{"id":2120,"date":"2015-04-22T20:53:24","date_gmt":"2015-04-22T20:53:24","guid":{"rendered":"https:\/\/courses.candelalearning.com\/oschemtemp\/?post_type=chapter&#038;p=2120"},"modified":"2016-10-27T15:56:43","modified_gmt":"2016-10-27T15:56:43","slug":"non-ideal-gas-behavior","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/chapter\/non-ideal-gas-behavior\/","title":{"raw":"Non-Ideal Gas Behavior","rendered":"Non-Ideal Gas Behavior"},"content":{"raw":"<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>Describe the physical factors that lead to deviations from ideal gas behavior<\/li>\r\n \t<li>Explain how these factors are represented in the van der Waals equation<\/li>\r\n \t<li>Define compressibility (Z) and describe how its variation with pressure reflects non-ideal behavior<\/li>\r\n \t<li>Quantify non-ideal behavior by comparing computations of gas properties using the ideal gas law and the van der Waals equation<\/li>\r\n<\/ul>\r\n<\/div>\r\nThus far, the ideal gas law, <em>PV = nRT<\/em>, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and molecular formula problems to determining the density and molar mass of a gas. As mentioned in the previous modules of this chapter, however, the behavior of a gas is often non-ideal, meaning that the observed relationships between its pressure, volume, and temperature are not accurately described by the gas laws. In this section, the reasons for these deviations from ideal gas behavior are considered.\r\n\r\nOne way in which the accuracy of <em>PV = nRT<\/em> can be judged is by comparing the actual volume of 1 mole of gas (its molar volume, <em>V<\/em><sub>m<\/sub>) to the molar volume of an ideal gas at the same temperature and pressure. This ratio is called the <strong>compressibility factor<\/strong>, <strong>Z<\/strong>, with:\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\text{Z}=\\frac{\\text{molar volume of gas at same }T\\text{ and }P}{\\text{molar volume of ideal gas at same }T\\text{ and }P}={\\left(\\frac{P{V}_{m}}{RT}\\right)}_{\\text{measured}}[\/latex]<\/p>\r\nIdeal gas behavior is therefore indicated when this ratio is equal to 1, and any deviation from 1 is an indication of non-ideal behavior. Figure\u00a01 shows plots of Z over a large pressure range for several common gases.\r\n\r\n<figure>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"697\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212111\/CNX_Chem_09_06_ZvsPgraph1.jpg\" width=\"697\" height=\"463\" \/> Figure\u00a01. A graph of the compressibility factor (Z) vs. pressure shows that gases can exhibit significant deviations from the behavior predicted by the ideal gas law.[\/caption]\r\n\r\n<\/figure>As is apparent from Figure\u00a01, the ideal gas law does not describe gas behavior well at relatively high pressures. To determine why this is, consider the differences between real gas properties and what is expected of a hypothetical ideal gas.\r\n\r\nParticles of a hypothetical ideal gas have no significant volume and do not attract or repel each other. In general, real gases approximate this behavior at relatively low pressures and high temperatures. However, at high pressures, the molecules of a gas are crowded closer together, and the amount of empty space between the molecules is reduced. At these higher pressures, the volume of the gas molecules themselves becomes appreciable relative to the total volume occupied by the gas (Figure\u00a02). The gas therefore becomes less compressible at these high pressures, and although its volume continues to decrease with increasing pressure, this decrease is not <em>proportional<\/em> as predicted by Boyle\u2019s law.\r\n\r\n<figure>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"1300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212112\/CNX_Chem_09_06_RealGas31.jpg\" alt=\"This figure includes three diagrams. In a, a cylinder with 9 purple spheres with trails indicating motion are shown. Above the cylinder, the label, \u201cParticles ideal gas,\u201d is connected to two of the spheres with line segments extending into the square. The label \u201cAssumes\u201d is beneath the square. In b, a cylinder and piston is shown. A relatively small open space is shaded lavender with 9 purple spheres packed close together. No motion trails are present on the spheres. Above the piston, a downward arrow labeled \u201cPressure\u201d is directed toward the enclosed area. In c, the cylinder is exactly the same as the first, but the number of molecules has doubled.\" width=\"1300\" height=\"528\" \/> Figure 2. Raising the pressure of a gas increases the fraction of its volume that is occupied by the gas molecules and makes the gas less compressible.[\/caption]\r\n\r\n<\/figure>At relatively low pressures, gas molecules have practically no attraction for one another because they are (on average) so far apart, and they behave almost like particles of an ideal gas. At higher pressures, however, the force of attraction is also no longer insignificant. This force pulls the molecules a little closer together, slightly decreasing the pressure (if the volume is constant) or decreasing the volume (at constant pressure) (Figure\u00a03). This change is more pronounced at low temperatures because the molecules have lower KE relative to the attractive forces, and so they are less effective in overcoming these attractions after colliding with one another.\r\n\r\n<figure>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"1300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212113\/CNX_Chem_09_06_RealGas21.jpg\" alt=\"This figure includes two diagrams. Each involves two lavender shaded boxes that contain 14 relatively evenly distributed, purple spheres. In the first box in a, a nearly centrally located purple sphere has 6 double-headed arrows extending outward from it to nearby spheres. A single purple arrow is pointing right into open space. This box is labeled, \u201creal.\u201d There is a second box that looks slightly larger than the first box in a. It has the same number of particles but no arrows. This box is labeled, \u201cideal.