice: <em>P<\/em><sub>0<\/sub><em>V<\/em><sub>0<\/sub>\u00a0= NkT<sub>0<\/sub> and <em>P<\/em><sub>f<\/sub><em>V<\/em><sub>f<\/sub>\u00a0= NkT<sub>f<\/sub>. If we divide <em>P<\/em><sub>f<\/sub><em>V<\/em><sub>f<\/sub> by <em>P<\/em><sub>0<\/sub><em>V<\/em><sub>0<\/sub> we can come up with an equation that allows us to solve for <em>P<\/em><sub>f<\/sub>.\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-1-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-1\" class=\"mjx-math\"><span id=\"MJXc-Node-2\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-3\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-4\" class=\"mjx-mrow\"><span id=\"MJXc-Node-5\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-6\" class=\"mjx-mrow\"><span id=\"MJXc-Node-7\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-8\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-9\" class=\"mjx-texatom\"><span id=\"MJXc-Node-10\" class=\"mjx-mrow\"><span id=\"MJXc-Node-11\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-12\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-13\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-14\" class=\"mjx-texatom\"><span id=\"MJXc-Node-15\" class=\"mjx-mrow\"><span id=\"MJXc-Node-16\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-17\" class=\"mjx-mrow\"><span id=\"MJXc-Node-18\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-19\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-20\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-21\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-22\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-23\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-24\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-25\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-26\" class=\"mjx-mrow\"><span id=\"MJXc-Node-27\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-28\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-29\" class=\"mjx-texatom\"><span id=\"MJXc-Node-30\" class=\"mjx-mrow\"><span id=\"MJXc-Node-31\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-32\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-33\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-34\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-35\" class=\"mjx-texatom\"><span id=\"MJXc-Node-36\" class=\"mjx-mrow\"><span id=\"MJXc-Node-37\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-38\" class=\"mjx-mrow\"><span id=\"MJXc-Node-39\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-40\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-41\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-42\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-43\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-44\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-45\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-46\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nSince the volume is constant, <em>V<\/em><sub>f<\/sub> and <em>V<\/em><sub>0<\/sub> are the same and they cancel out. The same is true for <em>N<\/em><sub>f<\/sub> and <em>N<\/em><sub>0<\/sub>, and <em>k<\/em>, which is a constant. Therefore,\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-2-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-47\" class=\"mjx-math\"><span id=\"MJXc-Node-48\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-49\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-50\" class=\"mjx-mrow\"><span id=\"MJXc-Node-51\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-52\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-53\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-54\" class=\"mjx-texatom\"><span id=\"MJXc-Node-55\" class=\"mjx-mrow\"><span id=\"MJXc-Node-56\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-57\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-58\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-59\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-60\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-61\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-62\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-63\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-64\" class=\"mjx-texatom\"><span id=\"MJXc-Node-65\" class=\"mjx-mrow\"><span id=\"MJXc-Node-66\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-67\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-68\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-69\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-70\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nWe can then rearrange this to solve for <em>P<\/em><sub>f<\/sub>: <span class=\"mj\"><span id=\"MathJax-Element-3-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-71\" class=\"mjx-math\"><span id=\"MJXc-Node-72\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-73\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-74\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-75\" class=\"mjx-texatom\"><span id=\"MJXc-Node-76\" class=\"mjx-mrow\"><span id=\"MJXc-Node-77\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-78\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-79\" class=\"mjx-msubsup MJXc-space3\"><span class=\"mjx-base\"><span id=\"MJXc-Node-80\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-81\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-82\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-83\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-84\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-85\" class=\"mjx-texatom\"><span id=\"MJXc-Node-86\" class=\"mjx-mrow\"><span id=\"MJXc-Node-87\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-88\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-89\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-90\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-91\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\n,\u00a0where the temperature must be in units of kelvins, because <em>T<\/em><sub>0<\/sub> and <em>T<\/em><sub>f<\/sub> are absolute temperatures.\r\n<h4>Solution<\/h4>\r\nConvert temperatures from Celsius to Kelvin:\r\n\r\n<em>T<\/em><sub>0<\/sub> = (18.0 + 273)K = 291 K\r\n\r\n<em>T<\/em><sub>f<\/sub> = (35.0 + 273)K = 308\u00a0K\r\n\r\nSubstitute the known values into the equation.\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-4-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-92\" class=\"mjx-math\"><span id=\"MJXc-Node-93\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-94\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-95\" class=\"mjx-mrow\"><span id=\"MJXc-Node-96\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-97\" class=\"mjx-texatom\"><span id=\"MJXc-Node-98\" class=\"mjx-mrow\"><span id=\"MJXc-Node-99\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-100\" class=\"mjx-texatom\"><span id=\"MJXc-Node-101\" class=\"mjx-mrow\"><span id=\"MJXc-Node-102\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-103\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-104\" class=\"mjx-msubsup MJXc-space3\"><span class=\"mjx-base\"><span id=\"MJXc-Node-105\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-106\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-107\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-108\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-109\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-110\" class=\"mjx-texatom\"><span id=\"MJXc-Node-111\" class=\"mjx-mrow\"><span id=\"MJXc-Node-112\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-113\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-114\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-115\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-116\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-117\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">7.00<\/span><\/span><span id=\"MJXc-Node-118\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-119\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-120\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-121\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-122\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><span id=\"MJXc-Node-123\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-124\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">(<\/span><\/span><span id=\"MJXc-Node-125\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-126\" class=\"mjx-mrow\"><span id=\"MJXc-Node-127\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">308<\/span><\/span><span id=\"MJXc-Node-128\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-129\" class=\"mjx-mrow\"><span id=\"MJXc-Node-130\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">291<\/span><\/span><span id=\"MJXc-Node-131\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-132\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-133\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-134\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">7.41<\/span><\/span><span id=\"MJXc-Node-135\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-136\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-137\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-138\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-139\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-140\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n<div class=\"textbox examples\">\r\n<h4>Discussion<\/h4>\r\nThe final temperature is about 6% greater than the original temperature, so the final pressure is about 6% greater as well. Note that <em>absolute<\/em> pressure and <em>absolute<\/em> temperature must be used in the ideal gas law.\r\n\r\n<\/div>\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Making Connections: Take-Home Experiment\u2014Refrigerating a Balloon<\/h3>\r\nInflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why?\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2. Calculating the Number of Molecules in a Cubic Meter of Gas<\/h3>\r\nHow many molecules are in a typical object, such as gas in a tire or water in a drink? We can use the ideal gas law to give us an idea of how large <em>N<\/em> typically is.\r\n\r\nCalculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP), which is defined to be 0\u00baC and atmospheric pressure.\r\n<h4>Strategy<\/h4>\r\nBecause pressure, volume, and temperature are all specified, we can use the ideal gas law <em>PV<\/em> =\u00a0<em>NkT<\/em>, to find <em>N<\/em>.\r\n<h4>Solution<\/h4>\r\nIdentify the knowns:\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-5-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-141\" class=\"mjx-math\"><span id=\"MJXc-Node-142\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-143\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-144\" class=\"mjx-mtr\"><span id=\"MJXc-Node-145\" class=\"mjx-mtd\"><span id=\"MJXc-Node-146\" class=\"mjx-mrow\"><span id=\"MJXc-Node-147\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-148\" class=\"mjx-mtd\"><span id=\"MJXc-Node-149\" class=\"mjx-mrow\"><span id=\"MJXc-Node-150\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-151\" class=\"mjx-mtd\"><span id=\"MJXc-Node-152\" class=\"mjx-mrow\"><span id=\"MJXc-Node-153\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-154\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-155\" class=\"mjx-texatom\"><span id=\"MJXc-Node-156\" class=\"mjx-mrow\"><span id=\"MJXc-Node-157\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-158\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">C<\/span><\/span><span id=\"MJXc-Node-159\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-160\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">273<\/span><\/span><span id=\"MJXc-Node-161\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-162\" class=\"mjx-mtr\"><span id=\"MJXc-Node-163\" class=\"mjx-mtd\"><span id=\"MJXc-Node-164\" class=\"mjx-mrow\"><span id=\"MJXc-Node-165\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-166\" class=\"mjx-mtd\"><span id=\"MJXc-Node-167\" class=\"mjx-mrow\"><span id=\"MJXc-Node-168\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-169\" class=\"mjx-mtd\"><span id=\"MJXc-Node-170\" class=\"mjx-mrow\"><span id=\"MJXc-Node-171\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.01<\/span><\/span><span id=\"MJXc-Node-172\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-173\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-174\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-175\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-176\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-177\" class=\"mjx-mtr\"><span id=\"MJXc-Node-178\" class=\"mjx-mtd\"><span id=\"MJXc-Node-179\" class=\"mjx-mrow\"><span id=\"MJXc-Node-180\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-181\" class=\"mjx-mtd\"><span id=\"MJXc-Node-182\" class=\"mjx-mrow\"><span id=\"MJXc-Node-183\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-184\" class=\"mjx-mtd\"><span id=\"MJXc-Node-185\" class=\"mjx-mrow\"><span id=\"MJXc-Node-186\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.00<\/span><\/span><span id=\"MJXc-Node-187\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-188\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-189\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-190\" class=\"mjx-mtr\"><span id=\"MJXc-Node-191\" class=\"mjx-mtd\"><span id=\"MJXc-Node-192\" class=\"mjx-mrow\"><span id=\"MJXc-Node-193\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-194\" class=\"mjx-mtd\"><span id=\"MJXc-Node-195\" class=\"mjx-mrow\"><span id=\"MJXc-Node-196\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-197\" class=\"mjx-mtd\"><span id=\"MJXc-Node-198\" class=\"mjx-mrow\"><span id=\"MJXc-Node-199\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.38<\/span><\/span><span id=\"MJXc-Node-200\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-201\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-202\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-203\" class=\"mjx-texatom\"><span id=\"MJXc-Node-204\" class=\"mjx-mrow\"><span id=\"MJXc-Node-205\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-206\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">23<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-207\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0J\/K<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-208\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\n<\/div>\r\nIdentify the unknown: number of molecules, <em>N<\/em>.\r\n\r\nRearrange the ideal gas law to solve for <em>N<\/em>:\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-6-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-209\" class=\"mjx-math\"><span id=\"MJXc-Node-210\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-211\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-212\" class=\"mjx-mtr\"><span id=\"MJXc-Node-213\" class=\"mjx-mtd\"><span id=\"MJXc-Node-214\" class=\"mjx-mrow\"><span id=\"MJXc-Node-215\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-216\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-217\" class=\"mjx-mtd\"><span id=\"MJXc-Node-218\" class=\"mjx-mrow\"><span id=\"MJXc-Node-219\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-220\" class=\"mjx-mtd\"><span id=\"MJXc-Node-221\" class=\"mjx-mrow\"><span id=\"MJXc-Node-222\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><span id=\"MJXc-Node-223\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-224\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-225\" class=\"mjx-mtr\"><span id=\"MJXc-Node-226\" class=\"mjx-mtd\"><span id=\"MJXc-Node-227\" class=\"mjx-mrow\"><span id=\"MJXc-Node-228\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-229\" class=\"mjx-mtd\"><span id=\"MJXc-Node-230\" class=\"mjx-mrow\"><span id=\"MJXc-Node-231\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-232\" class=\"mjx-mtd\"><span id=\"MJXc-Node-233\" class=\"mjx-mrow\"><span id=\"MJXc-Node-234\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-235\" class=\"mjx-mrow\"><span id=\"MJXc-Node-236\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-237\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-238\" class=\"mjx-mrow\"><span id=\"MJXc-Node-239\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-240\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-241\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nSubstitute the known values into the equation and solve for <em>N<\/em>:\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-7-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-242\" class=\"mjx-math\"><span id=\"MJXc-Node-243\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-244\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-245\" class=\"mjx-mrow\"><span id=\"MJXc-Node-246\" class=\"mjx-texatom\"><span id=\"MJXc-Node-247\" class=\"mjx-mrow\"><span id=\"MJXc-Node-248\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-249\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-250\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-251\" class=\"mjx-mrow\"><span id=\"MJXc-Node-252\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-253\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-254\" class=\"mjx-mrow\"><span id=\"MJXc-Node-255\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-256\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-257\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-258\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-259\" class=\"mjx-mrow\"><span id=\"MJXc-Node-260\" class=\"mjx-mrow\"><span id=\"MJXc-Node-261\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-262\" class=\"mjx-mrow\"><span