{"id":2202,"date":"2015-04-22T21:01:33","date_gmt":"2015-04-22T21:01:33","guid":{"rendered":"https:\/\/courses.candelalearning.com\/oschemtemp\/?post_type=chapter&#038;p=2202"},"modified":"2016-10-27T16:01:18","modified_gmt":"2016-10-27T16:01:18","slug":"rate-laws","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/chapter\/rate-laws\/","title":{"raw":"Rate Laws","rendered":"Rate Laws"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this module, you will be able to:\r\n<ul>\r\n \t<li>Explain the form and function of a rate law<\/li>\r\n \t<li>Use rate laws to calculate reaction rates<\/li>\r\n \t<li>Use rate and concentration data to identify reaction orders and derive rate laws<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs described in the previous module, the rate of a reaction is affected by the concentrations of reactants. Rate laws or rate equations are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants. In general, a rate law (or differential rate law, as it is sometimes called) takes this form:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[A\\right]}^{m}{\\left[B\\right]}^{n}{\\left[C\\right]}^{p}\\dots [\/latex]<\/p>\r\nin which [<em>A<\/em>], [<em>B<\/em>], and [<em>C<\/em>] represent the molar concentrations of reactants, and <em>k<\/em> is the rate constant, which is specific for a particular reaction at a particular temperature. The exponents <em>m<\/em>, <em>n<\/em>, and <em>p<\/em> are usually positive integers (although it is possible for them to be fractions or negative numbers). The rate constant <em>k<\/em> and the exponents <em>m<\/em>, <em>n<\/em>, and <em>p<\/em> must be determined experimentally by observing how the rate of a reaction changes as the concentrations of the reactants are changed. The rate constant <em>k<\/em> is independent of the concentration of <em>A<\/em>, <em>B<\/em>, or <em>C<\/em>, but it does vary with temperature and surface area.\r\n\r\nThe exponents in a rate law describe the effects of the reactant concentrations on the reaction rate and define the reaction order. Consider a reaction for which the rate law is:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[A\\right]}^{m}{\\left[B\\right]}^{n}[\/latex]<\/p>\r\nIf the exponent <em>m<\/em> is 1, the reaction is first order with respect to <em>A<\/em>. If <em>m<\/em> is 2, the reaction is second order with respect to <em>A<\/em>. If <em>n<\/em> is 1, the reaction is first order in <em>B<\/em>. If <em>n<\/em> is 2, the reaction is second order in <em>B<\/em>. If <em>m<\/em> or <em>n<\/em> is zero, the reaction is zero order in <em>A<\/em> or <em>B<\/em>, respectively, and the rate of the reaction is not affected by the concentration of that reactant. The overall reaction order is the sum of the orders with respect to each reactant. If <em>m<\/em> = 1 and <em>n<\/em> = 1, the overall order of the reaction is second order (<em>m<\/em> + <em>n<\/em> = 1 + 1 = 2).\r\n\r\nThe rate law:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k\\left[{\\text{H}}_{2}{\\text{O}}_{2}\\right][\/latex]<\/p>\r\ndescribes a reaction that is first order in hydrogen peroxide and first order overall. The rate law:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[{\\text{C}}_{4}{\\text{H}}_{6}\\right]}^{2}[\/latex]<\/p>\r\ndescribes a reaction that is second order in C<sub>4<\/sub>H<sub>6<\/sub> and second order overall. The rate law:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k\\left[{\\text{H}}^{+}\\right]\\left[{\\text{OH}}^{-}\\right][\/latex]<\/p>\r\ndescribes a reaction that is first order in H<sup>+<\/sup>, first order in OH<sup>-<\/sup>, and second order overall.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1:\u00a0Writing Rate Laws from Reaction Orders<\/h3>\r\nAn experiment shows that the reaction of nitrogen dioxide with carbon monoxide:\r\n<p style=\"text-align: center;\">[latex]{\\text{NO}}_{2}\\text{(}g\\text{)}+\\text{CO(}g\\text{)}\\rightarrow\\text{NO(}g\\text{)}+{\\text{CO}}_{2}\\left(g\\right)[\/latex]<\/p>\r\nis second order in NO<sub>2<\/sub> and zero order in CO at 100 \u00b0C. What is the rate law for the reaction?