\u201d In b, the first box has a purple sphere at the right side which has 4 double-headed arrows radiating out to the top, bottom, and left to other spheres. A single purple arrow points right through open space to the edge of the box. This box has no spheres positioned near its right edge This box is labeled, \u201creal.\u201d The second box is the same size as the first box and contains the same number of particles. There are no arrows in it, except for the purple arrow which appears to be bigger and bolder. This box is labeled, \u201cideal.\u201d\" width=\"1300\" height=\"483\" \/> Figure 3. (a) Attractions between gas molecules serve to decrease the gas volume at constant pressure compared to an ideal gas whose molecules experience no attractive forces. (b) These attractive forces will decrease the force of collisions between the molecules and container walls, therefore reducing the pressure exerted compared to an ideal gas.[\/caption]\r\n\r\n<\/figure>There are several different equations that better approximate gas behavior than does the ideal gas law. The first, and simplest, of these was developed by the Dutch scientist Johannes van der Waals in 1879. The <strong>van der Waals equation<\/strong> improves upon the ideal gas law by adding two terms: one to account for the volume of the gas molecules and another for the attractive forces between them. <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212115\/CNX_Chem_09_06_vanderWaals_img1.jpg\" \/>\r\n\r\nThe constant <em>a<\/em> corresponds to the strength of the attraction between molecules of a particular gas, and the constant <em>b<\/em> corresponds to the size of the molecules of a particular gas. The \u201ccorrection\u201d to the pressure term in the ideal gas law is [latex]\\frac{{n}^{2}a}{{V}^{2}},[\/latex] and the \u201ccorrection\u201d to the volume is <em>nb<\/em>. Note that when <em>V<\/em> is relatively large and <em>n<\/em> is relatively small, both of these correction terms become negligible, and the van der Waals equation reduces to the ideal gas law, <em>PV = nRT<\/em>. Such a condition corresponds to a gas in which a relatively low number of molecules is occupying a relatively large volume, that is, a gas at a relatively low pressure. Experimental values for the van der Waals constants of some common gases are given in Table\u00a01.\r\n<table id=\"fs-idm15100464\" class=\"span-all\" summary=\"This table has three columns and seven rows. The first row is a header, and it labels each column, \u201cGas,\u201d \u201ca ( L to the second power a t m divided by m o l to the second power ),\u201d \u201cb ( L divided by m o l ).\u201d Under \u201cGas\u201d are the following: N subscript 2, O subscript 2, C O subscript 2, H subscript 2 O, H e, and C C l subscript 4. Under \u201ca ( L to the second power a t m divided by m o l to the second power )\u201d are the following: 1.39, 1.36, 3.59, 5.46, 0.0342, and 20.4. Under \u201cb ( L divided by m o l )\u201d are the following: 0.0391, 0.0318, 0.0427, 0.0305, 0.0237, and 0.1383.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"3\" scope=\"col\" data-align=\"center\">Table 1. Values of van der Waals Constants for Some Common Gases<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th scope=\"col\" data-align=\"left\">Gas<\/th>\r\n<th scope=\"col\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> (L<sup>2<\/sup> atm\/mol<sup>2<\/sup>)<\/th>\r\n<th scope=\"col\" data-align=\"left\"><em data-effect=\"italics\">b<\/em> (L\/mol)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">N<sub>2<\/sub><\/td>\r\n<td data-align=\"left\">1.39<\/td>\r\n<td data-align=\"left\">0.0391<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">O<sub>2<\/sub><\/td>\r\n<td data-align=\"left\">1.36<\/td>\r\n<td data-align=\"left\">0.0318<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">CO<sub>2<\/sub><\/td>\r\n<td data-align=\"left\">3.59<\/td>\r\n<td data-align=\"left\">0.0427<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">H<sub>2<\/sub>O<\/td>\r\n<td data-align=\"left\">5.46<\/td>\r\n<td data-align=\"left\">0.0305<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">He<\/td>\r\n<td data-align=\"left\">0.0342<\/td>\r\n<td data-align=\"left\">0.0237<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">CCl<sub>4<\/sub><\/td>\r\n<td data-align=\"left\">20.4<\/td>\r\n<td data-align=\"left\">0.1383<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAt low pressures, the correction for intermolecular attraction, <em>a<\/em>, is more important than the one for molecular volume, <em>b<\/em>. At high pressures and small volumes, the correction for the volume of the molecules becomes important, because the molecules themselves are incompressible and constitute an appreciable fraction of the total volume. At some intermediate pressure, the two corrections have opposing influences and the gas appears to follow the relationship given by <em>PV = nRT<\/em> over a small range of pressures. This behavior is reflected by the \u201cdips\u201d in several of the compressibility curves shown in Figure\u00a09.35. The attractive force between molecules initially makes the gas more compressible than an ideal gas, as pressure is raised (Z decreases with increasing <em>P<\/em>). At very high pressures, the gas becomes less compressible (Z increases with <em>P<\/em>), as the gas molecules begin to occupy an increasingly significant fraction of the total gas volume.\r\n\r\nStrictly speaking, the ideal gas equation functions well when intermolecular attractions between gas molecules are negligible and the gas molecules themselves do not occupy an appreciable part of the whole volume. These criteria are satisfied under conditions of <em>low pressure and high temperature<\/em>. Under such conditions, the gas is said to behave ideally, and deviations from the gas laws are small enough that they may be disregarded\u2014this is, however, very often not the case.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1:\u00a0Comparison of Ideal Gas Law and van der Waals Equation<\/h3>\r\nA 4.25-L flask contains 3.46 mol CO<sub>2<\/sub> at 229 \u00b0C. Calculate the pressure of this sample of CO<sub>2<\/sub>:\r\n<ol>\r\n \t<li>from the ideal gas law<\/li>\r\n \t<li>from the van der Waals equation<\/li>\r\n \t<li>Explain the reason(s) for the difference.