id=\"MJXc-Node-263\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.01<\/span><\/span><span id=\"MJXc-Node-264\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-265\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-266\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-267\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-268\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><span id=\"MJXc-Node-269\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-270\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-271\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-272\" class=\"mjx-mrow\"><span id=\"MJXc-Node-273\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.00<\/span><\/span><span id=\"MJXc-Node-274\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-275\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-276\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-277\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-278\" class=\"mjx-mrow\"><span id=\"MJXc-Node-279\" class=\"mjx-mrow\"><span id=\"MJXc-Node-280\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-281\" class=\"mjx-mrow\"><span id=\"MJXc-Node-282\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.38<\/span><\/span><span id=\"MJXc-Node-283\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-284\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-285\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-286\" class=\"mjx-texatom\"><span id=\"MJXc-Node-287\" class=\"mjx-mrow\"><span id=\"MJXc-Node-288\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-289\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">23<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-290\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0J\/K<\/span><\/span><\/span><span id=\"MJXc-Node-291\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-292\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-293\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-294\" class=\"mjx-mrow\"><span id=\"MJXc-Node-295\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">273<\/span><\/span><span id=\"MJXc-Node-296\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><span id=\"MJXc-Node-297\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-298\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-299\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2.68<\/span><\/span><span id=\"MJXc-Node-300\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-301\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-302\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-303\" class=\"mjx-texatom\"><span id=\"MJXc-Node-304\" class=\"mjx-mrow\"><span id=\"MJXc-Node-305\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">25<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-306\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-307\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n<div class=\"textbox examples\">\r\n<h4>Discussion<\/h4>\r\nThis number is undeniably large, considering that a gas is mostly empty space. <em>N<\/em> is huge, even in small volumes. For example, 1 cm<sup>3<\/sup>\u00a0of a gas at STP has 2.68 \u00d7 10<sup>19<\/sup> molecules in it. Once again, note that <em>N<\/em> is the same for all types or mixtures of gases.\r\n\r\n<\/div>\r\n<h2>Moles and Avogadro\u2019s Number<\/h2>\r\nIt is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A <em>mole<\/em> (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called <em>Avogadro\u2019s number<\/em> (<em>N<\/em><sub>A<\/sub>), in recognition of Italian scientist Amedeo Avogadro (1776\u20131856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, contain equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro\u2019s number is\u00a0<em>N<\/em><sub>A<\/sub> =\u00a06.02 \u00d7 10<sup>23<\/sup> mol<sup>\u22121<\/sup>.\r\n<div class=\"textbox shaded\">\r\n<h3>Avogadro\u2019s Number<\/h3>\r\nOne mole always contains 6.02 \u00d7 10<sup>23<\/sup> particles (atoms or molecules), independent of the element or substance. A mole of any substance has a mass in grams equal to its molecular mass, which can be calculated from the atomic masses given in the periodic table of elements.\r\n\r\n<em>N<\/em><sub>A<\/sub> =\u00a06.02 \u00d7 10<sup>23<\/sup> mol<sup>\u22121<\/sup>\r\n\r\n<\/div>\r\n<div class=\"wp-caption aligncenter\">\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20104332\/Figure_14_03_03.jpg\" alt=\"The illustration shows relatively flat land with a solitary mountain, labeled Mt. Everest, and blue sky above. A double-headed vertical arrow stretches between the land and a point in the sky that is well above the peak of the mountain. The arrow, labeled table tennis balls, serves to indicate that a column of one mole of table tennis balls would reach a point in the sky that is much higher than the peak of Mt. Everest.\" width=\"750\" height=\"320\" \/>\r\n<p class=\"wp-caption-text\">Figure 4. How big is a mole? On a macroscopic level, one mole of table tennis balls would cover the Earth to a depth of about 40 km.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Check Your Understanding<\/h3>\r\nThe active ingredient in a Tylenol pill is 325 mg of acetaminophen (C<sub>8<\/sub>H<sub>9<\/sub>NO<sub>2<\/sub>). Find the number of active molecules of acetaminophen in a single pill.\r\n<h4>Solution<\/h4>\r\nWe first need to calculate the molar mass (the mass of one mole) of acetaminophen. To do this, we need to multiply the number of atoms of each element by the element\u2019s atomic mass.\r\n\r\n(8 moles of carbon)(12 grams\/mole) + (9 moles hydrogen)(1 gram\/mole) + (1 mole nitrogen)(14 grams\/mole) + (2 moles oxygen)(16 grams\/mole) = 151 g\r\n\r\nThen we need to calculate the number of moles in 325 mg.\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-8-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-308\" class=\"mjx-math\"><span id=\"MJXc-Node-309\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-310\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-311\" class=\"mjx-mrow\"><span id=\"MJXc-Node-312\" class=\"mjx-mrow\"><span id=\"MJXc-Node-313\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">(<\/span><\/span><span id=\"MJXc-Node-314\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-315\" class=\"mjx-mrow\"><span id=\"MJXc-Node-316\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">325<\/span><\/span><span id=\"MJXc-Node-317\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mg<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-318\" class=\"mjx-mrow\"><span id=\"MJXc-Node-319\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">151<\/span><\/span><span id=\"MJXc-Node-320\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0grams\/mole<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-321\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-322\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-323\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">(<\/span><\/span><span id=\"MJXc-Node-324\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-325\" class=\"mjx-mrow\"><span id=\"MJXc-Node-326\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-327\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0gram<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-328\" class=\"mjx-mrow\"><span id=\"MJXc-Node-329\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1000<\/span><\/span><span id=\"MJXc-Node-330\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mg<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-331\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-332\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-333\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2.15<\/span><\/span><span id=\"MJXc-Node-334\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-335\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-336\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-337\" class=\"mjx-texatom\"><span id=\"MJXc-Node-338\" class=\"mjx-mrow\"><span id=\"MJXc-Node-339\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-340\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-341\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0moles<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-342\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n\r\nThen use Avogadro\u2019s number to calculate the number of molecules.\r\n\r\n<em>N<\/em> = (2.15 \u00d7 10<sup>\u22123<\/sup> moles)(6.02 \u00d7 10<sup>23<\/sup> molecules\/mole) = 1.30\u00a0\u00d7 10<sup>21<\/sup> molecules\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3. Calculating Moles per Cubic Meter and Liters per Mole<\/h3>\r\nCalculate the following:\r\n<ol>\r\n \t<li>The number of moles in 1.00 m<sup>3<\/sup>\u00a0of gas at STP<\/li>\r\n \t<li>The number of liters of gas per mole.<\/li>\r\n<\/ol>\r\n<h4>Strategy and Solution<\/h4>\r\n<ol>\r\n \t<li>We are asked to find the number of moles per cubic meter, and we know from Example 2\u00a0that the number of molecules per cubic meter at STP is 2.68 \u00d7 10<sup>25<\/sup>. The number of moles can be found by dividing the number of molecules by Avogadro\u2019s number. We let <em>n<\/em> stand for the number of moles,\r\n<span class=\"mj\"><span id=\"MathJax-Element-9-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-343\" class=\"mjx-math\"><span id=\"MJXc-Node-344\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-345\" class=\"mjx-texatom\"><span id=\"MJXc-Node-346\" class=\"mjx-mrow\"><span id=\"MJXc-Node-347\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">m<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-348\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-349\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mol\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-350\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-351\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-352\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-353\" class=\"mjx-mrow\"><span id=\"MJXc-Node-354\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><span id=\"MJXc-Node-355\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-356\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-357\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-358\" class=\"mjx-mrow\"><span id=\"MJXc-Node-359\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6.02<\/span><\/span><span id=\"MJXc-Node-360\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-361\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-362\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-363\" class=\"mjx-texatom\"><span id=\"MJXc-Node-364\" class=\"mjx-mrow\"><span id=\"MJXc-Node-365\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">23<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-366\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules\/mol<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-367\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-368\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-369\" class=\"mjx-mrow\"><span id=\"MJXc-Node-370\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2.68<\/span><\/span><span id=\"MJXc-Node-371\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-372\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-373\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-374\" class=\"mjx-texatom\"><span id=\"MJXc-Node-375\" class=\"mjx-mrow\"><span id=\"MJXc-Node-376\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">25<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-377\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-378\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-379\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-380\" class=\"mjx-mrow\"><span id=\"MJXc-Node-381\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6.02<\/span><\/span><span id=\"MJXc-Node-382\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-383\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-384\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-385\" class=\"mjx-texatom\"><span id=\"MJXc-Node-386\" class=\"mjx-mrow\"><span id=\"MJXc-Node-387\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">23<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-388\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules\/mol<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-389\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-390\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">44.5<\/span><\/span><span id=\"MJXc-Node-391\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-392\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mol\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-393\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-394\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<ul>\r\n \t<li><\/li>\r\n \t<li>Using the value obtained for the number of moles in a cubic meter, and converting cubic meters to liters, we obtain <span class=\"mj\"><span id=\"MathJax-Element-10-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-395\" class=\"mjx-math\"><span id=\"MJXc-Node-396\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-397\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-398\" class=\"mjx-mrow\"><span id=\"MJXc-Node-399\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">(<\/span><\/span><span id=\"MJXc-Node-400\" class=\"mjx-mrow\"><span id=\"MJXc-Node-401\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-402\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-403\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-404\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-405\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0L\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-406\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-407\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">)<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-408\" class=\"mjx-mrow\"><span id=\"MJXc-Node-409\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">44.5<\/span><\/span><span id=\"MJXc-Node-410\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-411\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mol\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-412\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-413\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-414\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">22.5<\/span><\/span><span id=\"MJXc-Node-415\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0L\/mol<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-416\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\r\n<\/ul>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h4>Discussion<\/h4>\r\nThis value is very close to the accepted value of 22.4 L\/mol. The slight difference is due to rounding errors caused by using three-digit input. Again this number is the same for all gases. In other words, it is independent of the gas.\r\n\r\nThe (average) molar weight of air (approximately 80% N<sub>2<\/sub> and 20% O<sub>2<\/sub> is <em>M<\/em> = 28.8 g.\u00a0Thus the mass of one cubic meter of air is 1.28 kg. If a living room has dimensions 5 m \u00d7 5 m \u00d7 3 m,\u00a0the mass of air inside the room is 96 kg, which is the typical mass of a human.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Check Your Understanding<\/h3>\r\nThe density of air at standard conditions (<em>P<\/em> = 1 atm and\u00a0<em>T\u00a0<\/em>= 20\u00baC) is 1.28 kg\/m<sup>3<\/sup>. At what pressure is the density 0.64 kg\/m<sup>3<\/sup> if the temperature and number of molecules are kept constant?\r\n<h4>Solution<\/h4>\r\nThe best way to approach this question is to think about what is happening. If the density drops to half its original value and no molecules are lost, then the volume must double. If we look at the equation <em>PV<\/em> =\u00a0<em>NkT<\/em>, we see that when the temperature is constant, the pressure is inversely proportional to volume. Therefore, if the volume doubles, the pressure must drop to half its original value, and <em>P<\/em><sub>f<\/sub> = 0.50 atm.\r\n\r\n<\/div>\r\n<h2>The Ideal Gas Law Restated Using Moles<\/h2>\r\nA very common expression of the ideal gas law uses the number of moles, <em>n<\/em>, rather than the number of atoms and molecules, <em>N<\/em>. We start from the ideal gas law,\u00a0<em>PV<\/em> =\u00a0<em>NkT<\/em>,\u00a0and multiply and divide the equation by Avogadro\u2019s number <em>N<\/em><sub>A<\/sub>. This gives <span class=\"mj\"><span id=\"MathJax-Element-11-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-417\" class=\"mjx-math\"><span id=\"MJXc-Node-418\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-419\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-420\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><span id=\"MJXc-Node-421\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-422\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-423\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-424\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-425\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-426\" class=\"mjx-texatom\"><span id=\"MJXc-Node-427\" class=\"mjx-mrow\"><span id=\"MJXc-Node-428\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">A<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-429\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-430\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-431\" class=\"mjx-texatom\"><span id=\"MJXc-Node-432\" class=\"mjx-mrow\"><span id=\"MJXc-Node-433\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">A<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-434\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-435\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-436\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\n.