\r\n\r\n[reveal-answer q=\"341650\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"341650\"]\r\n\r\nThe reaction will have the form:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[{\\text{NO}}_{2}\\right]}^{m}{\\left[\\text{CO}\\right]}^{n}[\/latex]<\/p>\r\nThe reaction is second order in NO<sub>2<\/sub>; thus <em>m<\/em> = 2. The reaction is zero order in CO; thus <em>n<\/em> = 0. The rate law is:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[{\\text{NO}}_{2}\\right]}^{2}{\\left[\\text{CO}\\right]}^{0}=k{\\left[{\\text{NO}}_{2}\\right]}^{2}[\/latex]<\/p>\r\nRemember that a number raised to the zero power is equal to 1, thus [CO]<sup>0<\/sup> = 1, which is why we can simply drop the concentration of CO from the rate equation: the rate of reaction is solely dependent on the concentration of NO<sub>2<\/sub>. When we consider rate mechanisms later in this chapter, we will explain how a reactant\u2019s concentration can have no effect on a reaction despite being involved in the reaction.\r\n\r\n[\/hidden-answer]\r\n<h4>Check Your Learning<\/h4>\r\nThe rate law for the reaction:\r\n<p style=\"text-align: center;\">[latex]{\\text{H}}_{2}\\text{(}g\\text{)}+2\\text{NO(}g\\text{)}\\rightarrow{\\text{N}}_{2}\\text{O(}g\\text{)}+{\\text{H}}_{2}\\text{O(}g\\text{)}[\/latex]<\/p>\r\nhas been determined to be rate = <em>k<\/em>[NO]<sup>2<\/sup>[H<sub>2<\/sub>]. What are the orders with respect to each reactant, and what is the overall order of the reaction?\r\n[reveal-answer q=\"126680\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"126680\"]order in NO = 2; order in H<sub>2<\/sub> = 1; overall order = 3[\/hidden-answer]\r\n<h4>Check Your Learning<\/h4>\r\nIn a transesterification reaction, a triglyceride reacts with an alcohol to form an ester and glycerol. Many students learn about the reaction between methanol (CH<sub>3<\/sub>OH) and ethyl acetate (CH<sub>3<\/sub>CH<sub>2<\/sub>OCOCH<sub>3<\/sub>) as a sample reaction before studying the chemical reactions that produce biodiesel:\r\n\r\n[latex]{\\text{CH}}_{3}\\text{OH}+{\\text{CH}}_{3}{\\text{CH}}_{2}{\\text{OCOCH}}_{3}\\rightarrow{\\text{CH}}_{3}{\\text{OCOCH}}_{3}+{\\text{CH}}_{3}{\\text{CH}}_{2}\\text{OH}[\/latex]\r\n\r\nThe rate law for the reaction between methanol and ethyl acetate is, under certain conditions, determined to be:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k\\left[{\\text{CH}}_{3}\\text{OH}\\right][\/latex]<\/p>\r\nWhat is the order of reaction with respect to methanol and ethyl acetate, and what is the overall order of reaction?\r\n[reveal-answer q=\"36335\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"36335\"]order in CH<sub>3<\/sub>OH = 1; order in CH<sub>3<\/sub>CH<sub>2<\/sub>OCOCH<sub>3<\/sub> = 0; overall order = 1[\/hidden-answer]\r\n\r\n<\/div>\r\nIt is sometimes helpful to use a more explicit algebraic method, often referred to as the method of initial rates, to determine the orders in rate laws. To use this method, we select two sets of rate data that differ in the concentration of only one reactant and set up a ratio of the two rates and the two rate laws. After canceling terms that are equal, we are left with an equation that contains only one unknown, the coefficient of the concentration that varies. We then solve this equation for the coefficient.\r\n<div class=\"textbox examples\">\r\n<h3>Example 2:\u00a0Determining a Rate Law from Initial Rates<\/h3>\r\nOzone in the upper atmosphere is depleted when it reacts with nitrogen oxides. The rates of the reactions of nitrogen oxides with ozone are important factors in deciding how significant these reactions are in the formation of the ozone hole over Antarctica.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"352\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23212248\/CNX_Chem_12_03_OzoneHole1.jpg\" alt=\"A view of Earth\u2019s southern hemisphere is shown. A nearly circular region of approximately half the diameter of the image is shown in shades of purple, with Antarctica appearing in a slightly lighter color than the surrounding ocean areas. Immediately outside this region is a narrow bright blue zone followed by a bright green zone. In the top half of the figure, the purple region extends slightly outward from the circle and the blue zone extends more outward to the right of the center as compared to the lower half of the image. In the upper half of the image, the majority of the space outside the purple region is shaded green, with a few small strips of interspersed blue regions. The lower half however shows the majority of the space outside the central purple zone in yellow, orange, and red. The red zones appear in the lower central and left regions outside the purple zone. To the lower right of this image is a color scale that is labeled \u201cTotal Ozone (Dobsone units).