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"245157\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"245157\"]\r\n<ol>\r\n \t<li>From the ideal gas law:\r\n[latex]P=\\frac{nRT}{V}=\\frac{3.46\\cancel{\\text{mol}}\\times 0.08206\\cancel{\\text{L}}\\text{atm}\\cancel{{\\text{mol}}^{-\\text{1}}}\\cancel{{\\text{K}}^{-\\text{1}}}\\times 502\\cancel{\\text{K}}}{4.25\\cancel{\\text{L}}}=33.5\\text{atm}[\/latex]<\/li>\r\n \t<li>From the van der Waals equation:\r\n[latex]\\left(P+\\frac{{n}^{2}a}{{V}^{2}}\\right)\\times \\left(V-nb\\right)=nRT\\rightarrow P=\\frac{nRT}{\\left(V-nb\\right)}-\\frac{{n}^{2}a}{{V}^{2}}[\/latex]\r\n[latex]P=\\frac{3.46\\text{mol}\\times 0.08206\\text{L}\\text{atm}{\\text{mol}}^{-\\text{1}}{\\text{K}}^{-\\text{1}}\\times \\text{502 K}}{\\left(4.25\\text{L}-3.46\\text{mol}\\times 0.0427\\text{L}{\\text{mol}}^{-\\text{1}}\\right)}-\\frac{{\\left(3.46\\text{mol}\\right)}^{2}\\times 3.59{\\text{L}}^{2}\\text{atm}{\\text{mol}}^{2}}{{\\left(4.25\\text{L}\\right)}^{2}}[\/latex]\r\nThis finally yields <em>P<\/em> = 32.4 atm.<\/li>\r\n \t<li>This is not very different from the value from the ideal gas law, because the pressure is not very high and the temperature is not very low. The value is somewhat different because CO<sub>2<\/sub> molecules do have some volume and attractions between molecules, and the ideal gas law assumes they do not have volume or attractions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<h4><strong>Check your Learning<\/strong><\/h4>\r\nA 560-mL flask contains 21.3 g N<sub>2<\/sub> at 145 \u00b0C. Calculate the pressure of N<sub>2<\/sub>:\r\n<ol>\r\n \t<li>from the ideal gas law<\/li>\r\n \t<li>from the van der Waals equation<\/li>\r\n \t<li>Explain the reason(s) for the difference.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"451203\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"451203\"]\r\n<ol>\r\n \t<li>46.562 atm<\/li>\r\n \t<li>46.594 atm<\/li>\r\n \t<li>The van der Waals equation takes into account the volume of the gas molecules themselves as well as intermolecular attractions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Concepts and Summary<\/h3>\r\nGas molecules possess a finite volume and experience forces of attraction for one another. Consequently, gas behavior is not necessarily described well by the ideal gas law. Under conditions of low pressure and high temperature, these factors are negligible, the ideal gas equation is an accurate description of gas behavior, and the gas is said to exhibit ideal behavior. However, at lower temperatures and higher pressures, corrections for molecular volume and molecular attractions are required to account for finite molecular size and attractive forces. The van der Waals equation is a modified version of the ideal gas law that can be used to account for the non-ideal behavior of gases under conditions.\r\n<h4>Key Equations<\/h4>\r\n<ul>\r\n \t<li>[latex]\\text{Z}=\\frac{\\text{molar}\\text{volume of gas at same}T\\text{and}P}{\\text{molar volume of ideal gas at same}T\\text{and}P}={\\left(\\frac{P\\times {V}_{m}}{R\\times T}\\right)}_{\\text{measured}}[\/latex]<\/li>\r\n \t<li>[latex]\\left(P+\\frac{{n}^{2}a}{{V}^{2}}\\right)\\times \\left(V-nb\\right)=nRT[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nGraphs showing the behavior of several different gases follow. Which of these gases exhibit behavior significantly different from that expected for ideal gases?\r\n\r\n<img class=\"aligncenter size-large wp-image-6022\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/218\/2016\/10\/26214946\/CNX_Chem_09_06_Exercise1-1024x499.jpg\" alt=\"This figure includes 6 graphs. The first, which is labeled, \u201cGas A,\u201d has a horizontal axis labeled, \u201cTemperature,\u201d and a vertical axis labeled, \u201cVolume.\u201d A straight blue line segment extends from the lower left to the upper right of this graph. The open area in the lower right portion of the graph contains the label, \u201cn, P constant.\u201d The second, which is labeled, \u201cGas B,\u201d has a horizontal axis labeled, \u201cP,\u201d and a vertical axis labeled, \u201cP V.\u201d A straight blue line segment extends horizontally across the center of this graph. The open area in the lower right portion of the graph contains the label, \u201cn, T constant.\u201d The third, which is labeled, \u201cGas C,\u201d has a horizontal axis labeled,\u201cP V divided by R T,\u201d and a vertical axis labeled, \u201cMoles.\u201d A blue curve begins about halfway up the vertical axis, dips slightly, then increases steadily to the upper right region of the graph. The fourth, which is labeled, \u201cGas D,\u201d has a horizontal axis labeled, \u201cP V divided by R T,\u201d and a vertical axis labeled, \u201cMoles.\u201d A straight blue line segment extends horizontally across the center of this graph. The open area in the lower right portion of the graph contains the label \u201cn, P constant.\u201d The fifth, which is labeled, \u201cGas E,\u201d has a horizontal axis labeled, \u201cTemperature,\u201d and a vertical axis labeled, \u201cVolume.\u201d A blue curve extends from the lower left to the upper right of this graph. The open area in the lower right portion of the graph contains the label \u201cn, P constant.\u201d The sixth graph, which is labeled, \u201cGas F,\u201d has a horizontal axis labeled, \u201cTemperature,\u201d and a vertical axis labeled, \u201cPressure.\u201d A blue curve begins toward the lower left region of the graph, increases at a rapid rate, then continues to increase at a relatively slow rate moving left to right across the graph. The open area in the lower right portion of the graph contains the label, \u201cn, V constant.