\r\n\r\nNote that <span class=\"mj\"><span id=\"MathJax-Element-12-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-437\" class=\"mjx-math\"><span id=\"MJXc-Node-438\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-439\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">n<\/span><\/span><span id=\"MJXc-Node-440\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-441\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-442\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-443\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-444\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-445\" class=\"mjx-texatom\"><span id=\"MJXc-Node-446\" class=\"mjx-mrow\"><span id=\"MJXc-Node-447\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">A<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-448\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nis the number of moles. We define the universal gas constant <em>R<\/em>=<em>N<\/em><sub>A<\/sub><em>k<\/em>, and obtain the ideal gas law in terms of moles.\r\n<div class=\"textbox shaded\">\r\n<h3>Ideal Gas Law (in terms of moles)<\/h3>\r\nThe ideal gas law (in terms of moles) is\u00a0<em>PV<\/em> =\u00a0<em>nRT<\/em>.\r\n\r\nThe numerical value of <em>R<\/em> in SI units is\u00a0<em>R<\/em> =\u00a0<em>N<\/em><sub>A<\/sub><em>k<\/em> = (6.02 \u00d7 10<sup>23<\/sup> mol<sup>\u22121<\/sup>)(1.38 \u00d7 10<sup>\u221223<\/sup> J\/K) = 8.31 J\/mol\u00a0\u00b7 K.\r\n\r\nIn other units,\r\n\r\n<em>R<\/em> = 1.99 cal\/mol \u00b7 K\r\n\r\n<em>R<\/em> = 0.0821 L\u00a0\u00b7 atm\/mol\u00a0\u00b7 K\r\n\r\nYou can use whichever value of <em>R<\/em> is most convenient for a particular problem.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4. Calculating Number of Moles: Gas in a Bike Tire<\/h3>\r\nHow many moles of gas are in a bike tire with a volume of 2.00 \u00d7 10<sup>\u22123<\/sup> m<sup>3<\/sup>(2.00 L),\u00a0a pressure of 7.00 \u00d7 10<sup>5<\/sup> Pa\u00a0(a gauge pressure of just under 90.0 lb\/in<sup>2<\/sup>), and at a temperature of 18.0\u00baC?\r\n<h4>Strategy<\/h4>\r\nIdentify the knowns and unknowns, and choose an equation to solve for the unknown. In this case, we solve the ideal gas law, <em>PV<\/em> =\u00a0<em>nRT<\/em>, for the number of moles <em>n<\/em>.\r\n<h4>Solution<\/h4>\r\nIdentify the knowns:\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-13-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-449\" class=\"mjx-math\"><span id=\"MJXc-Node-450\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-451\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-452\" class=\"mjx-mtr\"><span id=\"MJXc-Node-453\" class=\"mjx-mtd\"><span id=\"MJXc-Node-454\" class=\"mjx-mrow\"><span id=\"MJXc-Node-455\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-456\" class=\"mjx-mtd\"><span id=\"MJXc-Node-457\" class=\"mjx-mrow\"><span id=\"MJXc-Node-458\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-459\" class=\"mjx-mtd\"><span id=\"MJXc-Node-460\" class=\"mjx-mrow\"><span id=\"MJXc-Node-461\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7.00<\/span><\/span><span id=\"MJXc-Node-462\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-463\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-464\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-465\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-466\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-467\" class=\"mjx-mtr\"><span id=\"MJXc-Node-468\" class=\"mjx-mtd\"><span id=\"MJXc-Node-469\" class=\"mjx-mrow\"><span id=\"MJXc-Node-470\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-471\" class=\"mjx-mtd\"><span id=\"MJXc-Node-472\" class=\"mjx-mrow\"><span id=\"MJXc-Node-473\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-474\" class=\"mjx-mtd\"><span id=\"MJXc-Node-475\" class=\"mjx-mrow\"><span id=\"MJXc-Node-476\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2.00<\/span><\/span><span id=\"MJXc-Node-477\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-478\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-479\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-480\" class=\"mjx-texatom\"><span id=\"MJXc-Node-481\" class=\"mjx-mrow\"><span id=\"MJXc-Node-482\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-483\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-484\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-485\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-486\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-487\" class=\"mjx-mtr\"><span id=\"MJXc-Node-488\" class=\"mjx-mtd\"><span id=\"MJXc-Node-489\" class=\"mjx-mrow\"><span id=\"MJXc-Node-490\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-491\" class=\"mjx-mtd\"><span id=\"MJXc-Node-492\" class=\"mjx-mrow\"><span id=\"MJXc-Node-493\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-494\" class=\"mjx-mtd\"><span id=\"MJXc-Node-495\" class=\"mjx-mrow\"><span id=\"MJXc-Node-496\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-497\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">18.0<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-498\" class=\"mjx-texatom\"><span id=\"MJXc-Node-499\" class=\"mjx-mrow\"><span id=\"MJXc-Node-500\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-501\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">C<\/span><\/span><span id=\"MJXc-Node-502\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-503\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">291<\/span><\/span><span id=\"MJXc-Node-504\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-505\" class=\"mjx-mtr\"><span id=\"MJXc-Node-506\" class=\"mjx-mtd\"><span id=\"MJXc-Node-507\" class=\"mjx-mrow\"><span id=\"MJXc-Node-508\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">R<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-509\" class=\"mjx-mtd\"><span id=\"MJXc-Node-510\" class=\"mjx-mrow\"><span id=\"MJXc-Node-511\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-512\" class=\"mjx-mtd\"><span id=\"MJXc-Node-513\" class=\"mjx-mrow\"><span id=\"MJXc-Node-514\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8.31<\/span><\/span><span id=\"MJXc-Node-515\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0J\/mol<\/span><\/span><span id=\"MJXc-Node-516\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-517\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-518\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\n<\/div>\r\nRearrange the equation to solve for <em>n<\/em> and substitute known values.\r\n\r\n<span class=\"mj\"><span id=\"MathJax-Element-14-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-519\" class=\"mjx-math\"><span id=\"MJXc-Node-520\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-521\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-522\" class=\"mjx-mtr\"><span id=\"MJXc-Node-523\" class=\"mjx-mtd\"><span id=\"MJXc-Node-524\" class=\"mjx-mrow\"><span id=\"MJXc-Node-525\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">n<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-526\" class=\"mjx-mtd\"><span id=\"MJXc-Node-527\" class=\"mjx-mrow\"><span id=\"MJXc-Node-528\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-529\" class=\"mjx-mtd\"><span id=\"MJXc-Node-530\" class=\"mjx-mrow\"><span id=\"MJXc-Node-531\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-532\" class=\"mjx-mrow\"><span id=\"MJXc-Node-533\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-534\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-535\" class=\"mjx-mrow\"><span id=\"MJXc-Node-536\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">R<\/span><\/span><span id=\"MJXc-Node-537\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-538\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-539\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-540\" class=\"mjx-mrow\"><span id=\"MJXc-Node-541\" class=\"mjx-mrow\"><span id=\"MJXc-Node-542\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-543\" class=\"mjx-mrow\"><span id=\"MJXc-Node-544\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7.00<\/span><\/span><span id=\"MJXc-Node-545\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-546\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-547\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-548\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-549\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><span id=\"MJXc-Node-550\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-551\" class=\"mjx-mrow\"><span id=\"MJXc-Node-552\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-553\" class=\"mjx-mrow\"><span id=\"MJXc-Node-554\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2.00<\/span><\/span><span id=\"MJXc-Node-555\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-556\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-557\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-558\" class=\"mjx-texatom\"><span id=\"MJXc-Node-559\" class=\"mjx-mrow\"><span id=\"MJXc-Node-560\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-561\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-562\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-563\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-564\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-565\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-566\" class=\"mjx-mrow\"><span id=\"MJXc-Node-567\" class=\"mjx-mrow\"><span id=\"MJXc-Node-568\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-569\" class=\"mjx-mrow\"><span id=\"MJXc-Node-570\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8.31<\/span><\/span><span id=\"MJXc-Node-571\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0J\/mol<\/span><\/span><span id=\"MJXc-Node-572\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-573\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><span id=\"MJXc-Node-574\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-575\" class=\"mjx-mrow\"><span id=\"MJXc-Node-576\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-577\" class=\"mjx-mrow\"><span id=\"MJXc-Node-578\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">291<\/span><\/span><span id=\"MJXc-Node-579\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><span id=\"MJXc-Node-580\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-581\" class=\"mjx-mtr\"><span id=\"MJXc-Node-582\" class=\"mjx-mtd\"><span id=\"MJXc-Node-583\" class=\"mjx-mrow\"><span id=\"MJXc-Node-584\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-585\" class=\"mjx-mtd\"><span id=\"MJXc-Node-586\" class=\"mjx-mrow\"><span id=\"MJXc-Node-587\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-588\" class=\"mjx-mtd\"><span id=\"MJXc-Node-589\" class=\"mjx-mrow\"><span id=\"MJXc-Node-590\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0.579<\/span><\/span><span id=\"MJXc-Node-591\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mol<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-592\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n<div class=\"textbox examples\">\r\n\r\n<strong>Discussion<\/strong>\r\n\r\nThe most convenient choice for <em>R<\/em> in this case is 8.31 J\/mol \u00b7\u00a0K,\u00a0because our known quantities are in SI units. The pressure and temperature are obtained from the initial conditions in Example 1, but we would get the same answer if we used the final values.\r\n\r\n<\/div>\r\nThe ideal gas law can be considered to be another manifestation of the law of conservation of energy (see Conservation of Energy). Work done on a gas results in an increase in its energy, increasing pressure and\/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas\u2019s atoms and molecules.\r\n<h2>The Ideal Gas Law and Energy<\/h2>\r\nLet us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.\r\n\r\nThe ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in <em>PV<\/em> =\u00a0<em>NkT<\/em>\u00a0is\u00a0<em>NkT<\/em>. This term is roughly the amount of translational kinetic energy of <em>N<\/em> atoms or molecules at an absolute temperature <em>T<\/em>, as we shall see formally in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature. The left-hand side of the ideal gas law is <em>PV<\/em>, which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands\u2014something we explore in Heat and Heat Transfer Methods\u2014similar to what occurs in gasoline or steam engines and turbines.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Problem-Solving Strategy: The Ideal Gas Law<\/h3>\r\n<strong>Step 1.<\/strong> Examine the situation to determine that an ideal gas is involved. Most gases are nearly ideal.\r\n\r\n<strong>Step 2.<\/strong>\u00a0Make a list of what quantities are given, or can be inferred from the problem as stated (identify the known quantities). Convert known values into proper SI units (K for temperature, Pa for pressure, m<sup>3<\/sup> for volume, molecules for <em>N<\/em>, and moles for <em>n<\/em>).\r\n\r\n<strong>Step 3.<\/strong> Identify exactly what needs to be determined in the problem (identify the unknown quantities). A written list is useful.\r\n\r\n<strong>Step 4.<\/strong> Determine whether the number of molecules or the number of moles is known, in order to decide which form of the ideal gas law to use. The first form is <em>PV<\/em> =\u00a0<em>NkT<\/em>\u00a0and involves <em>N<\/em>, the number of atoms or molecules. The second form is <em>PV\u00a0<\/em>=\u00a0<em>nRT<\/em>\u00a0and involves <em>n<\/em>, the number of moles.\r\n\r\n<strong>Step 5.<\/strong> Solve the ideal gas law for the quantity to be determined (the unknown quantity). You may need to take a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.\r\n\r\n<strong>Step 6.<\/strong> Substitute the known quantities, along with their units, into the appropriate equation, and obtain numerical solutions complete with units. Be certain to use absolute temperature and absolute pressure.\r\n\r\n<strong>Step 7.<\/strong> Check the answer to see if it is reasonable: Does it make sense?\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Check Your Understanding<\/h3>\r\nLiquids and solids have densities about 1000 times greater than gases. Explain how this implies that the distances between atoms and molecules in gases are about 10 times greater than the size of their atoms and molecules.\r\n<h4>Solution<\/h4>\r\nAtoms and molecules are close together in solids and liquids. In gases they are separated by empty space. Thus gases have lower densities than liquids and solids. Density is mass per unit volume, and volume is related to the size of a body (such as a sphere) cubed. So if the distance between atoms and molecules increases by a factor of 10, then the volume occupied increases by a factor of 1000, and the density decreases by a factor of 1000.\r\n\r\n<\/div>\r\n<h2>Section Summary<\/h2>\r\n<ul>\r\n \t<li>The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.<\/li>\r\n \t<li>The ideal gas law can be written in terms of the number of molecules of gas:\u00a0<em>PV\u00a0<\/em>= <em>NkT<\/em>,\u00a0where <em>P<\/em> is pressure, <em>V<\/em> is volume, <em>T<\/em> is temperature, <em>N<\/em> is number of molecules, and k is the Boltzmann constant\u00a0<em>k\u00a0<\/em>= 1.38 \u00d7 10<sup>\u201323<\/sup> J\/K.<\/li>\r\n \t<li>A mole is the number of atoms in a 12-g sample of carbon-12.<\/li>\r\n \t<li>The number of molecules in a mole is called Avogadro\u2019s number <em>NA<\/em>,\u00a0<em>NA\u00a0<\/em>= 6.02 \u00d7 10<sup>23<\/sup> mol<sup>\u22121<\/sup>.<\/li>\r\n \t<li>A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.<\/li>\r\n \t<li>The ideal gas law can also be written and solved in terms of the number of moles of gas:\u00a0<em>PV\u00a0<\/em>= <em>nRT<\/em>,\u00a0where n is number of moles and <em>R<\/em> is the universal gas constant,\u00a0<em>R<\/em> = 8.31 J\/mol \u22c5 K.<\/li>\r\n \t<li>The ideal gas law is generally valid at temperatures well above the boiling temperature.<\/li>\r\n<\/ul>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Conceptual Questions<\/h3>\r\nFind out the human population of Earth. Is there a mole of people inhabiting Earth? If the average mass of a person is 60 kg, calculate the mass of a mole of people. How does the mass of a mole of people compare with the mass of Earth?\r\n\r\nUnder what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?\r\n\r\nA constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Problems &amp; Exercises<\/h3>\r\n<ol>\r\n \t<li>The gauge pressure in your car tires is 2.50 \u00d7 10<sup>5<\/sup> N\/m<sup>2<\/sup> at a temperature of 35.0\u00baC when you drive it onto a ferry boat to Alaska. What is their gauge pressure later, when their temperature has dropped to \u201340.0\u00baC?<\/li>\r\n \t<li>Convert an absolute pressure of 7.00 \u00d7 10<sup>5<\/sup> N\/m<sup>2<\/sup> to gauge pressure in lb\/in<sup>2<\/sup>. (This value was stated to be just less than 90.0 lb\/in<sup>2<\/sup> in Example 4. Is it?)<\/li>\r\n \t<li>Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of 20.0\u00baC. (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is 60.0\u00baC (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks. (b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. What will the actual final pressure be, taking this into account? Is this a negligible difference?