\u201d This scale begins at 0 and increases by 100\u2019s up to 700. At the left end of the scale, the value 0 shows a very deep purple color, 100 is indigo, 200 is blue, 300 is green, 400 is a yellow-orange, 500 is red, 600 is pink, and 700 is white.\" width=\"352\" height=\"360\" data-media-type=\"image\/jpeg\" \/> Figure\u00a01. Over the past several years, the atmospheric ozone concentration over Antarctica has decreased during the winter. This map shows the decreased concentration as a purple area. (credit: NASA)[\/caption]\r\n\r\nOne such reaction is the combination of nitric oxide, NO, with ozone, O<sub>3<\/sub>:\r\n<p style=\"text-align: center;\">[latex]\\text{NO(}g\\text{)}+{\\text{O}}_{3}\\left(g\\right)\\rightarrow{\\text{NO}}_{2}\\text{(}g\\text{)}+{\\text{O}}_{2}\\left(g\\right)[\/latex]<\/p>\r\nThis reaction has been studied in the laboratory, and the following rate data were determined at 25 \u00b0C.\r\n<table id=\"fs-idm205685856\" class=\"medium unnumbered\" summary=\"This table has four columns and six rows. The first row is a header row, and it labels each column: \u201cTrial,\u201d \u201c[ N O ] ( mol \/ L),\u201d \u201c[ O subscript 3 ] ( mol \/ L ),\u201d and \u201ccapital delta [ N O subscript 2 ] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 ).\u201d Under the \u201cTrial\u201d column are the numbers: 1, 2, 3, 4, and 5. Under the \u201c[ N O ] ( mol \/ L)\u201d column are the numbers: 1.00 times ten to the negative six power; 1.00 times ten to the negative six power; 1.00 times ten to the negative six power; 2.00 times ten to the negative six power; and 3.00 times ten to the negative six power. Under the \u201c[ O subscript 3 ] ( mol \/ L )\u201d column are the numbers: 3.00 times ten to the negative six; 6.00 times ten to the negative six; 9.00 times ten to the negative six; 9.00 times ten to the negative six; and 9.00 times ten to the negative six. Under the column \u201ccapital delta [ N O subscript 2 ] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 )\u201d are the numbers: 6.60 times ten to the negative five; 1.32 times ten to the negative four; 1.98 times ten to the negative four; 3.96 times ten to the negative four; and 5.94 times ten to the negative four.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Trial<\/th>\r\n<th>[NO] (mol\/L)<\/th>\r\n<th>[O<sub>3<\/sub>] (mol\/L)<\/th>\r\n<th>[latex]\\frac{{\\text{Delta [NO}}_{\\text{2}}\\text{]}}{\\Delta t}\\left(\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>1<\/td>\r\n<td>1.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>3.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>6.60 \u00d7 10<sup>\u22125<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2<\/td>\r\n<td>1.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>6.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>1.32 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3<\/td>\r\n<td>1.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>9.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>1.98 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>4<\/td>\r\n<td>2.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>9.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>3.96 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>5<\/td>\r\n<td>3.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>9.00 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>5.94 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDetermine the rate law and the rate constant for the reaction at 25 \u00b0C.