\u201d\" width=\"1024\" height=\"499\" \/>\r\n\r\n[reveal-answer q=\"434062\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"434062\"]\r\n<ul>\r\n \t<li>Gas A: volume increases linearly as temperature increases with moles and pressure held constant, as expected by the ideal gas law <em>V<\/em> = (<em><strong>nR\/P<\/strong><\/em>)<em>T<\/em><\/li>\r\n \t<li>Gas B: <em>PV<\/em> stays constant as pressure increases with moles and temperature held constant, as expected by the ideal gas law <em>PV<\/em> = <strong>n<em>RT<\/em><\/strong><\/li>\r\n \t<li>Gas C: compressibility factor (Z) varies as <em>PV<\/em>\/<em>RT<\/em> increases, as expected of a real gas<\/li>\r\n \t<li>Gas D: compressibility factor (Z) stays constant as <em>PV<\/em>\/<em>RT<\/em> increases with moles and pressure held constant, as expected of an ideal gas<\/li>\r\n \t<li>Gas E: as temperature increases, volume increases, but not linearly with moles and pressure held constant, as would <strong>not<\/strong> be expected by the ideal gas law <em>V<\/em> = (<strong><em>nR<\/em>\/<em>P<\/em><\/strong>)<em>T,<\/em> as seen in Gas A<\/li>\r\n \t<li>Gas F: as temperature increases, pressure increases with moles and volume held constant, but not linearly, as would <strong>not<\/strong> be expected by the ideal gas law <em>P<\/em> = (<strong><em>nR<\/em>\/<em>V<\/em><\/strong>)<em>T,<\/em> as seen in Gas A<\/li>\r\n<\/ul>\r\nGases C, E, and F exhibit behavior significantly different from that expected for an ideal gas.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\n<ol>\r\n \t<li>Explain why the plot of <em>PV<\/em> for CO<sub>2<\/sub> differs from that of an ideal gas.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212118\/CNX_Chem_09_06_RealGases1.jpg\" width=\"399\" height=\"334\" \/><\/li>\r\n \t<li>Under which of the following sets of conditions does a real gas behave most like an ideal gas, and for which conditions is a real gas expected to deviate from ideal behavior? Explain.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>high pressure, small volume<\/li>\r\n \t<li>high temperature, low pressure<\/li>\r\n \t<li>low temperature, high pressure<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Describe the factors responsible for the deviation of the behavior of real gases from that of an ideal gas.<\/li>\r\n \t<li>For which of the following gases should the correction for the molecular volume be largest: CO, CO<sub>2<\/sub>, H<sub>2<\/sub>, He, NH<sub>3<\/sub>, SF<sub>6<\/sub>?<\/li>\r\n \t<li>A 0.245-L flask contains 0.467 mol CO<sub>2<\/sub> at 159 \u00b0C. Calculate the pressure:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>using the ideal gas law<\/li>\r\n \t<li>using the van der Waals equation<\/li>\r\n \t<li>Explain the reason for the difference.<\/li>\r\n \t<li>Identify which correction (that for P or V) is dominant and why.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Answer the following questions:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>If XX behaved as an ideal gas, what would its graph of Z vs. P look like?<\/li>\r\n \t<li>For most of this chapter, we performed calculations treating gases as ideal. Was this justified?<\/li>\r\n \t<li>What is the effect of the volume of gas molecules on Z? Under what conditions is this effect small? When is it large? Explain using an appropriate diagram.<\/li>\r\n \t<li>What is the effect of intermolecular attractions on the value of Z? Under what conditions is this effect small? When is it large? Explain using an appropriate diagram.<\/li>\r\n \t<li>In general, under what temperature conditions would you expect Z to have the largest deviations from the Z for an ideal gas?<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"339691\"]Selected Answers[\/reveal-answer]\r\n[hidden-answer a=\"339691\"]\r\n\r\n2.\u00a0The gas behavior most like an ideal gas will occur under the conditions in (b). Molecules have high speeds and move through greater distances between collision; they also have shorter contact times and interactions are less likely. Deviations occur with the conditions described in (a) and (c). Under conditions of (a), some gases may liquefy. Under conditions of (c), most gases will liquefy.\r\n\r\n4. We would expect the molecule with the largest volume to need the largest correction. SF<sub>6<\/sub> would need the largest correction.\r\n\r\n6. The answers are as follows:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>A straight horizontal line at 1.0;<\/li>\r\n \t<li>When real gases are at low pressures and high temperatures they behave close enough to ideal gases that they are approximated as such, however, in some cases, we see that at a high pressure and temperature, the ideal gas approximation breaks down and is significantly different from the pressure calculated by the van der Waals equation<\/li>\r\n \t<li>The greater the compressibility, the more the volume matters. At low pressures, the correction factor for intermolecular attractions is more significant, and the effect of the volume of the gas molecules on Z would be a small lowering compressibility. At higher pressures, the effect of the volume of the gas molecules themselves on Z would increase compressibility (see Figure\u00a01)<\/li>\r\n \t<li>Once again, at low pressures, the effect of intermolecular attractions on Z would be more important than the correction factor for the volume of the gas molecules themselves, though perhaps still small. At higher pressures and low temperatures, the effect of intermolecular attractions would be larger. See Figure\u00a01.<\/li>\r\n \t<li>low temperatures<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<strong>compressibility factor (Z):\u00a0<\/strong>ratio of the experimentally measured molar volume for a gas to its molar volume as computed from the ideal gas equation\r\n\r\n<strong>van der Waals equation:\u00a0<\/strong>modified version of the ideal gas equation containing additional terms to account for non-ideal gas behavior","rendered":"<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>Describe the physical factors that lead to deviations from ideal gas behavior<\/li>\n<li>Explain how these factors are represented in the van der Waals equation<\/li>\n<li>Define compressibility (Z) and describe how its variation with pressure reflects non-ideal behavior<\/li>\n<li>Quantify non-ideal behavior by comparing computations of gas properties using the ideal gas law and the van der Waals equation<\/li>\n<\/ul>\n<\/div>\n<p>Thus far, the ideal gas law, <em>PV = nRT<\/em>, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and molecular formula problems to determining the density and molar mass of a gas. As mentioned in the previous modules of this chapter, however, the behavior of a gas is often non-ideal, meaning that the observed relationships between its pressure, volume, and temperature are not accurately described by the gas laws. In this section, the reasons for these deviations from ideal gas behavior are considered.