<\/li>\r\n \t<li>Large helium-filled balloons are used to lift scientific equipment to high altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of 10.0\u00baC and rises to an altitude where its volume is twenty times the original volume and its temperature is \u201350.0\u00baC? (b) What is the gauge pressure? (Assume atmospheric pressure is constant.)<\/li>\r\n \t<li>Confirm that the units of nRT are those of energy for each value of R: (a) 8.31 J\/mol \u22c5 K, (b) 1.99 cal\/mol \u22c5 K, and (c) 0.0821 L \u22c5 atm\/mol \u22c5 K.<\/li>\r\n \t<li>In the text, it was shown that <em>N<\/em>\/<em>V\u00a0<\/em>= 2.68 \u00d7 10<sup>25<\/sup> m<sup>\u22123<\/sup> for gas at STP. (a) Show that this quantity is equivalent to <em>N<\/em>\/<em>V<\/em> = 2.68 \u00d7 10<sup>19<\/sup> cm<sup>\u22123<\/sup>, as stated. (b) About how many atoms are there in one \u03bcm<sup>3<\/sup> (a cubic micrometer) at STP? (c) What does your answer to part (b) imply about the separation of atoms and molecules?<\/li>\r\n \t<li>Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at 37.0\u00baC (body temperature).<\/li>\r\n \t<li>An airplane passenger has 100 cm<sup>3<\/sup> of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to 7.50 \u00d7 10<sup>4<\/sup> N\/m<sup>2<\/sup>?<\/li>\r\n \t<li>(a) What is the volume (in km<sup>3<\/sup>) of Avogadro\u2019s number of sand grains if each grain is a cube and has sides that are 1.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages 100 m in width and 10.0 m in depth? Neglect air spaces between grains.<\/li>\r\n \t<li>An expensive vacuum system can achieve a pressure as low as 1.00 \u00d7 10<sup>\u20137<\/sup> N\/m<sup>2<\/sup> at 20\u00baC. How many atoms are there in a cubic centimeter at this pressure and temperature?<\/li>\r\n \t<li>The number density of gas atoms at a certain location in the space above our planet is about 1.00 \u00d7 10<sup>11<\/sup> m<sup>\u22123<\/sup>, and the pressure is 2.75 \u00d7 10<sup>\u201310<\/sup> N\/m<sup>2<\/sup> in this space. What is the temperature there?<\/li>\r\n \t<li>A bicycle tire has a pressure of 7.00 \u00d7 10<sup>5<\/sup> N\/m<sup>2<\/sup> at a temperature of 18.0\u00baC and contains 2.00 L of gas. What will its pressure be if you let out an amount of air that has a volume of 100cm3 at atmospheric pressure? Assume tire temperature and volume remain constant.<\/li>\r\n \t<li>A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of 1.40 \u00d7 10<sup>7<\/sup> N\/m<sup>2<\/sup> and a temperature of 25.0\u00baC. Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperature (\u201378.5\u00baC) to reduce the leak rate and pressure so that it can be safely repaired. (a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank appear to be a practical solution?<\/li>\r\n \t<li>Find the number of moles in 2.00 L of gas at 35.0\u00baC and under 7.41 \u00d7 10<sup>7<\/sup> N\/m<sup>2<\/sup> of pressure.<\/li>\r\n \t<li>Calculate the depth to which Avogadro\u2019s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.<\/li>\r\n \t<li>(a) What is the gauge pressure in a 25.0\u00baC car tire containing 3.60 mol of gas in a 30.0 L volume? (b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and 25.0\u00baC? Assume the temperature returns to 25.0\u00baC and the volume remains constant.<\/li>\r\n \t<li>(a) In the deep space between galaxies, the density of atoms is as low as 10<sup>6<\/sup> atoms\/m<sup>3<\/sup>, and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in m<sup>3<\/sup>) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<strong>ideal gas law:<\/strong>\u00a0the physical law that relates the pressure and volume of a gas to the number of gas molecules or number of moles of gas and the temperature of the gas\r\n\r\n<strong>Boltzmann constant:<\/strong> <em>k<\/em>, a physical constant that relates energy to temperature; <em>k\u00a0<\/em>= 1.38 \u00d7 10<sup>\u201323<\/sup> J\/K\r\n\r\n<strong>Avogadro\u2019s number:<\/strong> <em>NA<\/em>, the number of molecules or atoms in one mole of a substance; <em>NA<\/em> = 6.02 \u00d7 10<sup>23<\/sup> particles\/mole\r\n\r\n<strong>mole:<\/strong>\u00a0the quantity of a substance whose mass (in grams) is equal to its molecular mass\r\n<div class=\"textbox exercises\">\r\n<h3>Selected Solutions to\u00a0Problems &amp; Exercises<\/h3>\r\n1.\u00a01.62 atm\r\n\r\n3.\u00a0(a) 0.136 atm;\u00a0(b) 0.135 atm. The difference between this value and the value from part (a) is negligible.\r\n\r\n5. (a) <span class=\"mj\"><span id=\"MathJax-Element-15-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-593\" class=\"mjx-math\"><span id=\"MJXc-Node-594\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-595\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">nRT<\/span><\/span><span id=\"MJXc-Node-596\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-597\" class=\"mjx-mrow MJXc-space3\"><span id=\"MJXc-Node-598\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-599\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">mol<\/span><\/span><span id=\"MJXc-Node-600\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-601\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-602\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-603\" class=\"mjx-mrow\"><span id=\"MJXc-Node-604\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">J\/mol<\/span><\/span><span id=\"MJXc-Node-605\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-606\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><\/span><span id=\"MJXc-Node-607\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-608\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-609\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-610\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><span id=\"MJXc-Node-611\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-612\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-613\" class=\"mjx-mtext MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">J<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-614\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\n<\/div>\r\n;\r\n\r\n(b) <span class=\"mj\"><span id=\"MathJax-Element-16-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-615\" class=\"mjx-math\"><span id=\"MJXc-Node-616\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-617\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">nRT<\/span><\/span><span id=\"MJXc-Node-618\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-619\" class=\"mjx-mrow MJXc-space3\"><span id=\"MJXc-Node-620\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-621\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">mol<\/span><\/span><span id=\"MJXc-Node-622\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-623\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-624\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-625\" class=\"mjx-mrow\"><span id=\"MJXc-Node-626\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">cal\/mol<\/span><\/span><span id=\"MJXc-Node-627\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-628\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><\/span><span id=\"MJXc-Node-629\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-630\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-631\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-632\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><span id=\"MJXc-Node-633\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-634\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-635\" class=\"mjx-mtext MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">cal<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-636\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\n;\r\n\r\n(c) <span class=\"mj\"><span id=\"MathJax-Element-17-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-637\" class=\"mjx-math\"><span id=\"MJXc-Node-638\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-639\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-640\" class=\"mjx-mtr\"><span id=\"MJXc-Node-641\" class=\"mjx-mtd\"><span id=\"MJXc-Node-642\" class=\"mjx-mrow\"><span id=\"MJXc-Node-643\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">nRT<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-644\" class=\"mjx-mtd\"><span id=\"MJXc-Node-645\" class=\"mjx-mrow\"><span id=\"MJXc-Node-646\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-647\" class=\"mjx-mtd\"><span id=\"MJXc-Node-648\" class=\"mjx-mrow\"><span id=\"MJXc-Node-649\" class=\"mjx-mrow\"><span id=\"MJXc-Node-650\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-651\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">mol<\/span><\/span><span id=\"MJXc-Node-652\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-653\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-654\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-655\" class=\"mjx-mrow\"><span id=\"MJXc-Node-656\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">L<\/span><\/span><span id=\"MJXc-Node-657\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-658\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">atm\/mol<\/span><\/span><span id=\"MJXc-Node-659\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-660\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><\/span><span id=\"MJXc-Node-661\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-662\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-663\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-664\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><span id=\"MJXc-Node-665\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-666\" class=\"mjx-mtr\"><span id=\"MJXc-Node-667\" class=\"mjx-mtd\"><span id=\"MJXc-Node-668\" class=\"mjx-mrow\"><\/span><\/span><span id=\"MJXc-Node-669\" class=\"mjx-mtd\"><span id=\"MJXc-Node-670\" class=\"mjx-mrow\"><span id=\"MJXc-Node-671\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-672\" class=\"mjx-mtd\"><span id=\"MJXc-Node-673\" class=\"mjx-mrow\"><span id=\"MJXc-Node-674\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">L<\/span><\/span><span id=\"MJXc-Node-675\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-676\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">atm<\/span><\/span><span id=\"MJXc-Node-677\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-678\" class=\"mjx-mrow MJXc-space3\"><span id=\"MJXc-Node-679\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-680\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-681\" class=\"mjx-texatom\"><span id=\"MJXc-Node-682\" class=\"mjx-mrow\"><span id=\"MJXc-Node-683\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">m<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-684\" class=\"mjx-texatom\"><span id=\"MJXc-Node-685\" class=\"mjx-mrow\"><span id=\"MJXc-Node-686\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-687\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-688\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-689\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">(<\/span><\/span><span id=\"MJXc-Node-690\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-691\" class=\"mjx-texatom\"><span id=\"MJXc-Node-692\" class=\"mjx-mrow\"><span id=\"MJXc-Node-693\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">N\/m<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-694\" class=\"mjx-texatom\"><span id=\"MJXc-Node-695\" class=\"mjx-mrow\"><span id=\"MJXc-Node-696\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-697\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">)<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-698\" class=\"mjx-mtr\"><span id=\"MJXc-Node-699\" class=\"mjx-mtd\"><span id=\"MJXc-Node-700\" class=\"mjx-mrow\"><\/span><\/span><span id=\"MJXc-Node-701\" class=\"mjx-mtd\"><span id=\"MJXc-Node-702\" class=\"mjx-mrow\"><span id=\"MJXc-Node-703\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-704\" class=\"mjx-mtd\"><span id=\"MJXc-Node-705\" class=\"mjx-mrow\"><span id=\"MJXc-Node-706\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">N<\/span><\/span><span id=\"MJXc-Node-707\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-708\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">m<\/span><\/span><span id=\"MJXc-Node-709\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-710\" class=\"mjx-mtext MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">J<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-711\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span>\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-2650\" class=\"standard post-2650 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox exercises\">\r\n\r\n7.\u00a07.86 \u00d7 10<sup>\u22122<\/sup> mol\r\n\r\n9.\u00a0(a) 6.02 \u00d7 10<sup>5<\/sup> km<sup>3<\/sup>;\u00a0(b) 6.02 \u00d7 10<sup>8<\/sup> km\r\n\r\n11.\u00a0\u221273.9\u00baC\r\n\r\n13.\u00a0(a) 9.14 \u00d7 10<sup>6<\/sup> N\/m<sup>2<\/sup>;\u00a0(b) 8.23 \u00d7 10<sup>6<\/sup> N\/m<sup>2<\/sup>;\u00a0(c) 2.16 K;\u00a0(d) No. The final temperature needed is much too low to be easily achieved for a large object.\r\n\r\n15.\u00a041 km\r\n\r\n17. \u00a0(a) 3.7 \u00d7 10<sup>\u221217<\/sup> Pa;\u00a0(b) 6.0 \u00d7 10<sup>17<\/sup> m<sup>3<\/sup>;\u00a0(c) 8.4 \u00d7 10<sup>2<\/sup> km\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section class=\" focusable\" role=\"contentinfo\">\r\n<div class=\"post-citations sidebar\"><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"footer\"><\/div>","rendered":"<h1 class=\"entry-title\">The Ideal Gas Law<\/h1>\n<div class=\"difficulty\"><\/div>\n<div id=\"post-2650\" class=\"standard post-2650 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>State the ideal gas law in terms of molecules and in terms of moles.<\/li>\n<li>Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.<\/li>\n<li>Use Avogadro\u2019s number to convert between number of molecules and number of moles.<\/li>\n<\/ul>\n<\/div>\n<div class=\"wp-caption alignright\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20104241\/Figure_14_03_00.jpg\" alt=\"Figure (Figure_14_03_00.jpg)\" width=\"200\" height=\"553\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. The air inside this hot air balloon flying over Putrajaya, Malaysia, is hotter than the ambient air. As a result, the balloon experiences a buoyant force pushing it upward. (credit: Kevin Poh, Flickr)<\/p>\n<\/div>\n<p>In this section, we continue to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (Most gases, for example nitrogen, N<sub>2<\/sub>, and oxygen, O<sub>2<\/sub>, are composed of two or more atoms. We will primarily use the term \u201cmolecule\u201d in discussing a gas because the term can also be applied to monatomic gases, such as helium.)<\/p>\n<p>Gases are easily compressed. We can see evidence of this in <a href=\"https:\/\/courses.lumenlearning.com\/physics\/chapter\/13-3-the-ideal-gas-law\/chapter\/13-2-thermal-expansion-of-solids-and-liquids\/\" target=\"_blank\" rel=\"noopener\">Table 1 in\u00a0Thermal Expansion of Solids and Liquids<\/a>, where you will note that gases have the <em>largest<\/em> coefficients of volume expansion. The large coefficients mean that gases expand and contract very rapidly with temperature changes. In addition, you will note that most gases expand at the <em>same<\/em> rate, or have the same <em>\u03b2<\/em>. This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates.<\/p>\n<p>The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in Figure 2. Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.<\/p>\n<div class=\"wp-caption aligncenter\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20104246\/Figure_14_03_01.jpg\" alt=\"Spheres representing atoms and molecules; the spheres are relatively far apart and are distributed randomly.\" width=\"599\" height=\"279\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. Atoms and molecules in a gas are typically widely separated, as shown. Because the forces between them are quite weak at these distances, the properties of a gas depend more on the number of atoms per unit volume and on temperature than on the type of atom.<\/p>\n<\/div>\n<p>To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into an initially deflated tire. The tire\u2019s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If we continue to pump air into it, the pressure increases. The pressure will further increase when the car is driven and the tires move. Most manufacturers specify optimal tire pressure for cold tires. (See Figure 3.)<\/p>\n<div class=\"wp-caption aligncenter\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20104251\/Figure_14_03_02.jpg\" alt=\"The figure has three parts, each part showing a pair of tires, and each tire connected to a pressure gauge. Each pair of tires represents the before and after images of a single tire, along with a change in pressure in that tire. In part a, the tire pressure is initially zero. After some air is added, represented by an arrow labeled Add air, the pressure rises to slightly above zero. In part b, the tire pressure is initially at the half-way mark. After some air is added, represented by an arrow labeled Add air, the pressure rises to the three-fourths mark. In part c, the tire pressure is initially at the three-fourths mark. After the temperature is raised, represented by an arrow labeled Increase temperature, the pressure rises to nearly the full mark.\" width=\"750\" height=\"189\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. (a) When air is pumped into a deflated tire, its volume first increases without much increase in pressure. (b) When the tire is filled to a certain point, the tire walls resist further expansion and the pressure increases with more air. (c) Once the tire is inflated, its pressure increases with temperature.