\r\n\r\n[reveal-answer q=\"646205\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"646205\"]\r\n\r\nThe rate law will have the form:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{m}{\\left[{\\text{O}}_{3}\\right]}^{n}[\/latex]<\/p>\r\nWe can determine the values of <em>m<\/em>, <em>n<\/em>, and <em>k<\/em> from the experimental data using the following three-part process:\r\n<ol>\r\n \t<li>Determine the value of <em>m<\/em> from the data in which [NO] varies and [O<sub>3<\/sub>] is constant:\u00a0 In the last three experiments, [NO] varies while [O<sub>3<\/sub>] remains constant. When [NO] doubles from trial 3 to 4, the rate doubles, and when [NO] triples from trial 3 to 5, the rate also triples. Thus, the rate is also directly proportional to [NO], and <em>m<\/em> in the rate law is equal to 1.<\/li>\r\n \t<li>Determine the value of <em>n<\/em> from data in which [O<sub>3<\/sub>] varies and [NO] is constant: In the first three experiments, [NO] is constant and [O<sub>3<\/sub>] varies. The reaction rate changes in direct proportion to the change in [O<sub>3<\/sub>]. When [O<sub>3<\/sub>] doubles from trial 1 to 2, the rate doubles; when [O<sub>3<\/sub>] triples from trial 1 to 3, the rate increases also triples. Thus, the rate is directly proportional to [O<sub>3<\/sub>], and <em>n<\/em> is equal to 1.The rate law is thus:[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{1}{\\left[{\\text{O}}_{3}\\right]}^{1}=k\\left[\\text{NO}\\right]\\left[{\\text{O}}_{3}\\right][\/latex]<\/li>\r\n \t<li>Determine the value of <em>k<\/em> from one set of concentrations and the corresponding rate:[latex]\\begin{array}{cc}\\hfill k&amp; =\\frac{\\text{rate}}{\\left[\\text{NO}\\right]\\left[{\\text{O}}_{3}\\right]}\\hfill \\\\ &amp; =\\frac{0.660\\times {10}^{-4}\\cancel{{\\text{mol L}}^{-1}}{\\text{s}}^{-1}}{\\left(1.00\\times {10}^{-6}\\cancel{{\\text{mol L}}^{-1}}\\right)\\left(3.00\\times {10}^{-6}\\text{mol}{\\text{L}}^{-1}\\right)}\\hfill \\\\ &amp; =\\text{2}.\\text{2}0\\times {10}^{7}\\text{L}{\\text{mol}}^{-1}{\\text{s}}^{-1}\\hfill \\end{array}[\/latex]The large value of <em>k<\/em> tells us that this is a fast reaction that could play an important role in ozone depletion if [NO] is large enough.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<h4>Check Your Learning<\/h4>\r\nAcetaldehyde decomposes when heated to yield methane and carbon monoxide according to the equation:\r\n<p style=\"text-align: center;\">[latex]{\\text{CH}}_{3}\\text{CHO(}g\\text{)}\\rightarrow{\\text{CH}}_{4}\\text{(}g\\text{)}+\\text{CO(}g\\text{)}[\/latex]<\/p>\r\nDetermine the rate law and the rate constant for the reaction from the following experimental data:\r\n<table id=\"fs-idm276791216\" class=\"medium unnumbered\" summary=\"This table has three columns and four rows. The first row is a header row, and it labels each column, \u201cTrial,\u201d \u201c[ C H subscript 3 C H O ] ( mol \/ L),\u201d and \u201cnegative capital delta [ C H subscript 3 C H O ] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 ).\u201d Under the \u201cTrial\u201d column are the numbers: 1, 2, and 3. Under the \u201c[ C H subscript 3 C H O ] ( mol \/ L)\u201d are the numbers: 1.75 times ten to the negative three; 3.50 times ten to the negative three; and 7.00 times ten to the negative three. Under the column \u201cnegative capital delta [ C H subscript 3 C H O ] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 )\u201d are the numbers: 2.06 times ten to the negative 11; 8.24 times ten to the negative 11; and 3.30 times ten to the negative ten.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Trial<\/th>\r\n<th>[CH<sub>3<\/sub>CHO] (mol\/L)<\/th>\r\n<th>[latex]-\\frac{{\\text{Delta [CH}}_{3}\\text{CHO]}}{\\Delta t}\\text{(mol}{\\text{L}}^{-1}{\\text{s}}^{-1}\\text{)}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>1<\/td>\r\n<td>1.75 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<td>2.06 \u00d7 10<sup>\u221211<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2<\/td>\r\n<td>3.50 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<td>8.24 \u00d7 10<sup>\u221211<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3<\/td>\r\n<td>7.00 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<td>3.30 \u00d7 10<sup>\u221210<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"950345\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"950345\"][latex]\\text{rate}=k{\\left[{\\text{CH}}_{3}\\text{CHO}\\right]}^{2}[\/latex] with <em>k<\/em> = 