<\/p>\n<p>One way in which the accuracy of <em>PV = nRT<\/em> can be judged is by comparing the actual volume of 1 mole of gas (its molar volume, <em>V<\/em><sub>m<\/sub>) to the molar volume of an ideal gas at the same temperature and pressure. This ratio is called the <strong>compressibility factor<\/strong>, <strong>Z<\/strong>, with:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\text{Z}=\\frac{\\text{molar volume of gas at same }T\\text{ and }P}{\\text{molar volume of ideal gas at same }T\\text{ and }P}={\\left(\\frac{P{V}_{m}}{RT}\\right)}_{\\text{measured}}[\/latex]<\/p>\n<p>Ideal gas behavior is therefore indicated when this ratio is equal to 1, and any deviation from 1 is an indication of non-ideal behavior. Figure\u00a01 shows plots of Z over a large pressure range for several common gases.<\/p>\n<figure>\n<div style=\"width: 707px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212111\/CNX_Chem_09_06_ZvsPgraph1.jpg\" width=\"697\" height=\"463\" alt=\"image\" \/><\/p>\n<p class=\"wp-caption-text\">Figure\u00a01. A graph of the compressibility factor (Z) vs. pressure shows that gases can exhibit significant deviations from the behavior predicted by the ideal gas law.<\/p>\n<\/div>\n<\/figure>\n<p>As is apparent from Figure\u00a01, the ideal gas law does not describe gas behavior well at relatively high pressures. To determine why this is, consider the differences between real gas properties and what is expected of a hypothetical ideal gas.<\/p>\n<p>Particles of a hypothetical ideal gas have no significant volume and do not attract or repel each other. In general, real gases approximate this behavior at relatively low pressures and high temperatures. However, at high pressures, the molecules of a gas are crowded closer together, and the amount of empty space between the molecules is reduced. At these higher pressures, the volume of the gas molecules themselves becomes appreciable relative to the total volume occupied by the gas (Figure\u00a02). The gas therefore becomes less compressible at these high pressures, and although its volume continues to decrease with increasing pressure, this decrease is not <em>proportional<\/em> as predicted by Boyle\u2019s law.<\/p>\n<figure>\n<div style=\"width: 1310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212112\/CNX_Chem_09_06_RealGas31.jpg\" alt=\"This figure includes three diagrams. In a, a cylinder with 9 purple spheres with trails indicating motion are shown. Above the cylinder, the label, \u201cParticles ideal gas,\u201d is connected to two of the spheres with line segments extending into the square. The label \u201cAssumes\u201d is beneath the square. In b, a cylinder and piston is shown. A relatively small open space is shaded lavender with 9 purple spheres packed close together. No motion trails are present on the spheres. Above the piston, a downward arrow labeled \u201cPressure\u201d is directed toward the enclosed area. In c, the cylinder is exactly the same as the first, but the number of molecules has doubled.\" width=\"1300\" height=\"528\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. Raising the pressure of a gas increases the fraction of its volume that is occupied by the gas molecules and makes the gas less compressible.<\/p>\n<\/div>\n<\/figure>\n<p>At relatively low pressures, gas molecules have practically no attraction for one another because they are (on average) so far apart, and they behave almost like particles of an ideal gas. At higher pressures, however, the force of attraction is also no longer insignificant. This force pulls the molecules a little closer together, slightly decreasing the pressure (if the volume is constant) or decreasing the volume (at constant pressure) (Figure\u00a03). This change is more pronounced at low temperatures because the molecules have lower KE relative to the attractive forces, and so they are less effective in overcoming these attractions after colliding with one another.<\/p>\n<figure>\n<div style=\"width: 1310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212113\/CNX_Chem_09_06_RealGas21.jpg\" alt=\"This figure includes two diagrams. Each involves two lavender shaded boxes that contain 14 relatively evenly distributed, purple spheres. In the first box in a, a nearly centrally located purple sphere has 6 double-headed arrows extending outward from it to nearby spheres. A single purple arrow is pointing right into open space. This box is labeled, \u201creal.\u201d There is a second box that looks slightly larger than the first box in a. It has the same number of particles but no arrows. This box is labeled, \u201cideal.\u201d In b, the first box has a purple sphere at the right side which has 4 double-headed arrows radiating out to the top, bottom, and left to other spheres. A single purple arrow points right through open space to the edge of the box. This box has no spheres positioned near its right edge This box is labeled, \u201creal.\u201d The second box is the same size as the first box and contains the same number of particles. There are no arrows in it, except for the purple arrow which appears to be bigger and bolder. This box is labeled, \u201cideal.\u201d\" width=\"1300\" height=\"483\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. (a) Attractions between gas molecules serve to decrease the gas volume at constant pressure compared to an ideal gas whose molecules experience no attractive forces. (b) These attractive forces will decrease the force of collisions between the molecules and container walls, therefore reducing the pressure exerted compared to an ideal gas.