<\/p>\n<\/div>\n<p>At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.<\/p>\n<div class=\"textbox shaded\">\n<h3>Ideal Gas Law<\/h3>\n<p>The <em>ideal gas law<\/em> states that\u00a0<em>PV<\/em> =\u00a0<em>NkT<\/em>,\u00a0where <em>P<\/em> is the absolute pressure of a gas, <em>V<\/em> is the volume it occupies, <em>N<\/em> is the number of atoms and molecules in the gas, and <em>T<\/em> is its absolute temperature. The constant <em>k<\/em> is called the <em>Boltzmann constant<\/em> in honor of Austrian physicist Ludwig Boltzmann (1844\u20131906) and has the value\u00a0<em>k<\/em> = 1.38\u00a0\u00d7 10<sup>\u221223<\/sup> J\/K.<\/p>\n<\/div>\n<p>The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles\u2019 law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle\u2019s law (that for a fixed temperature, the product <em>PV<\/em>\u00a0is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of <em>V<\/em>. The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that <em>N<\/em> is the total number of atoms and molecules, independent of the type of gas.)<\/p>\n<p>Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure <em>P<\/em> is essentially equal to atmospheric pressure, and the volume <em>V<\/em> increases in direct proportion to the number of atoms and molecules <em>N<\/em> put into the tire. Once the volume of the tire is constant, the equation <em>PV<\/em> =\u00a0<em>NkT<\/em>\u00a0predicts that the pressure should increase in proportion to <em>the number N of atoms and molecules<\/em>.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1. Calculating Pressure Changes Due to Temperature Changes: Tire Pressure<\/h3>\n<p>Suppose your bicycle tire is fully inflated, with an absolute pressure of 7.00 \u00d7 10<sup>5<\/sup> Pa\u00a0(a gauge pressure of just under 90.0 lb\/in<sup>2<\/sup>) at a temperature of 18.0\u00baC. What is the pressure after its temperature has risen to 35.0\u00baC? Assume that there are no appreciable leaks or changes in volume.<\/p>\n<h4>Strategy<\/h4>\n<p>The pressure in the tire is changing only because of changes in temperature. First we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown.<\/p>\n<p>We know the initial pressure <em>P<\/em><sub>0<\/sub> = 7.00\u00a0\u00d7 10<sup>5<\/sup> Pa, the initial temperature <em>T<\/em><sub>0<\/sub>\u00a0= 18.0\u00baC, and the final temperature <em>T<\/em><sub>f<\/sub>\u00a0= 35.0\u00baC. We must find the final pressure <em>P<\/em><sub>f<\/sub>. How can we use the equation <em>PV<\/em> = <em>NkT<\/em>? At first, it may seem that not enough information is given, because the volume <em>V<\/em> and number of atoms <em>N<\/em> are not specified. What we can do is use the equation twice: <em>P<\/em><sub>0<\/sub><em>V<\/em><sub>0<\/sub>\u00a0= NkT<sub>0<\/sub> and <em>P<\/em><sub>f<\/sub><em>V<\/em><sub>f<\/sub>\u00a0= NkT<sub>f<\/sub>. If we divide <em>P<\/em><sub>f<\/sub><em>V<\/em><sub>f<\/sub> by <em>P<\/em><sub>0<\/sub><em>V<\/em><sub>0<\/sub> we can come up with an equation that allows us to solve for <em>P<\/em><sub>f<\/sub>.<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-1-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-1\" class=\"mjx-math\"><span id=\"MJXc-Node-2\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-3\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-4\" class=\"mjx-mrow\"><span id=\"MJXc-Node-5\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-6\" class=\"mjx-mrow\"><span id=\"MJXc-Node-7\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-8\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-9\" class=\"mjx-texatom\"><span id=\"MJXc-Node-10\" class=\"mjx-mrow\"><span id=\"MJXc-Node-11\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-12\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-13\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-14\" class=\"mjx-texatom\"><span id=\"MJXc-Node-15\" class=\"mjx-mrow\"><span id=\"MJXc-Node-16\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-17\" class=\"mjx-mrow\"><span id=\"MJXc-Node-18\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-19\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-20\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-21\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-22\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-23\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-24\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-25\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-26\" class=\"mjx-mrow\"><span id=\"MJXc-Node-27\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-28\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-29\" class=\"mjx-texatom\"><span id=\"MJXc-Node-30\" class=\"mjx-mrow\"><span id=\"MJXc-Node-31\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-32\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-33\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-34\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-35\" class=\"mjx-texatom\"><span id=\"MJXc-Node-36\" class=\"mjx-mrow\"><span id=\"MJXc-Node-37\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-38\" class=\"mjx-mrow\"><span id=\"MJXc-Node-39\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-40\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-41\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-42\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-43\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-44\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-45\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-46\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Since the volume is constant, <em>V<\/em><sub>f<\/sub> and <em>V<\/em><sub>0<\/sub> are the same and they cancel out. The same is true for <em>N<\/em><sub>f<\/sub> and <em>N<\/em><sub>0<\/sub>, and <em>k<\/em>, which is a constant. Therefore,<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-2-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-47\" class=\"mjx-math\"><span id=\"MJXc-Node-48\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-49\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-50\" class=\"mjx-mrow\"><span id=\"MJXc-Node-51\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-52\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-53\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-54\" class=\"mjx-texatom\"><span id=\"MJXc-Node-55\" class=\"mjx-mrow\"><span id=\"MJXc-Node-56\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-57\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-58\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-59\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-60\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-61\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-62\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-63\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-64\" class=\"mjx-texatom\"><span id=\"MJXc-Node-65\" class=\"mjx-mrow\"><span id=\"MJXc-Node-66\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-67\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-68\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-69\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-70\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>We can then rearrange this to solve for <em>P<\/em><sub>f<\/sub>: <span class=\"mj\"><span id=\"MathJax-Element-3-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-71\" class=\"mjx-math\"><span id=\"MJXc-Node-72\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-73\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-74\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-75\" class=\"mjx-texatom\"><span id=\"MJXc-Node-76\" class=\"mjx-mrow\"><span id=\"MJXc-Node-77\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-78\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-79\" class=\"mjx-msubsup MJXc-space3\"><span class=\"mjx-base\"><span id=\"MJXc-Node-80\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-81\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-82\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-83\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-84\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-85\" class=\"mjx-texatom\"><span id=\"MJXc-Node-86\" class=\"mjx-mrow\"><span id=\"MJXc-Node-87\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-88\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-89\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-90\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-91\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>,\u00a0where the temperature must be in units of kelvins, because <em>T<\/em><sub>0<\/sub> and <em>T<\/em><sub>f<\/sub> are absolute temperatures.<\/p>\n<h4>Solution<\/h4>\n<p>Convert temperatures from Celsius to Kelvin:<\/p>\n<p><em>T<\/em><sub>0<\/sub> = (18.0 + 273)K = 291 K<\/p>\n<p><em>T<\/em><sub>f<\/sub> = (35.0 + 273)K = 308\u00a0K<\/p>\n<p>Substitute the known values into the equation.<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-4-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-92\" class=\"mjx-math\"><span id=\"MJXc-Node-93\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-94\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-95\" class=\"mjx-mrow\"><span id=\"MJXc-Node-96\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-97\" class=\"mjx-texatom\"><span id=\"MJXc-Node-98\" class=\"mjx-mrow\"><span id=\"MJXc-Node-99\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-100\" class=\"mjx-texatom\"><span id=\"MJXc-Node-101\" class=\"mjx-mrow\"><span id=\"MJXc-Node-102\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-103\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-104\" class=\"mjx-msubsup MJXc-space3\"><span class=\"mjx-base\"><span id=\"MJXc-Node-105\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-106\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-107\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-108\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-109\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-110\" class=\"mjx-texatom\"><span id=\"MJXc-Node-111\" class=\"mjx-mrow\"><span id=\"MJXc-Node-112\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-113\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-114\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-115\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-116\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-117\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">7.00<\/span><\/span><span id=\"MJXc-Node-118\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-119\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-120\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-121\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-122\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><span id=\"MJXc-Node-123\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-124\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">(<\/span><\/span><span id=\"MJXc-Node-125\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-126\" class=\"mjx-mrow\"><span id=\"MJXc-Node-127\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">308<\/span><\/span><span id=\"MJXc-Node-128\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-129\" class=\"mjx-mrow\"><span id=\"MJXc-Node-130\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">291<\/span><\/span><span id=\"MJXc-Node-131\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-132\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-133\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-134\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">7.41<\/span><\/span><span id=\"MJXc-Node-135\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-136\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-137\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-138\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-139\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-140\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<div class=\"textbox examples\">\n<h4>Discussion<\/h4>\n<p>The final temperature is about 6% greater than the original temperature, so the final pressure is about 6% greater as well. Note that <em>absolute<\/em> pressure and <em>absolute<\/em> temperature must be used in the ideal gas law.<\/p>\n<\/div>\n<div class=\"textbox learning-objectives\">\n<h3>Making Connections: Take-Home Experiment\u2014Refrigerating a Balloon<\/h3>\n<p>Inflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why?<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2. Calculating the Number of Molecules in a Cubic Meter of Gas<\/h3>\n<p>How many molecules are in a typical object, such as gas in a tire or water in a drink? We can use the ideal gas law to give us an idea of how large <em>N<\/em> typically is.<\/p>\n<p>Calculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP), which is defined to be 0\u00baC and atmospheric pressure.<\/p>\n<h4>Strategy<\/h4>\n<p>Because pressure, volume, and temperature are all specified, we can use the ideal gas law <em>PV<\/em> =\u00a0<em>NkT<\/em>, to find <em>N<\/em>.<\/p>\n<h4>Solution<\/h4>\n<p>Identify the knowns:<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-5-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-141\" class=\"mjx-math\"><span id=\"MJXc-Node-142\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-143\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-144\" class=\"mjx-mtr\"><span id=\"MJXc-Node-145\" class=\"mjx-mtd\"><span id=\"MJXc-Node-146\" class=\"mjx-mrow\"><span id=\"MJXc-Node-147\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-148\" class=\"mjx-mtd\"><span id=\"MJXc-Node-149\" class=\"mjx-mrow\"><span id=\"MJXc-Node-150\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-151\" class=\"mjx-mtd\"><span id=\"MJXc-Node-152\" class=\"mjx-mrow\"><span id=\"MJXc-Node-153\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-154\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-155\" class=\"mjx-texatom\"><span id=\"MJXc-Node-156\" class=\"mjx-mrow\"><span id=\"MJXc-Node-157\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-158\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">C<\/span><\/span><span id=\"MJXc-Node-159\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-160\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">273<\/span><\/span><span id=\"MJXc-Node-161\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-162\" class=\"mjx-mtr\"><span id=\"MJXc-Node-163\" class=\"mjx-mtd\"><span id=\"MJXc-Node-164\" class=\"mjx-mrow\"><span id=\"MJXc-Node-165\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-166\" class=\"mjx-mtd\"><span id=\"MJXc-Node-167\" class=\"mjx-mrow\"><span id=\"MJXc-Node-168\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-169\" class=\"mjx-mtd\"><span id=\"MJXc-Node-170\" class=\"mjx-mrow\"><span id=\"MJXc-Node-171\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.01<\/span><\/span><span id=\"MJXc-Node-172\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-173\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-174\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-175\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-176\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-177\" class=\"mjx-mtr\"><span id=\"MJXc-Node-178\" class=\"mjx-mtd\"><span id=\"MJXc-Node-179\" class=\"mjx-mrow\"><span id=\"MJXc-Node-180\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-181\" class=\"mjx-mtd\"><span id=\"MJXc-Node-182\" class=\"mjx-mrow\"><span id=\"MJXc-Node-183\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-184\" class=\"mjx-mtd\"><span id=\"MJXc-Node-185\" class=\"mjx-mrow\"><span id=\"MJXc-Node-186\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.00<\/span><\/span><span id=\"MJXc-Node-187\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-188\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-189\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-190\" class=\"mjx-mtr\"><span id=\"MJXc-Node-191\" class=\"mjx-mtd\"><span id=\"MJXc-Node-192\" class=\"mjx-mrow\"><span id=\"MJXc-Node-193\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-194\" class=\"mjx-mtd\"><span id=\"MJXc-Node-195\" class=\"mjx-mrow\"><span id=\"MJXc-Node-196\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-197\" class=\"mjx-mtd\"><span id=\"MJXc-Node-198\" class=\"mjx-mrow\"><span id=\"MJXc-Node-199\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.38<\/span><\/span><span id=\"MJXc-Node-200\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-201\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-202\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-203\" class=\"mjx-texatom\"><span id=\"MJXc-Node-204\" class=\"mjx-mrow\"><span id=\"MJXc-Node-205\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-206\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">23<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-207\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0J\/K<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-208\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/div>\n<p>Identify the unknown: number of molecules, <em>N<\/em>.