6.73 \u00d7 10<sup>-6<\/sup> L\/mol\/s[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3:\u00a0Determining Rate Laws from Initial Rates<\/h3>\r\nUsing the initial rates method and the experimental data, determine the rate law and the value of the rate constant for this reaction:\r\n<p style=\"text-align: center;\">[latex]\\text{2NO(}g\\text{)}+{\\text{Cl}}_{2}\\left(g\\right)\\rightarrow\\text{2NOCl(}g\\text{)}[\/latex]<\/p>\r\n\r\n<table id=\"fs-idm285249664\" class=\"medium unnumbered\" summary=\"This table has four columns and four rows. The first row is a header row, and it labels the columns, \u201cTrial,\u201d \u201c[ N O ] ( mol \/ L),\u201d \u201c[ C l subscript 2 ] ( mol \/ L ),\u201d and \u201cnegative capital delta [ N O] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 ).\u201d Under the column \u201cTrial\u201d are the numbers: 1, 2, and 3. Under the column \u201c[ N O ] ( mol \/ L)\u201d are the numbers: 0.10, 0.10, and 0.15. Under the column \u201c[ C l subscript 2 ] ( mol \/ L )\u201d are the numbers: 0.10, 0.15, and 0.10. Under the column \u201cnegative delta [ N O] divided by delta t ( mol L superscript negative 1 s superscript negative 1\u201d are the numbers: 0.00300, 0.00450, 0.00675.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Trial<\/th>\r\n<th>[NO] (mol\/L)<\/th>\r\n<th>[Cl<sub>2<\/sub>] (mol\/L)<\/th>\r\n<th>[latex]-\\frac{\\Delta\\left[\\text{NO}\\right]}{\\Delta t}\\left(\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>1<\/td>\r\n<td>0.10<\/td>\r\n<td>0.10<\/td>\r\n<td>0.00300<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2<\/td>\r\n<td>0.10<\/td>\r\n<td>0.15<\/td>\r\n<td>0.00450<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3<\/td>\r\n<td>0.15<\/td>\r\n<td>0.10<\/td>\r\n<td>0.00675<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"64050\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"64050\"]\r\n\r\nThe rate law for this reaction will have the form:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{m}{\\left[{\\text{Cl}}_{2}\\right]}^{n}[\/latex]<\/p>\r\nAs in Example 2, we can approach this problem in a stepwise fashion, determining the values of <em>m<\/em> and <em>n<\/em> from the experimental data and then using these values to determine the value of <em>k<\/em>. In this example, however, we will use a different approach to determine the values of <em>m<\/em> and <em>n<\/em>:\r\n<ol>\r\n \t<li>Determine the value of <em>m<\/em> from the data in which [NO] varies and [Cl<sub>2<\/sub>] is constant: We can write the ratios with the subscripts <em>x<\/em> and <em>y<\/em> to indicate data from two different trials:[latex]\\frac{{\\text{rate}}_{x}}{{\\text{rate}}_{y}}=\\frac{k{\\text{[}\\text{NO}\\text{]}}_{x}^{m}{\\text{[}{\\text{Cl}}_{2}\\text{]}}_{x}^{n}}{k{\\text{[}\\text{NO}\\text{]}}_{y}^{m}{\\text{[}{\\text{Cl}}_{2}\\text{]}}_{y}^{n}}[\/latex]Using the third trial and the first trial, in which [Cl<sub>2<\/sub>] does not vary, gives:[latex]\\frac{\\text{rate 3}}{\\text{rate 1}}=\\frac{0.00675}{0.00300}=\\frac{k{\\left(0.15\\right)}^{m}{\\left(0.10\\right)}^{n}}{k{\\left(0.10\\right)}^{m}{\\left(0.10\\right)}^{n}}[\/latex]\r\n\r\nAfter canceling equivalent terms in the numerator and denominator, we are left with:\r\n\r\n[latex]\\frac{0.00675}{0.00300}=\\frac{{\\left(0.15\\right)}^{m}}{{\\left(0.10\\right)}^{m}}[\/latex]\r\n\r\nwhich simplifies to:\r\n\r\n[latex]2.25={\\left(1.5\\right)}^{m}[\/latex]\r\n\r\nWe can use natural logs to determine the value of the exponent <em>m<\/em>:\r\n\r\n[latex]\\begin{array}{ccc}\\hfill \\text{ln}\\left(2.25\\right)&amp; =&amp; m\\text{ln}\\left(1.5\\right)\\hfill \\\\ \\hfill \\frac{\\text{ln}\\left(2.25\\right)}{\\text{ln}\\left(1.5\\right)}&amp; =&amp; m\\hfill \\\\ \\hfill 2&amp; =&amp; m\\hfill \\end{array}[\/latex]\r\n\r\nWe can confirm the result easily, since:\r\n\r\n1.5<sup>2<\/sup> = 2.25<\/li>\r\n \t<li>Determine the value of <em>n<\/em> from data in which [Cl<sub>2<\/sub>] varies and [NO] is constant:[latex]\\frac{\\text{rate 2}}{\\text{rate 1}}=\\frac{0.00450}{0.00300}=\\frac{k{\\left(0.10\\right)}^{m}{\\left(0.15\\right)}^{n}}{k{\\left(0.10\\right)}^{m}{\\left(0.10\\right)}^{n}}[\/latex]Cancelation