<\/p>\n<\/div>\n<\/figure>\n<p>There are several different equations that better approximate gas behavior than does the ideal gas law. The first, and simplest, of these was developed by the Dutch scientist Johannes van der Waals in 1879. The <strong>van der Waals equation<\/strong> improves upon the ideal gas law by adding two terms: one to account for the volume of the gas molecules and another for the attractive forces between them. <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212115\/CNX_Chem_09_06_vanderWaals_img1.jpg\" alt=\"image\" \/><\/p>\n<p>The constant <em>a<\/em> corresponds to the strength of the attraction between molecules of a particular gas, and the constant <em>b<\/em> corresponds to the size of the molecules of a particular gas. The \u201ccorrection\u201d to the pressure term in the ideal gas law is [latex]\\frac{{n}^{2}a}{{V}^{2}},[\/latex] and the \u201ccorrection\u201d to the volume is <em>nb<\/em>. Note that when <em>V<\/em> is relatively large and <em>n<\/em> is relatively small, both of these correction terms become negligible, and the van der Waals equation reduces to the ideal gas law, <em>PV = nRT<\/em>. Such a condition corresponds to a gas in which a relatively low number of molecules is occupying a relatively large volume, that is, a gas at a relatively low pressure. Experimental values for the van der Waals constants of some common gases are given in Table\u00a01.<\/p>\n<table id=\"fs-idm15100464\" class=\"span-all\" summary=\"This table has three columns and seven rows. The first row is a header, and it labels each column, \u201cGas,\u201d \u201ca ( L to the second power a t m divided by m o l to the second power ),\u201d \u201cb ( L divided by m o l ).\u201d Under \u201cGas\u201d are the following: N subscript 2, O subscript 2, C O subscript 2, H subscript 2 O, H e, and C C l subscript 4. Under \u201ca ( L to the second power a t m divided by m o l to the second power )\u201d are the following: 1.39, 1.36, 3.59, 5.46, 0.0342, and 20.4. Under \u201cb ( L divided by m o l )\u201d are the following: 0.0391, 0.0318, 0.0427, 0.0305, 0.0237, and 0.1383.\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" scope=\"col\" data-align=\"center\">Table 1. Values of van der Waals Constants for Some Common Gases<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th scope=\"col\" data-align=\"left\">Gas<\/th>\n<th scope=\"col\" data-align=\"left\"><em data-effect=\"italics\">a<\/em> (L<sup>2<\/sup> atm\/mol<sup>2<\/sup>)<\/th>\n<th scope=\"col\" data-align=\"left\"><em data-effect=\"italics\">b<\/em> (L\/mol)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">N<sub>2<\/sub><\/td>\n<td data-align=\"left\">1.39<\/td>\n<td data-align=\"left\">0.0391<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">O<sub>2<\/sub><\/td>\n<td data-align=\"left\">1.36<\/td>\n<td data-align=\"left\">0.0318<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">CO<sub>2<\/sub><\/td>\n<td data-align=\"left\">3.59<\/td>\n<td data-align=\"left\">0.0427<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">H<sub>2<\/sub>O<\/td>\n<td data-align=\"left\">5.46<\/td>\n<td data-align=\"left\">0.0305<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">He<\/td>\n<td data-align=\"left\">0.0342<\/td>\n<td data-align=\"left\">0.0237<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">CCl<sub>4<\/sub><\/td>\n<td data-align=\"left\">20.4<\/td>\n<td data-align=\"left\">0.1383<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>At low pressures, the correction for intermolecular attraction, <em>a<\/em>, is more important than the one for molecular volume, <em>b<\/em>. At high pressures and small volumes, the correction for the volume of the molecules becomes important, because the molecules themselves are incompressible and constitute an appreciable fraction of the total volume. At some intermediate pressure, the two corrections have opposing influences and the gas appears to follow the relationship given by <em>PV = nRT<\/em> over a small range of pressures. This behavior is reflected by the \u201cdips\u201d in several of the compressibility curves shown in Figure\u00a09.35. The attractive force between molecules initially makes the gas more compressible than an ideal gas, as pressure is raised (Z decreases with increasing <em>P<\/em>). At very high pressures, the gas becomes less compressible (Z increases with <em>P<\/em>), as the gas molecules begin to occupy an increasingly significant fraction of the total gas volume.<\/p>\n<p>Strictly speaking, the ideal gas equation functions well when intermolecular attractions between gas molecules are negligible and the gas molecules themselves do not occupy an appreciable part of the whole volume. These criteria are satisfied under conditions of <em>low pressure and high temperature<\/em>. Under such conditions, the gas is said to behave ideally, and deviations from the gas laws are small enough that they may be disregarded\u2014this is, however, very often not the case.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1:\u00a0Comparison of Ideal Gas Law and van der Waals Equation<\/h3>\n<p>A 4.25-L flask contains 3.46 mol CO<sub>2<\/sub> at 229 \u00b0C. Calculate the pressure of this sample of CO<sub>2<\/sub>:<\/p>\n<ol>\n<li>from the ideal gas law<\/li>\n<li>from the van der Waals equation<\/li>\n<li>Explain the reason(s) for the difference.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q245157\">Show Answer<\/span><\/p>\n<div id=\"q245157\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>From the ideal gas law:<br \/>\n[latex]P=\\frac{nRT}{V}=\\frac{3.46\\cancel{\\text{mol}}\\times 0.08206\\cancel{\\text{L}}\\text{atm}\\cancel{{\\text{mol}}^{-\\text{1}}}\\cancel{{\\text{K}}^{-\\text{1}}}\\times 502\\cancel{\\text{K}}}{4.25\\cancel{\\text{L}}}=33.5\\text{atm}[\/latex]<\/li>\n<li>From the van der Waals equation:<br \/>\n[latex]\\left(P+\\frac{{n}^{2}a}{{V}^{2}}\\right)\\times \\left(V-nb\\right)=nRT\\rightarrow P=\\frac{nRT}{\\left(V-nb\\right)}-\\frac{{n}^{2}a}{{V}^{2}}[\/latex]<br \/>\n[latex]P=\\frac{3.46\\text{mol}\\times 0.08206\\text{L}\\text{atm}{\\text{mol}}^{-\\text{1}}{\\text{K}}^{-\\text{1}}\\times \\text{502 K}}{\\left(4.25\\text{L}-3.46\\text{mol}\\times 0.0427\\text{L}{\\text{mol}}^{-\\text{1}}\\right)}-\\frac{{\\left(3.46\\text{mol}\\right)}^{2}\\times 3.59{\\text{L}}^{2}\\text{atm}{\\text{mol}}^{2}}{{\\left(4.25\\text{L}\\right)}^{2}}[\/latex]<br \/>\nThis finally yields <em>P<\/em> = 32.4 atm.