<\/p>\n<p>Rearrange the ideal gas law to solve for <em>N<\/em>:<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-6-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-209\" class=\"mjx-math\"><span id=\"MJXc-Node-210\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-211\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-212\" class=\"mjx-mtr\"><span id=\"MJXc-Node-213\" class=\"mjx-mtd\"><span id=\"MJXc-Node-214\" class=\"mjx-mrow\"><span id=\"MJXc-Node-215\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-216\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-217\" class=\"mjx-mtd\"><span id=\"MJXc-Node-218\" class=\"mjx-mrow\"><span id=\"MJXc-Node-219\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-220\" class=\"mjx-mtd\"><span id=\"MJXc-Node-221\" class=\"mjx-mrow\"><span id=\"MJXc-Node-222\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><span id=\"MJXc-Node-223\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-224\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-225\" class=\"mjx-mtr\"><span id=\"MJXc-Node-226\" class=\"mjx-mtd\"><span id=\"MJXc-Node-227\" class=\"mjx-mrow\"><span id=\"MJXc-Node-228\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-229\" class=\"mjx-mtd\"><span id=\"MJXc-Node-230\" class=\"mjx-mrow\"><span id=\"MJXc-Node-231\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-232\" class=\"mjx-mtd\"><span id=\"MJXc-Node-233\" class=\"mjx-mrow\"><span id=\"MJXc-Node-234\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-235\" class=\"mjx-mrow\"><span id=\"MJXc-Node-236\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-237\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-238\" class=\"mjx-mrow\"><span id=\"MJXc-Node-239\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-240\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-241\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>Substitute the known values into the equation and solve for <em>N<\/em>:<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-7-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-242\" class=\"mjx-math\"><span id=\"MJXc-Node-243\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-244\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-245\" class=\"mjx-mrow\"><span id=\"MJXc-Node-246\" class=\"mjx-texatom\"><span id=\"MJXc-Node-247\" class=\"mjx-mrow\"><span id=\"MJXc-Node-248\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-249\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-250\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-251\" class=\"mjx-mrow\"><span id=\"MJXc-Node-252\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-253\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-254\" class=\"mjx-mrow\"><span id=\"MJXc-Node-255\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-256\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-257\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-258\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-259\" class=\"mjx-mrow\"><span id=\"MJXc-Node-260\" class=\"mjx-mrow\"><span id=\"MJXc-Node-261\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-262\" class=\"mjx-mrow\"><span id=\"MJXc-Node-263\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.01<\/span><\/span><span id=\"MJXc-Node-264\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-265\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-266\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-267\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-268\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><span id=\"MJXc-Node-269\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-270\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-271\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-272\" class=\"mjx-mrow\"><span id=\"MJXc-Node-273\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.00<\/span><\/span><span id=\"MJXc-Node-274\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-275\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-276\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-277\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-278\" class=\"mjx-mrow\"><span id=\"MJXc-Node-279\" class=\"mjx-mrow\"><span id=\"MJXc-Node-280\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-281\" class=\"mjx-mrow\"><span id=\"MJXc-Node-282\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1.38<\/span><\/span><span id=\"MJXc-Node-283\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-284\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-285\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-286\" class=\"mjx-texatom\"><span id=\"MJXc-Node-287\" class=\"mjx-mrow\"><span id=\"MJXc-Node-288\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-289\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">23<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-290\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0J\/K<\/span><\/span><\/span><span id=\"MJXc-Node-291\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-292\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-293\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-294\" class=\"mjx-mrow\"><span id=\"MJXc-Node-295\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">273<\/span><\/span><span id=\"MJXc-Node-296\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><span id=\"MJXc-Node-297\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-298\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-299\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2.68<\/span><\/span><span id=\"MJXc-Node-300\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-301\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-302\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-303\" class=\"mjx-texatom\"><span id=\"MJXc-Node-304\" class=\"mjx-mrow\"><span id=\"MJXc-Node-305\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">25<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-306\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-307\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<div class=\"textbox examples\">\n<h4>Discussion<\/h4>\n<p>This number is undeniably large, considering that a gas is mostly empty space. <em>N<\/em> is huge, even in small volumes. For example, 1 cm<sup>3<\/sup>\u00a0of a gas at STP has 2.68 \u00d7 10<sup>19<\/sup> molecules in it. Once again, note that <em>N<\/em> is the same for all types or mixtures of gases.<\/p>\n<\/div>\n<h2>Moles and Avogadro\u2019s Number<\/h2>\n<p>It is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A <em>mole<\/em> (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called <em>Avogadro\u2019s number<\/em> (<em>N<\/em><sub>A<\/sub>), in recognition of Italian scientist Amedeo Avogadro (1776\u20131856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, contain equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro\u2019s number is\u00a0<em>N<\/em><sub>A<\/sub> =\u00a06.02 \u00d7 10<sup>23<\/sup> mol<sup>\u22121<\/sup>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Avogadro\u2019s Number<\/h3>\n<p>One mole always contains 6.02 \u00d7 10<sup>23<\/sup> particles (atoms or molecules), independent of the element or substance. A mole of any substance has a mass in grams equal to its molecular mass, which can be calculated from the atomic masses given in the periodic table of elements.<\/p>\n<p><em>N<\/em><sub>A<\/sub> =\u00a06.02 \u00d7 10<sup>23<\/sup> mol<sup>\u22121<\/sup><\/p>\n<\/div>\n<div class=\"wp-caption aligncenter\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20104332\/Figure_14_03_03.jpg\" alt=\"The illustration shows relatively flat land with a solitary mountain, labeled Mt. Everest, and blue sky above. A double-headed vertical arrow stretches between the land and a point in the sky that is well above the peak of the mountain. The arrow, labeled table tennis balls, serves to indicate that a column of one mole of table tennis balls would reach a point in the sky that is much higher than the peak of Mt. Everest.\" width=\"750\" height=\"320\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. How big is a mole? On a macroscopic level, one mole of table tennis balls would cover the Earth to a depth of about 40 km.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Check Your Understanding<\/h3>\n<p>The active ingredient in a Tylenol pill is 325 mg of acetaminophen (C<sub>8<\/sub>H<sub>9<\/sub>NO<sub>2<\/sub>). Find the number of active molecules of acetaminophen in a single pill.<\/p>\n<h4>Solution<\/h4>\n<p>We first need to calculate the molar mass (the mass of one mole) of acetaminophen. To do this, we need to multiply the number of atoms of each element by the element\u2019s atomic mass.<\/p>\n<p>(8 moles of carbon)(12 grams\/mole) + (9 moles hydrogen)(1 gram\/mole) + (1 mole nitrogen)(14 grams\/mole) + (2 moles oxygen)(16 grams\/mole) = 151 g<\/p>\n<p>Then we need to calculate the number of moles in 325 mg.<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-8-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-308\" class=\"mjx-math\"><span id=\"MJXc-Node-309\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-310\" class=\"mjx-mstyle\"><span id=\"MJXc-Node-311\" class=\"mjx-mrow\"><span id=\"MJXc-Node-312\" class=\"mjx-mrow\"><span id=\"MJXc-Node-313\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">(<\/span><\/span><span id=\"MJXc-Node-314\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-315\" class=\"mjx-mrow\"><span id=\"MJXc-Node-316\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">325<\/span><\/span><span id=\"MJXc-Node-317\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mg<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-318\" class=\"mjx-mrow\"><span id=\"MJXc-Node-319\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">151<\/span><\/span><span id=\"MJXc-Node-320\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0grams\/mole<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-321\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-322\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-323\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">(<\/span><\/span><span id=\"MJXc-Node-324\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-325\" class=\"mjx-mrow\"><span id=\"MJXc-Node-326\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><span id=\"MJXc-Node-327\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0gram<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-328\" class=\"mjx-mrow\"><span id=\"MJXc-Node-329\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1000<\/span><\/span><span id=\"MJXc-Node-330\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mg<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-331\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size3-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-332\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-333\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">2.15<\/span><\/span><span id=\"MJXc-Node-334\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-335\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-336\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-337\" class=\"mjx-texatom\"><span id=\"MJXc-Node-338\" class=\"mjx-mrow\"><span id=\"MJXc-Node-339\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-340\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-341\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0moles<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-342\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<p>Then use Avogadro\u2019s number to calculate the number of molecules.<\/p>\n<p><em>N<\/em> = (2.15 \u00d7 10<sup>\u22123<\/sup> moles)(6.02 \u00d7 10<sup>23<\/sup> molecules\/mole) = 1.30\u00a0\u00d7 10<sup>21<\/sup> molecules<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3. Calculating Moles per Cubic Meter and Liters per Mole<\/h3>\n<p>Calculate the following:<\/p>\n<ol>\n<li>The number of moles in 1.00 m<sup>3<\/sup>\u00a0of gas at STP<\/li>\n<li>The number of liters of gas per mole.<\/li>\n<\/ol>\n<h4>Strategy and Solution<\/h4>\n<ol>\n<li>We are asked to find the number of moles per cubic meter, and we know from Example 2\u00a0that the number of molecules per cubic meter at STP is 2.68 \u00d7 10<sup>25<\/sup>. The number of moles can be found by dividing the number of molecules by Avogadro\u2019s number. We let <em>n<\/em> stand for the number of moles,<br \/>\n<span class=\"mj\"><span id=\"MathJax-Element-9-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-343\" class=\"mjx-math\"><span id=\"MJXc-Node-344\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-345\" class=\"mjx-texatom\"><span id=\"MJXc-Node-346\" class=\"mjx-mrow\"><span id=\"MJXc-Node-347\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">m<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-348\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-349\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mol\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-350\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-351\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-352\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-353\" class=\"mjx-mrow\"><span id=\"MJXc-Node-354\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><span id=\"MJXc-Node-355\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-356\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-357\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-358\" class=\"mjx-mrow\"><span id=\"MJXc-Node-359\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6.02<\/span><\/span><span id=\"MJXc-Node-360\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-361\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-362\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-363\" class=\"mjx-texatom\"><span id=\"MJXc-Node-364\" class=\"mjx-mrow\"><span id=\"MJXc-Node-365\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">23<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-366\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules\/mol<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-367\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-368\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-369\" class=\"mjx-mrow\"><span id=\"MJXc-Node-370\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2.68<\/span><\/span><span id=\"MJXc-Node-371\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-372\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-373\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-374\" class=\"mjx-texatom\"><span id=\"MJXc-Node-375\" class=\"mjx-mrow\"><span id=\"MJXc-Node-376\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">25<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-377\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-378\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-379\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-380\" class=\"mjx-mrow\"><span id=\"MJXc-Node-381\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6.02<\/span><\/span><span id=\"MJXc-Node-382\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-383\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-384\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-385\" class=\"mjx-texatom\"><span id=\"MJXc-Node-386\" class=\"mjx-mrow\"><span id=\"MJXc-Node-387\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">23<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-388\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0molecules\/mol<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-389\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-390\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">44.5<\/span><\/span><span id=\"MJXc-Node-391\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-392\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mol\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-393\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-394\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<ul>\n<li><\/li>\n<li>Using the value obtained for the number of moles in a cubic meter, and converting cubic meters to liters, we obtain <span class=\"mj\"><span id=\"MathJax-Element-10-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-395\" class=\"mjx-math\"><span id=\"MJXc-Node-396\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-397\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-398\" class=\"mjx-mrow\"><span id=\"MJXc-Node-399\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">(<\/span><\/span><span id=\"MJXc-Node-400\" class=\"mjx-mrow\"><span id=\"MJXc-Node-401\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-402\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-403\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-404\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-405\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0L\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-406\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-407\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">)<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-408\" class=\"mjx-mrow\"><span id=\"MJXc-Node-409\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">44.5<\/span><\/span><span id=\"MJXc-Node-410\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-411\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mol\/m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-412\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-413\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-414\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">22.5<\/span><\/span><span id=\"MJXc-Node-415\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0L\/mol<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-416\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h4>Discussion<\/h4>\n<p>This value is very close to the accepted value of 22.4 L\/mol. The slight difference is due to rounding errors caused by using three-digit input. Again this number is the same for all gases. In other words, it is independent of the gas.