gives:[latex]\\frac{0.0045}{0.0030}=\\frac{{\\left(0.15\\right)}^{n}}{{\\left(0.10\\right)}^{n}}[\/latex]\r\n\r\nwhich simplifies to:\r\n\r\n1.5 = (1.5)<sup><em>n<\/em><\/sup>\r\n\r\nThus <em>n<\/em> must be 1, and the form of the rate law is:\r\n\r\n[latex]\\text{Rate}=k{\\left[\\text{NO}\\right]}^{m}{\\left[{\\text{Cl}}_{2}\\right]}^{n}=k{\\left[\\text{NO}\\right]}^{2}\\left[{\\text{Cl}}_{2}\\right][\/latex]<\/li>\r\n \t<li>Determine the numerical value of the rate constant <em>k<\/em> with appropriate units:\u00a0The units for the rate of a reaction are mol\/L\/ s. The units for <em>k<\/em> are whatever is needed so that substituting into the rate law expression affords the appropriate units for the rate. In this example, the concentration units are mol<sup>3<\/sup>\/L<sup>3<\/sup>. The units for <em>k<\/em> should be mol<sup>-2<\/sup> L<sup>2\/<\/sup>s so that the rate is in terms of mol\/L\/ s.To determine the value of <em>k<\/em> once the rate law expression has been solved, simply plug in values from the first experimental trial and solve for <em>k<\/em>:\r\n[latex]\\begin{array}{ccc}\\hfill 0.00300\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}&amp; =&amp; k{\\left(0.10\\text{mol}{\\text{L}}^{-1}\\right)}^{2}{\\left(0.10\\text{mol}{\\text{L}}^{-1}\\right)}^{1}\\hfill \\\\ \\hfill k&amp; =&amp; 3.0{\\text{mol}}^{-2}{\\text{L}}^{2}{\\text{s}}^{-1}\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<h4>Check Your Learning<\/h4>\r\nUse the provided initial rate data to derive the rate law for the reaction whose equation is:\r\n<p style=\"text-align: center;\">[latex]{\\text{OCl}}^{-}\\left(aq\\right)+{\\text{I}}^{-}\\left(aq\\right)\\rightarrow{\\text{OI}}^{-}\\left(aq\\right)+{\\text{Cl}}^{-}\\left(aq\\right)[\/latex]<\/p>\r\n\r\n<table id=\"fs-idm275492048\" class=\"medium unnumbered\" summary=\"This table has four columns and four rows. The first row is a header row and it labels each column, \u201cTrial,\u201d \u201c[ O C l superscript negative sign ] ( mol \/ L),\u201d \u201c[ I superscript negative sign ] ( mol \/ l ),\u201d and \u201cInitial Rate ( mol \/ L \/ s).\u201d Under the column \u201cTrial\u201d are the numbers: 1, 2, and 3. Under the column \u201c[ O C l superscript negative sign ] ( mol \/ L)\u201d are the numbers 0.0040, 0.0020, and 0.0020. Under the column \u201c[ I superscript negative sign ] ( mol \/ l )\u201d are the numbers: 0.0020, 0.0040, and 0.0020. Under the column \u201cInitial Rate ( mol \/ L \/ s)\u201d are the numbers: 0.00184, 0.00092, and 0.00046.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Trial<\/th>\r\n<th>[OCl<sup>-<\/sup>] (mol\/L)<\/th>\r\n<th>[I<sup>-<\/sup>] (mol\/L)<\/th>\r\n<th>Initial Rate (mol\/L\/s)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>1<\/td>\r\n<td>0.0040<\/td>\r\n<td>0.0020<\/td>\r\n<td>0.00184<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2<\/td>\r\n<td>0.0020<\/td>\r\n<td>0.0040<\/td>\r\n<td>0.00092<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3<\/td>\r\n<td>0.0020<\/td>\r\n<td>0.0020<\/td>\r\n<td>0.00046<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDetermine the rate law expression and the value of the rate constant <em>k<\/em> with appropriate units for this reaction.\r\n[reveal-answer q=\"858130\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"858130\"][latex]\\frac{\\text{rate 2}}{\\text{rate 3}}=\\frac{0.00092}{0.00046}=\\frac{k{\\left(0.0020\\right)}^{x}{\\left(0.0040\\right)}^{y}}{k{\\left(0.0020\\right)}^{x}{\\left(0.0020\\right)}^{y}}[\/latex]\r\n\r\n2.00 = 2.00<sup><em>y<\/em><\/sup>\r\n\r\ny = 1\r\n\r\n[latex]\\frac{\\text{rate 1}}{\\text{rate 2}}=\\frac{0.00184}{0.00092}=\\frac{k{\\left(0.0040\\right)}^{x}{\\left(0.0020\\right)}^{y}}{k{\\left(0.0020\\right)}^{x}{\\left(0.0040\\right)}^{y}}[\/latex]\r\n\r\n[latex]\\begin{array}{ccc}\\hfill 2.00&amp; =&amp; \\frac{{2}^{x}}{{2}^{y}}\\hfill \\\\ \\hfill 2.00&amp; =&amp; \\frac{{2}^{x}}{{2}^{1}}\\hfill \\\\ \\hfill 4.00&amp; =&amp; {2}^{x}\\hfill \\\\ x&amp; =&amp; 2\\hfill \\end{array}[\/latex]\r\n\r\nSubstituting the concentration data from trial 1 and solving for <em>k<\/em> yields:\r\n\r\n[latex]\\begin{array}{ccc}\\hfill \\text{rate}&amp; =&amp; k{\\left[{\\text{OCl}}^{-}\\right]}^{2}{\\left[{\\text{I}}^{-}\\right]}^{1}\\hfill \\\\ 0.00184\\hfill &amp; =&amp; k{\\left(0.0040\\right)}^{2}{\\left(0.0020\\right)}^{1}\\hfill \\\\ \\hfill k&amp; =&amp; 5.75{10}^{4}{\\text{mol}}^{-2}{\\text{L}}^{2}{\\text{s}}^{-1}\\hfill \\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reaction Order and Rate Constant Units<\/h2>\r\nIn some of our examples, the reaction orders in the rate law happen to be the same as the coefficients in the chemical equation for the reaction. This is merely a coincidence and very often not the case.