<\/li>\n<li>This is not very different from the value from the ideal gas law, because the pressure is not very high and the temperature is not very low. The value is somewhat different because CO<sub>2<\/sub> molecules do have some volume and attractions between molecules, and the ideal gas law assumes they do not have volume or attractions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h4><strong>Check your Learning<\/strong><\/h4>\n<p>A 560-mL flask contains 21.3 g N<sub>2<\/sub> at 145 \u00b0C. Calculate the pressure of N<sub>2<\/sub>:<\/p>\n<ol>\n<li>from the ideal gas law<\/li>\n<li>from the van der Waals equation<\/li>\n<li>Explain the reason(s) for the difference.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q451203\">Show Answer<\/span><\/p>\n<div id=\"q451203\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>46.562 atm<\/li>\n<li>46.594 atm<\/li>\n<li>The van der Waals equation takes into account the volume of the gas molecules themselves as well as intermolecular attractions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts and Summary<\/h3>\n<p>Gas molecules possess a finite volume and experience forces of attraction for one another. Consequently, gas behavior is not necessarily described well by the ideal gas law. Under conditions of low pressure and high temperature, these factors are negligible, the ideal gas equation is an accurate description of gas behavior, and the gas is said to exhibit ideal behavior. However, at lower temperatures and higher pressures, corrections for molecular volume and molecular attractions are required to account for finite molecular size and attractive forces. The van der Waals equation is a modified version of the ideal gas law that can be used to account for the non-ideal behavior of gases under conditions.<\/p>\n<h4>Key Equations<\/h4>\n<ul>\n<li>[latex]\\text{Z}=\\frac{\\text{molar}\\text{volume of gas at same}T\\text{and}P}{\\text{molar volume of ideal gas at same}T\\text{and}P}={\\left(\\frac{P\\times {V}_{m}}{R\\times T}\\right)}_{\\text{measured}}[\/latex]<\/li>\n<li>[latex]\\left(P+\\frac{{n}^{2}a}{{V}^{2}}\\right)\\times \\left(V-nb\\right)=nRT[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Graphs showing the behavior of several different gases follow. Which of these gases exhibit behavior significantly different from that expected for ideal gases?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6022\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/218\/2016\/10\/26214946\/CNX_Chem_09_06_Exercise1-1024x499.jpg\" alt=\"This figure includes 6 graphs. The first, which is labeled, \u201cGas A,\u201d has a horizontal axis labeled, \u201cTemperature,\u201d and a vertical axis labeled, \u201cVolume.\u201d A straight blue line segment extends from the lower left to the upper right of this graph. The open area in the lower right portion of the graph contains the label, \u201cn, P constant.\u201d The second, which is labeled, \u201cGas B,\u201d has a horizontal axis labeled, \u201cP,\u201d and a vertical axis labeled, \u201cP V.\u201d A straight blue line segment extends horizontally across the center of this graph. The open area in the lower right portion of the graph contains the label, \u201cn, T constant.\u201d The third, which is labeled, \u201cGas C,\u201d has a horizontal axis labeled,\u201cP V divided by R T,\u201d and a vertical axis labeled, \u201cMoles.\u201d A blue curve begins about halfway up the vertical axis, dips slightly, then increases steadily to the upper right region of the graph. The fourth, which is labeled, \u201cGas D,\u201d has a horizontal axis labeled, \u201cP V divided by R T,\u201d and a vertical axis labeled, \u201cMoles.\u201d A straight blue line segment extends horizontally across the center of this graph. The open area in the lower right portion of the graph contains the label \u201cn, P constant.\u201d The fifth, which is labeled, \u201cGas E,\u201d has a horizontal axis labeled, \u201cTemperature,\u201d and a vertical axis labeled, \u201cVolume.\u201d A blue curve extends from the lower left to the upper right of this graph. The open area in the lower right portion of the graph contains the label \u201cn, P constant.\u201d The sixth graph, which is labeled, \u201cGas F,\u201d has a horizontal axis labeled, \u201cTemperature,\u201d and a vertical axis labeled, \u201cPressure.\u201d A blue curve begins toward the lower left region of the graph, increases at a rapid rate, then continues to increase at a relatively slow rate moving left to right across the graph. The open area in the lower right portion of the graph contains the label, \u201cn, V constant.\u201d\" width=\"1024\" height=\"499\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q434062\">Show Answer<\/span><\/p>\n<div id=\"q434062\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Gas A: volume increases linearly as temperature increases with moles and pressure held constant, as expected by the ideal gas law <em>V<\/em> = (<em><strong>nR\/P<\/strong><\/em>)<em>T<\/em><\/li>\n<li>Gas B: <em>PV<\/em> stays constant as pressure increases with moles and temperature held constant, as expected by the ideal gas law <em>PV<\/em> = <strong>n<em>RT<\/em><\/strong><\/li>\n<li>Gas C: compressibility factor (Z) varies as <em>PV<\/em>\/<em>RT<\/em> increases, as expected of a real gas<\/li>\n<li>Gas D: compressibility factor (Z) stays constant as <em>PV<\/em>\/<em>RT<\/em> increases with moles and pressure held constant, as expected of an ideal gas<\/li>\n<li>Gas E: as temperature increases, volume increases, but not linearly with moles and pressure held constant, as would <strong>not<\/strong> be expected by the ideal gas law <em>V<\/em> = (<strong><em>nR<\/em>\/<em>P<\/em><\/strong>)<em>T,<\/em> as seen in Gas A<\/li>\n<li>Gas F: as temperature increases, pressure increases with moles and volume held constant, but not linearly, as would <strong>not<\/strong> be expected by the ideal gas law <em>P<\/em> = (<strong><em>nR<\/em>\/<em>V<\/em><\/strong>)<em>T,<\/em> as seen in Gas A<\/li>\n<\/ul>\n<p>Gases C, E, and F exhibit behavior significantly different from that expected for an ideal gas.