<\/p>\n<p>The (average) molar weight of air (approximately 80% N<sub>2<\/sub> and 20% O<sub>2<\/sub> is <em>M<\/em> = 28.8 g.\u00a0Thus the mass of one cubic meter of air is 1.28 kg. If a living room has dimensions 5 m \u00d7 5 m \u00d7 3 m,\u00a0the mass of air inside the room is 96 kg, which is the typical mass of a human.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Check Your Understanding<\/h3>\n<p>The density of air at standard conditions (<em>P<\/em> = 1 atm and\u00a0<em>T\u00a0<\/em>= 20\u00baC) is 1.28 kg\/m<sup>3<\/sup>. At what pressure is the density 0.64 kg\/m<sup>3<\/sup> if the temperature and number of molecules are kept constant?<\/p>\n<h4>Solution<\/h4>\n<p>The best way to approach this question is to think about what is happening. If the density drops to half its original value and no molecules are lost, then the volume must double. If we look at the equation <em>PV<\/em> =\u00a0<em>NkT<\/em>, we see that when the temperature is constant, the pressure is inversely proportional to volume. Therefore, if the volume doubles, the pressure must drop to half its original value, and <em>P<\/em><sub>f<\/sub> = 0.50 atm.<\/p>\n<\/div>\n<h2>The Ideal Gas Law Restated Using Moles<\/h2>\n<p>A very common expression of the ideal gas law uses the number of moles, <em>n<\/em>, rather than the number of atoms and molecules, <em>N<\/em>. We start from the ideal gas law,\u00a0<em>PV<\/em> =\u00a0<em>NkT<\/em>,\u00a0and multiply and divide the equation by Avogadro\u2019s number <em>N<\/em><sub>A<\/sub>. This gives <span class=\"mj\"><span id=\"MathJax-Element-11-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-417\" class=\"mjx-math\"><span id=\"MJXc-Node-418\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-419\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-420\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><span id=\"MJXc-Node-421\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-422\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-423\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-424\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-425\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-426\" class=\"mjx-texatom\"><span id=\"MJXc-Node-427\" class=\"mjx-mrow\"><span id=\"MJXc-Node-428\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">A<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-429\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-430\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-431\" class=\"mjx-texatom\"><span id=\"MJXc-Node-432\" class=\"mjx-mrow\"><span id=\"MJXc-Node-433\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">A<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-434\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">k<\/span><\/span><span id=\"MJXc-Node-435\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-436\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>.<\/p>\n<p>Note that <span class=\"mj\"><span id=\"MathJax-Element-12-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-437\" class=\"mjx-math\"><span id=\"MJXc-Node-438\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-439\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">n<\/span><\/span><span id=\"MJXc-Node-440\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-441\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-442\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-443\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-444\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">N<\/span><\/span><\/span><span class=\"mjx-sub\"><span id=\"MJXc-Node-445\" class=\"mjx-texatom\"><span id=\"MJXc-Node-446\" class=\"mjx-mrow\"><span id=\"MJXc-Node-447\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">A<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-448\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>is the number of moles. We define the universal gas constant <em>R<\/em>=<em>N<\/em><sub>A<\/sub><em>k<\/em>, and obtain the ideal gas law in terms of moles.<\/p>\n<div class=\"textbox shaded\">\n<h3>Ideal Gas Law (in terms of moles)<\/h3>\n<p>The ideal gas law (in terms of moles) is\u00a0<em>PV<\/em> =\u00a0<em>nRT<\/em>.<\/p>\n<p>The numerical value of <em>R<\/em> in SI units is\u00a0<em>R<\/em> =\u00a0<em>N<\/em><sub>A<\/sub><em>k<\/em> = (6.02 \u00d7 10<sup>23<\/sup> mol<sup>\u22121<\/sup>)(1.38 \u00d7 10<sup>\u221223<\/sup> J\/K) = 8.31 J\/mol\u00a0\u00b7 K.<\/p>\n<p>In other units,<\/p>\n<p><em>R<\/em> = 1.99 cal\/mol \u00b7 K<\/p>\n<p><em>R<\/em> = 0.0821 L\u00a0\u00b7 atm\/mol\u00a0\u00b7 K<\/p>\n<p>You can use whichever value of <em>R<\/em> is most convenient for a particular problem.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4. Calculating Number of Moles: Gas in a Bike Tire<\/h3>\n<p>How many moles of gas are in a bike tire with a volume of 2.00 \u00d7 10<sup>\u22123<\/sup> m<sup>3<\/sup>(2.00 L),\u00a0a pressure of 7.00 \u00d7 10<sup>5<\/sup> Pa\u00a0(a gauge pressure of just under 90.0 lb\/in<sup>2<\/sup>), and at a temperature of 18.0\u00baC?<\/p>\n<h4>Strategy<\/h4>\n<p>Identify the knowns and unknowns, and choose an equation to solve for the unknown. In this case, we solve the ideal gas law, <em>PV<\/em> =\u00a0<em>nRT<\/em>, for the number of moles <em>n<\/em>.<\/p>\n<h4>Solution<\/h4>\n<p>Identify the knowns:<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-13-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-449\" class=\"mjx-math\"><span id=\"MJXc-Node-450\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-451\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-452\" class=\"mjx-mtr\"><span id=\"MJXc-Node-453\" class=\"mjx-mtd\"><span id=\"MJXc-Node-454\" class=\"mjx-mrow\"><span id=\"MJXc-Node-455\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-456\" class=\"mjx-mtd\"><span id=\"MJXc-Node-457\" class=\"mjx-mrow\"><span id=\"MJXc-Node-458\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-459\" class=\"mjx-mtd\"><span id=\"MJXc-Node-460\" class=\"mjx-mrow\"><span id=\"MJXc-Node-461\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7.00<\/span><\/span><span id=\"MJXc-Node-462\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-463\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-464\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-465\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-466\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-467\" class=\"mjx-mtr\"><span id=\"MJXc-Node-468\" class=\"mjx-mtd\"><span id=\"MJXc-Node-469\" class=\"mjx-mrow\"><span id=\"MJXc-Node-470\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-471\" class=\"mjx-mtd\"><span id=\"MJXc-Node-472\" class=\"mjx-mrow\"><span id=\"MJXc-Node-473\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-474\" class=\"mjx-mtd\"><span id=\"MJXc-Node-475\" class=\"mjx-mrow\"><span id=\"MJXc-Node-476\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2.00<\/span><\/span><span id=\"MJXc-Node-477\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-478\" class=\"mjx-msubsup MJXc-space2\"><span class=\"mjx-base\"><span id=\"MJXc-Node-479\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-480\" class=\"mjx-texatom\"><span id=\"MJXc-Node-481\" class=\"mjx-mrow\"><span id=\"MJXc-Node-482\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-483\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-484\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-485\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-486\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-487\" class=\"mjx-mtr\"><span id=\"MJXc-Node-488\" class=\"mjx-mtd\"><span id=\"MJXc-Node-489\" class=\"mjx-mrow\"><span id=\"MJXc-Node-490\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-491\" class=\"mjx-mtd\"><span id=\"MJXc-Node-492\" class=\"mjx-mrow\"><span id=\"MJXc-Node-493\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-494\" class=\"mjx-mtd\"><span id=\"MJXc-Node-495\" class=\"mjx-mrow\"><span id=\"MJXc-Node-496\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-497\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">18.0<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-498\" class=\"mjx-texatom\"><span id=\"MJXc-Node-499\" class=\"mjx-mrow\"><span id=\"MJXc-Node-500\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-501\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">C<\/span><\/span><span id=\"MJXc-Node-502\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-503\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">291<\/span><\/span><span id=\"MJXc-Node-504\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-505\" class=\"mjx-mtr\"><span id=\"MJXc-Node-506\" class=\"mjx-mtd\"><span id=\"MJXc-Node-507\" class=\"mjx-mrow\"><span id=\"MJXc-Node-508\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">R<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-509\" class=\"mjx-mtd\"><span id=\"MJXc-Node-510\" class=\"mjx-mrow\"><span id=\"MJXc-Node-511\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-512\" class=\"mjx-mtd\"><span id=\"MJXc-Node-513\" class=\"mjx-mrow\"><span id=\"MJXc-Node-514\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8.31<\/span><\/span><span id=\"MJXc-Node-515\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0J\/mol<\/span><\/span><span id=\"MJXc-Node-516\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-517\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-518\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/div>\n<p>Rearrange the equation to solve for <em>n<\/em> and substitute known values.<\/p>\n<p><span class=\"mj\"><span id=\"MathJax-Element-14-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-519\" class=\"mjx-math\"><span id=\"MJXc-Node-520\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-521\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-522\" class=\"mjx-mtr\"><span id=\"MJXc-Node-523\" class=\"mjx-mtd\"><span id=\"MJXc-Node-524\" class=\"mjx-mrow\"><span id=\"MJXc-Node-525\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">n<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-526\" class=\"mjx-mtd\"><span id=\"MJXc-Node-527\" class=\"mjx-mrow\"><span id=\"MJXc-Node-528\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-529\" class=\"mjx-mtd\"><span id=\"MJXc-Node-530\" class=\"mjx-mrow\"><span id=\"MJXc-Node-531\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-532\" class=\"mjx-mrow\"><span id=\"MJXc-Node-533\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">P<\/span><\/span><span id=\"MJXc-Node-534\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">V<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-535\" class=\"mjx-mrow\"><span id=\"MJXc-Node-536\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">R<\/span><\/span><span id=\"MJXc-Node-537\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">T<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-538\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-539\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-540\" class=\"mjx-mrow\"><span id=\"MJXc-Node-541\" class=\"mjx-mrow\"><span id=\"MJXc-Node-542\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-543\" class=\"mjx-mrow\"><span id=\"MJXc-Node-544\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">7.00<\/span><\/span><span id=\"MJXc-Node-545\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-546\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-547\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-548\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-549\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0Pa<\/span><\/span><\/span><span id=\"MJXc-Node-550\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-551\" class=\"mjx-mrow\"><span id=\"MJXc-Node-552\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-553\" class=\"mjx-mrow\"><span id=\"MJXc-Node-554\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2.00<\/span><\/span><span id=\"MJXc-Node-555\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-556\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-557\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">10<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-558\" class=\"mjx-texatom\"><span id=\"MJXc-Node-559\" class=\"mjx-mrow\"><span id=\"MJXc-Node-560\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-561\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-562\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-563\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0m<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-564\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-565\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-566\" class=\"mjx-mrow\"><span id=\"MJXc-Node-567\" class=\"mjx-mrow\"><span id=\"MJXc-Node-568\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-569\" class=\"mjx-mrow\"><span id=\"MJXc-Node-570\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8.31<\/span><\/span><span id=\"MJXc-Node-571\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0J\/mol<\/span><\/span><span id=\"MJXc-Node-572\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-573\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><span id=\"MJXc-Node-574\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-575\" class=\"mjx-mrow\"><span id=\"MJXc-Node-576\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-577\" class=\"mjx-mrow\"><span id=\"MJXc-Node-578\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">291<\/span><\/span><span id=\"MJXc-Node-579\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0K<\/span><\/span><\/span><span id=\"MJXc-Node-580\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-581\" class=\"mjx-mtr\"><span id=\"MJXc-Node-582\" class=\"mjx-mtd\"><span id=\"MJXc-Node-583\" class=\"mjx-mrow\"><span id=\"MJXc-Node-584\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-585\" class=\"mjx-mtd\"><span id=\"MJXc-Node-586\" class=\"mjx-mrow\"><span id=\"MJXc-Node-587\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-588\" class=\"mjx-mtd\"><span id=\"MJXc-Node-589\" class=\"mjx-mrow\"><span id=\"MJXc-Node-590\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">0.579<\/span><\/span><span id=\"MJXc-Node-591\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00a0mol<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-592\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<div class=\"textbox examples\">\n<p><strong>Discussion<\/strong><\/p>\n<p>The most convenient choice for <em>R<\/em> in this case is 8.31 J\/mol \u00b7\u00a0K,\u00a0because our known quantities are in SI units. The pressure and temperature are obtained from the initial conditions in Example 1, but we would get the same answer if we used the final values.<\/p>\n<\/div>\n<p>The ideal gas law can be considered to be another manifestation of the law of conservation of energy (see Conservation of Energy). Work done on a gas results in an increase in its energy, increasing pressure and\/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas\u2019s atoms and molecules.<\/p>\n<h2>The Ideal Gas Law and Energy<\/h2>\n<p>Let us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.<\/p>\n<p>The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in <em>PV<\/em> =\u00a0<em>NkT<\/em>\u00a0is\u00a0<em>NkT<\/em>. This term is roughly the amount of translational kinetic energy of <em>N<\/em> atoms or molecules at an absolute temperature <em>T<\/em>, as we shall see formally in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature. The left-hand side of the ideal gas law is <em>PV<\/em>, which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands\u2014something we explore in Heat and Heat Transfer Methods\u2014similar to what occurs in gasoline or steam engines and turbines.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Problem-Solving Strategy: The Ideal Gas Law<\/h3>\n<p><strong>Step 1.<\/strong> Examine the situation to determine that an ideal gas is involved. Most gases are nearly ideal.<\/p>\n<p><strong>Step 2.<\/strong>\u00a0Make a list of what quantities are given, or can be inferred from the problem as stated (identify the known quantities). Convert known values into proper SI units (K for temperature, Pa for pressure, m<sup>3<\/sup> for volume, molecules for <em>N<\/em>, and moles for <em>n<\/em>).<\/p>\n<p><strong>Step 3.<\/strong> Identify exactly what needs to be determined in the problem (identify the unknown quantities). A written list is useful.<\/p>\n<p><strong>Step 4.<\/strong> Determine whether the number of molecules or the number of moles is known, in order to decide which form of the ideal gas law to use. The first form is <em>PV<\/em> =\u00a0<em>NkT<\/em>\u00a0and involves <em>N<\/em>, the number of atoms or molecules. The second form is <em>PV\u00a0<\/em>=\u00a0<em>nRT<\/em>\u00a0and involves <em>n<\/em>, the number of moles.<\/p>\n<p><strong>Step 5.<\/strong> Solve the ideal gas law for the quantity to be determined (the unknown quantity). You may need to take a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.<\/p>\n<p><strong>Step 6.<\/strong> Substitute the known quantities, along with their units, into the appropriate equation, and obtain numerical solutions complete with units. Be certain to use absolute temperature and absolute pressure.<\/p>\n<p><strong>Step 7.<\/strong> Check the answer to see if it is reasonable: Does it make sense?<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Check Your Understanding<\/h3>\n<p>Liquids and solids have densities about 1000 times greater than gases. Explain how this implies that the distances between atoms and molecules in gases are about 10 times greater than the size of their atoms and molecules.