\r\n\r\nRate laws may exhibit fractional orders for some reactants, and negative reaction orders are sometimes observed when an increase in the concentration of one reactant causes a decrease in reaction rate. A few examples illustrating these points are provided:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\\\ {\\text{NO}}_{2}+\\text{CO}\\rightarrow\\text{NO}+{\\text{CO}}_{\\text{2}}\\text{rate}=\\text{k}{\\left[{\\text{NO}}_{2}\\right]}^{2}\\\\ {\\text{CH}}_{3}\\text{CHO}\\rightarrow{\\text{CH}}_{4}+\\text{CO}\\text{rate}=\\text{k}{\\left[{\\text{CH}}_{3}\\text{CHO}\\right]}^{2}\\\\ {\\text{2N}}_{2}{\\text{O}}_{5}\\rightarrow{\\text{2NO}}_{2}+{\\text{O}}_{\\text{2}}\\text{rate}=\\text{k}\\left[{\\text{N}}_{2}{\\text{O}}_{5}\\right]\\\\ {\\text{2NO}}_{2}+{\\text{F}}_{2}\\rightarrow{\\text{2NO}}_{2}\\text{F}\\text{rate}=\\text{k}\\left[{\\text{NO}}_{2}\\right]\\left[{\\text{F}}_{2}\\right]\\\\ {\\text{2NO}}_{2}\\text{Cl}\\rightarrow{\\text{2NO}}_{2}+{\\text{Cl}}_{2}\\text{rate}=\\text{k}\\left[{\\text{NO}}_{2}\\text{Cl}\\right]\\end{array}[\/latex]<\/p>\r\nIt is important to note that <em>rate laws are determined by experiment only and are not reliably predicted by reaction stoichiometry.<\/em>\r\n\r\nReaction orders also play a role in determining the units for the rate constant <em>k<\/em>. In Example 2, a second-order reaction, we found the units for <em>k<\/em> to be [latex]\\text{L}{\\text{mol}}^{-4}{\\text{s}}^{-1},[\/latex] whereas in Example 3, a third order reaction, we found the units for <em>k<\/em> to be mol<sup>\u22122<\/sup> L<sup>2<\/sup>\/s. More generally speaking, the units for the rate constant for a reaction of order [latex]\\left(m+n\\right)[\/latex] are [latex]{\\text{mol}}^{1-\\left(m+n\\right)}{\\text{L}}^{\\left(m+n\\right)-1}{\\text{s}}^{-1}[\/latex]. Table\u00a01\u00a0summarizes the rate constant units for common reaction orders.\r\n<table id=\"fs-idm233742304\" summary=\"This table has two columns and four rows. The first row is a header row and it labels each column, \u201cReaction order,\u201d and \u201cUnits of k.\u201d Under the \u201cReaction Order\u201d column are: \u201c( m + n ),\u201d \u201czero,\u201d \u201cfirst,\u201d \u201csecond,\u201d and, \u201cthird.\u201d Under the \u201cUnits of k\u201d column are: \u201cmol superscript 1 minus ( m + n) L superscript ( m + n) minus 1 s superscript negative one,\u201d \u201cmol \/ L \/ s,\u201d \u201cs superscript negative 1,\u201d \u201cL \/ mol \/ s,\u201d and \u201cmol superscript negative 2 L superscript 2 s superscript negative 1.\u201d\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\">Table\u00a01. Rate Constants for Common Reaction Orders<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<th>Reaction Order<\/th>\r\n<th>Units of <em>k<\/em><\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\left(m+n\\right)[\/latex]<\/td>\r\n<td>[latex]{\\text{mol}}^{1-\\left(m+n\\right)}{\\text{L}}^{\\left(m+n\\right)-1}{\\text{s}}^{-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>zero<\/td>\r\n<td>mol\/L\/s<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>first<\/td>\r\n<td>s<sup>\u22121<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>second<\/td>\r\n<td>L\/mol\/s<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>third<\/td>\r\n<td>mol<sup>\u22122<\/sup> L<sup>2<\/sup> s<sup>\u22121<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote that the units in the table can also be expressed in terms of molarity (<em>M<\/em>) instead of mol\/L. Also, units of time other than the second (such as minutes, hours, days) may be used, depending on the situation.