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<ol>\n<li>Explain why the plot of <em>PV<\/em> for CO<sub>2<\/sub> differs from that of an ideal gas.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212118\/CNX_Chem_09_06_RealGases1.jpg\" width=\"399\" height=\"334\" alt=\"image\" \/><\/li>\n<li>Under which of the following sets of conditions does a real gas behave most like an ideal gas, and for which conditions is a real gas expected to deviate from ideal behavior? Explain.\n<ol style=\"list-style-type: lower-alpha;\">\n<li>high pressure, small volume<\/li>\n<li>high temperature, low pressure<\/li>\n<li>low temperature, high pressure<\/li>\n<\/ol>\n<\/li>\n<li>Describe the factors responsible for the deviation of the behavior of real gases from that of an ideal gas.<\/li>\n<li>For which of the following gases should the correction for the molecular volume be largest: CO, CO<sub>2<\/sub>, H<sub>2<\/sub>, He, NH<sub>3<\/sub>, SF<sub>6<\/sub>?<\/li>\n<li>A 0.245-L flask contains 0.467 mol CO<sub>2<\/sub> at 159 \u00b0C. Calculate the pressure:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>using the ideal gas law<\/li>\n<li>using the van der Waals equation<\/li>\n<li>Explain the reason for the difference.<\/li>\n<li>Identify which correction (that for P or V) is dominant and why.<\/li>\n<\/ol>\n<\/li>\n<li>Answer the following questions:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>If XX behaved as an ideal gas, what would its graph of Z vs. P look like?<\/li>\n<li>For most of this chapter, we performed calculations treating gases as ideal. Was this justified?<\/li>\n<li>What is the effect of the volume of gas molecules on Z? Under what conditions is this effect small? When is it large? Explain using an appropriate diagram.<\/li>\n<li>What is the effect of intermolecular attractions on the value of Z? Under what conditions is this effect small? When is it large? Explain using an appropriate diagram.<\/li>\n<li>In general, under what temperature conditions would you expect Z to have the largest deviations from the Z for an ideal gas?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q339691\">Selected Answers<\/span><\/p>\n<div id=\"q339691\" class=\"hidden-answer\" style=\"display: none\">\n<p>2.\u00a0The gas behavior most like an ideal gas will occur under the conditions in (b). Molecules have high speeds and move through greater distances between collision; they also have shorter contact times and interactions are less likely. Deviations occur with the conditions described in (a) and (c). Under conditions of (a), some gases may liquefy. Under conditions of (c), most gases will liquefy.<\/p>\n<p>4. We would expect the molecule with the largest volume to need the largest correction. SF<sub>6<\/sub> would need the largest correction.<\/p>\n<p>6. The answers are as follows:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>A straight horizontal line at 1.0;<\/li>\n<li>When real gases are at low pressures and high temperatures they behave close enough to ideal gases that they are approximated as such, however, in some cases, we see that at a high pressure and temperature, the ideal gas approximation breaks down and is significantly different from the pressure calculated by the van der Waals equation<\/li>\n<li>The greater the compressibility, the more the volume matters. At low pressures, the correction factor for intermolecular attractions is more significant, and the effect of the volume of the gas molecules on Z would be a small lowering compressibility. At higher pressures, the effect of the volume of the gas molecules themselves on Z would increase compressibility (see Figure\u00a01)<\/li>\n<li>Once again, at low pressures, the effect of intermolecular attractions on Z would be more important than the correction factor for the volume of the gas molecules themselves, though perhaps still small. At higher pressures and low temperatures, the effect of intermolecular attractions would be larger. See Figure\u00a01.<\/li>\n<li>low temperatures<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Glossary<\/h2>\n<p><strong>compressibility factor (Z):\u00a0<\/strong>ratio of the experimentally measured molar volume for a gas to its molar volume as computed from the ideal gas equation<\/p>\n<p><strong>van der Waals equation:\u00a0<\/strong>modified version of the ideal gas equation containing additional terms to account for non-ideal gas behavior<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2120\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Chemistry. <strong>Provided by<\/strong>: OpenStax College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/openstaxcollege.org\">http:\/\/openstaxcollege.org<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Chemistry\",\"author\":\"\",\"organization\":\"OpenStax College\",\"url\":\"http:\/\/openstaxcollege.org\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2120","chapter","type-chapter","status-publish","hentry"],"part":3001,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/pressbooks\/v2\/chapters\/2120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/wp\/v2\/users\/17"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/pressbooks\/v2\/chapters\/2120\/revisions"}],"predecessor-version":[{"id":6023,"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/pressbooks\/v2\/chapters\/2120\/revisions\/6023"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/pressbooks\/v2\/parts\/3001"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/pressbooks\/v2\/chapters\/2120\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/wp\/v2\/media?parent=2120"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/pressbooks\/v2\/chapter-type?post=2120"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/wp\/v2\/contributor?post=2120"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-binghamton-chemistry\/wp-json\/wp\/v2\/license?post=2120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}