<\/p>\n<h4>Solution<\/h4>\n<p>Atoms and molecules are close together in solids and liquids. In gases they are separated by empty space. Thus gases have lower densities than liquids and solids. Density is mass per unit volume, and volume is related to the size of a body (such as a sphere) cubed. So if the distance between atoms and molecules increases by a factor of 10, then the volume occupied increases by a factor of 1000, and the density decreases by a factor of 1000.<\/p>\n<\/div>\n<h2>Section Summary<\/h2>\n<ul>\n<li>The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.<\/li>\n<li>The ideal gas law can be written in terms of the number of molecules of gas:\u00a0<em>PV\u00a0<\/em>= <em>NkT<\/em>,\u00a0where <em>P<\/em> is pressure, <em>V<\/em> is volume, <em>T<\/em> is temperature, <em>N<\/em> is number of molecules, and k is the Boltzmann constant\u00a0<em>k\u00a0<\/em>= 1.38 \u00d7 10<sup>\u201323<\/sup> J\/K.<\/li>\n<li>A mole is the number of atoms in a 12-g sample of carbon-12.<\/li>\n<li>The number of molecules in a mole is called Avogadro\u2019s number <em>NA<\/em>,\u00a0<em>NA\u00a0<\/em>= 6.02 \u00d7 10<sup>23<\/sup> mol<sup>\u22121<\/sup>.<\/li>\n<li>A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.<\/li>\n<li>The ideal gas law can also be written and solved in terms of the number of moles of gas:\u00a0<em>PV\u00a0<\/em>= <em>nRT<\/em>,\u00a0where n is number of moles and <em>R<\/em> is the universal gas constant,\u00a0<em>R<\/em> = 8.31 J\/mol \u22c5 K.<\/li>\n<li>The ideal gas law is generally valid at temperatures well above the boiling temperature.<\/li>\n<\/ul>\n<div class=\"textbox key-takeaways\">\n<h3>Conceptual Questions<\/h3>\n<p>Find out the human population of Earth. Is there a mole of people inhabiting Earth? If the average mass of a person is 60 kg, calculate the mass of a mole of people. How does the mass of a mole of people compare with the mass of Earth?<\/p>\n<p>Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?<\/p>\n<p>A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol>\n<li>The gauge pressure in your car tires is 2.50 \u00d7 10<sup>5<\/sup> N\/m<sup>2<\/sup> at a temperature of 35.0\u00baC when you drive it onto a ferry boat to Alaska. What is their gauge pressure later, when their temperature has dropped to \u201340.0\u00baC?<\/li>\n<li>Convert an absolute pressure of 7.00 \u00d7 10<sup>5<\/sup> N\/m<sup>2<\/sup> to gauge pressure in lb\/in<sup>2<\/sup>. (This value was stated to be just less than 90.0 lb\/in<sup>2<\/sup> in Example 4. Is it?)<\/li>\n<li>Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of 20.0\u00baC. (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is 60.0\u00baC (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks. (b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. What will the actual final pressure be, taking this into account? Is this a negligible difference?<\/li>\n<li>Large helium-filled balloons are used to lift scientific equipment to high altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of 10.0\u00baC and rises to an altitude where its volume is twenty times the original volume and its temperature is \u201350.0\u00baC? (b) What is the gauge pressure? (Assume atmospheric pressure is constant.)<\/li>\n<li>Confirm that the units of nRT are those of energy for each value of R: (a) 8.31 J\/mol \u22c5 K, (b) 1.99 cal\/mol \u22c5 K, and (c) 0.0821 L \u22c5 atm\/mol \u22c5 K.<\/li>\n<li>In the text, it was shown that <em>N<\/em>\/<em>V\u00a0<\/em>= 2.68 \u00d7 10<sup>25<\/sup> m<sup>\u22123<\/sup> for gas at STP. (a) Show that this quantity is equivalent to <em>N<\/em>\/<em>V<\/em> = 2.68 \u00d7 10<sup>19<\/sup> cm<sup>\u22123<\/sup>, as stated. (b) About how many atoms are there in one \u03bcm<sup>3<\/sup> (a cubic micrometer) at STP? (c) What does your answer to part (b) imply about the separation of atoms and molecules?<\/li>\n<li>Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at 37.0\u00baC (body temperature).<\/li>\n<li>An airplane passenger has 100 cm<sup>3<\/sup> of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to 7.50 \u00d7 10<sup>4<\/sup> N\/m<sup>2<\/sup>?<\/li>\n<li>(a) What is the volume (in km<sup>3<\/sup>) of Avogadro\u2019s number of sand grains if each grain is a cube and has sides that are 1.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages 100 m in width and 10.0 m in depth? Neglect air spaces between grains.<\/li>\n<li>An expensive vacuum system can achieve a pressure as low as 1.00 \u00d7 10<sup>\u20137<\/sup> N\/m<sup>2<\/sup> at 20\u00baC. How many atoms are there in a cubic centimeter at this pressure and temperature?<\/li>\n<li>The number density of gas atoms at a certain location in the space above our planet is about 1.00 \u00d7 10<sup>11<\/sup> m<sup>\u22123<\/sup>, and the pressure is 2.75 \u00d7 10<sup>\u201310<\/sup> N\/m<sup>2<\/sup> in this space. What is the temperature there?<\/li>\n<li>A bicycle tire has a pressure of 7.00 \u00d7 10<sup>5<\/sup> N\/m<sup>2<\/sup> at a temperature of 18.0\u00baC and contains 2.00 L of gas. What will its pressure be if you let out an amount of air that has a volume of 100cm3 at atmospheric pressure? Assume tire temperature and volume remain constant.<\/li>\n<li>A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of 1.40 \u00d7 10<sup>7<\/sup> N\/m<sup>2<\/sup> and a temperature of 25.0\u00baC. Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperature (\u201378.5\u00baC) to reduce the leak rate and pressure so that it can be safely repaired. (a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank appear to be a practical solution?<\/li>\n<li>Find the number of moles in 2.00 L of gas at 35.0\u00baC and under 7.41 \u00d7 10<sup>7<\/sup> N\/m<sup>2<\/sup> of pressure.<\/li>\n<li>Calculate the depth to which Avogadro\u2019s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.<\/li>\n<li>(a) What is the gauge pressure in a 25.0\u00baC car tire containing 3.60 mol of gas in a 30.0 L volume? (b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and 25.0\u00baC? Assume the temperature returns to 25.0\u00baC and the volume remains constant.<\/li>\n<li>(a) In the deep space between galaxies, the density of atoms is as low as 10<sup>6<\/sup> atoms\/m<sup>3<\/sup>, and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in m<sup>3<\/sup>) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?<\/li>\n<\/ol>\n<\/div>\n<h2>Glossary<\/h2>\n<p><strong>ideal gas law:<\/strong>\u00a0the physical law that relates the pressure and volume of a gas to the number of gas molecules or number of moles of gas and the temperature of the gas<\/p>\n<p><strong>Boltzmann constant:<\/strong> <em>k<\/em>, a physical constant that relates energy to temperature; <em>k\u00a0<\/em>= 1.38 \u00d7 10<sup>\u201323<\/sup> J\/K<\/p>\n<p><strong>Avogadro\u2019s number:<\/strong> <em>NA<\/em>, the number of molecules or atoms in one mole of a substance; <em>NA<\/em> = 6.02 \u00d7 10<sup>23<\/sup> particles\/mole<\/p>\n<p><strong>mole:<\/strong>\u00a0the quantity of a substance whose mass (in grams) is equal to its molecular mass<\/p>\n<div class=\"textbox exercises\">\n<h3>Selected Solutions to\u00a0Problems &amp; Exercises<\/h3>\n<p>1.\u00a01.62 atm<\/p>\n<p>3.\u00a0(a) 0.136 atm;\u00a0(b) 0.135 atm. The difference between this value and the value from part (a) is negligible.<\/p>\n<p>5. (a) <span class=\"mj\"><span id=\"MathJax-Element-15-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-593\" class=\"mjx-math\"><span id=\"MJXc-Node-594\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-595\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">nRT<\/span><\/span><span id=\"MJXc-Node-596\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-597\" class=\"mjx-mrow MJXc-space3\"><span id=\"MJXc-Node-598\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-599\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">mol<\/span><\/span><span id=\"MJXc-Node-600\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-601\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-602\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-603\" class=\"mjx-mrow\"><span id=\"MJXc-Node-604\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">J\/mol<\/span><\/span><span id=\"MJXc-Node-605\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-606\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><\/span><span id=\"MJXc-Node-607\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-608\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-609\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-610\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><span id=\"MJXc-Node-611\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-612\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-613\" class=\"mjx-mtext MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">J<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-614\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/div>\n<p>;<\/p>\n<p>(b) <span class=\"mj\"><span id=\"MathJax-Element-16-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-615\" class=\"mjx-math\"><span id=\"MJXc-Node-616\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-617\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">nRT<\/span><\/span><span id=\"MJXc-Node-618\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-619\" class=\"mjx-mrow MJXc-space3\"><span id=\"MJXc-Node-620\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-621\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">mol<\/span><\/span><span id=\"MJXc-Node-622\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-623\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-624\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-625\" class=\"mjx-mrow\"><span id=\"MJXc-Node-626\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">cal\/mol<\/span><\/span><span id=\"MJXc-Node-627\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-628\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><\/span><span id=\"MJXc-Node-629\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-630\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-631\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-632\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><span id=\"MJXc-Node-633\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-634\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-635\" class=\"mjx-mtext MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">cal<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-636\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>;<\/p>\n<p>(c) <span class=\"mj\"><span id=\"MathJax-Element-17-Frame\" class=\"mjx-full-width mjx-chtml MathJax_CHTML\" role=\"presentation\"><span id=\"MJXc-Node-637\" class=\"mjx-math\"><span id=\"MJXc-Node-638\" class=\"mjx-mrow\"><span class=\"mjx-stack\"><span class=\"mjx-block\"><span class=\"mjx-box\"><span id=\"MJXc-Node-639\" class=\"mjx-mtable\"><span class=\"mjx-table\"><span id=\"MJXc-Node-640\" class=\"mjx-mtr\"><span id=\"MJXc-Node-641\" class=\"mjx-mtd\"><span id=\"MJXc-Node-642\" class=\"mjx-mrow\"><span id=\"MJXc-Node-643\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">nRT<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-644\" class=\"mjx-mtd\"><span id=\"MJXc-Node-645\" class=\"mjx-mrow\"><span id=\"MJXc-Node-646\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-647\" class=\"mjx-mtd\"><span id=\"MJXc-Node-648\" class=\"mjx-mrow\"><span id=\"MJXc-Node-649\" class=\"mjx-mrow\"><span id=\"MJXc-Node-650\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-651\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">mol<\/span><\/span><span id=\"MJXc-Node-652\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-653\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-654\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-655\" class=\"mjx-mrow\"><span id=\"MJXc-Node-656\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">L<\/span><\/span><span id=\"MJXc-Node-657\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-658\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">atm\/mol<\/span><\/span><span id=\"MJXc-Node-659\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-660\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><\/span><span id=\"MJXc-Node-661\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-662\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-663\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-664\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">K<\/span><\/span><span id=\"MJXc-Node-665\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-666\" class=\"mjx-mtr\"><span id=\"MJXc-Node-667\" class=\"mjx-mtd\"><span id=\"MJXc-Node-668\" class=\"mjx-mrow\"><\/span><\/span><span id=\"MJXc-Node-669\" class=\"mjx-mtd\"><span id=\"MJXc-Node-670\" class=\"mjx-mrow\"><span id=\"MJXc-Node-671\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-672\" class=\"mjx-mtd\"><span id=\"MJXc-Node-673\" class=\"mjx-mrow\"><span id=\"MJXc-Node-674\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">L<\/span><\/span><span id=\"MJXc-Node-675\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-676\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">atm<\/span><\/span><span id=\"MJXc-Node-677\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-678\" class=\"mjx-mrow MJXc-space3\"><span id=\"MJXc-Node-679\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">(<\/span><\/span><span id=\"MJXc-Node-680\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-681\" class=\"mjx-texatom\"><span id=\"MJXc-Node-682\" class=\"mjx-mrow\"><span id=\"MJXc-Node-683\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">m<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-684\" class=\"mjx-texatom\"><span id=\"MJXc-Node-685\" class=\"mjx-mrow\"><span id=\"MJXc-Node-686\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-687\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size1-R\">)<\/span><\/span><\/span><span id=\"MJXc-Node-688\" class=\"mjx-mrow MJXc-space1\"><span id=\"MJXc-Node-689\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">(<\/span><\/span><span id=\"MJXc-Node-690\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-691\" class=\"mjx-texatom\"><span id=\"MJXc-Node-692\" class=\"mjx-mrow\"><span id=\"MJXc-Node-693\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">N\/m<\/span><\/span><\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-694\" class=\"mjx-texatom\"><span id=\"MJXc-Node-695\" class=\"mjx-mrow\"><span id=\"MJXc-Node-696\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-697\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">)<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-698\" class=\"mjx-mtr\"><span id=\"MJXc-Node-699\" class=\"mjx-mtd\"><span id=\"MJXc-Node-700\" class=\"mjx-mrow\"><\/span><\/span><span id=\"MJXc-Node-701\" class=\"mjx-mtd\"><span id=\"MJXc-Node-702\" class=\"mjx-mrow\"><span id=\"MJXc-Node-703\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-704\" class=\"mjx-mtd\"><span id=\"MJXc-Node-705\" class=\"mjx-mrow\"><span id=\"MJXc-Node-706\" class=\"mjx-mtext\"><span class=\"mjx-char MJXc-TeX-main-R\">N<\/span><\/span><span id=\"MJXc-Node-707\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-708\" class=\"mjx-mtext MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">m<\/span><\/span><span id=\"MJXc-Node-709\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-710\" class=\"mjx-mtext MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">J<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-711\" class=\"mjx-mspace\"><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-2650\" class=\"standard post-2650 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox exercises\">\n<p>7.\u00a07.86 \u00d7 10<sup>\u22122<\/sup> mol<\/p>\n<p>9.\u00a0(a) 6.02 \u00d7 10<sup>5<\/sup> km<sup>3<\/sup>;\u00a0(b) 6.02 \u00d7 10<sup>8<\/sup> km<\/p>\n<p>11.\u00a0\u221273.9\u00baC<\/p>\n<p>13.\u00a0(a) 9.14 \u00d7 10<sup>6<\/sup> N\/m<sup>2<\/sup>;\u00a0(b) 8.23 \u00d7 10<sup>6<\/sup> N\/m<sup>2<\/sup>;\u00a0(c) 2.16 K;\u00a0(d) No. The final temperature needed is much too low to be easily achieved for a large object.<\/p>\n<p>15.\u00a041 km<\/p>\n<p>17. \u00a0(a) 3.7 \u00d7 10<sup>\u221217<\/sup> Pa;\u00a0(b) 6.0 \u00d7 10<sup>17<\/sup> m<sup>3<\/sup>;\u00a0(c) 8.4 \u00d7 10<sup>2<\/sup> km<\/p>\n<\/div>\n<\/div>\n<\/div>\n<section class=\"focusable\" role=\"contentinfo\">\n<div class=\"post-citations sidebar\"><\/div>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"footer\"><\/div>\n","protected":false},"author":51812,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-64","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/pressbooks\/v2\/chapters\/64","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/wp\/v2\/users\/51812"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/pressbooks\/v2\/chapters\/64\/revisions"}],"predecessor-version":[{"id":69,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/pressbooks\/v2\/chapters\/64\/revisions\/69"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/pressbooks\/v2\/chapters\/64\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/wp\/v2\/media?parent=64"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/pressbooks\/v2\/chapter-type?post=64"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/wp\/v2\/contributor?post=64"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-hccc-generalscience\/wp-json\/wp\/v2\/license?post=64"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}