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Concepts and Summary<\/h3>\r\nRate laws provide a mathematical description of how changes in the amount of a substance affect the rate of a chemical reaction. Rate laws are determined experimentally and cannot be predicted by reaction stoichiometry. The order of reaction describes how much a change in the amount of each substance affects the overall rate, and the overall order of a reaction is the sum of the orders for each substance present in the reaction. Reaction orders are typically first order, second order, or zero order, but fractional and even negative orders are possible.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\n<ol>\r\n \t<li>How do the rate of a reaction and its rate constant differ?<\/li>\r\n \t<li>Doubling the concentration of a reactant increases the rate of a reaction four times. With this knowledge, answer the following questions:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>What is the order of the reaction with respect to that reactant?<\/li>\r\n \t<li>Tripling the concentration of a different reactant increases the rate of a reaction three times. What is the order of the reaction with respect to that reactant?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Tripling the concentration of a reactant increases the rate of a reaction nine times. With this knowledge, answer the following questions:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>What is the order of the reaction with respect to that reactant?<\/li>\r\n \t<li>Increasing the concentration of a reactant by a factor of four increases the rate of a reaction four times. What is the order of the reaction with respect to that reactant?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>How much and in what direction will each of the following affect the rate of the reaction: [latex]\\text{CO(}g\\text{)}+{\\text{NO}}_{2}\\text{(}g\\text{)}\\rightarrow{\\text{CO}}_{2}\\text{(}g\\text{)}+\\text{NO(}g\\text{)}[\/latex] if the rate law for the reaction is [latex]\\text{rate}=k{\\left[{\\text{NO}}_{2}\\right]}^{2}?[\/latex]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Decreasing the pressure of NO<sub>2<\/sub> from 0.50 atm to 0.250 atm.<\/li>\r\n \t<li>Increasing the concentration of CO from 0.01 <em>M<\/em> to 0.03 <em>M<\/em>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>How will each of the following affect the rate of the reaction: [latex]\\text{CO(}g\\text{)}+{\\text{NO}}_{2}\\text{(}g\\text{)}\\rightarrow{\\text{CO}}_{2}\\text{(}g\\text{)}+\\text{NO(}g\\text{)}[\/latex] if the rate law for the reaction is [latex]\\text{rate}=k\\left[{\\text{NO}}_{2}\\right]\\left[\\text{CO}\\right][\/latex] ?\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Increasing the pressure of NO<sub>2<\/sub> from 0.1 atm to 0.3 atm<\/li>\r\n \t<li>Increasing the concentration of CO from 0.02 <em>M<\/em> to 0.06 <em>M<\/em>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Regular flights of supersonic aircraft in the stratosphere are of concern because such aircraft produce nitric oxide, NO, as a byproduct in the exhaust of their engines. Nitric oxide reacts with ozone, and it has been suggested that this could contribute to depletion of the ozone layer. The reaction [latex]\\text{NO}+{\\text{O}}_{3}\\rightarrow{\\text{NO}}_{2}+{\\text{O}}_{2}[\/latex] is first order with respect to both NO and O<sub>3<\/sub> with a rate constant of 2.20 \u00d7 10<sup>7<\/sup> L\/mol\/s. What is the instantaneous rate of disappearance of NO when [NO] = 3.3 \u00d7 10<sup>\u22126<\/sup><em>M<\/em> and [O<sub>3<\/sub>] = 5.9 \u00d7 10<sup>\u22127<\/sup><em>M<\/em>?<\/li>\r\n \t<li>Radioactive phosphorus is used in the study of biochemical reaction mechanisms because phosphorus atoms are components of many biochemical molecules. The location of the phosphorus (and the location of the molecule it is bound in) can be detected from the electrons (beta particles) it produces:[latex]{}_{15}^{32}\\text{P}\\rightarrow{}_{16}^{32}\\text{S}+{\\text{e}}^{-}[\/latex]Rate = 4.85 \u00d7 10<sup>\u22122<\/sup> day<sup>-1<\/sup>[<sup>32<\/sup>P]What is the instantaneous rate of production of electrons in a sample with a phosphorus concentration of 0.0033 <em>M<\/em>?<\/li>\r\n \t<li>The rate constant for the radioactive decay of <sup>14<\/sup>C is 1.21 \u00d7 10<sup>\u22124<\/sup> year<sup>\u22121<\/sup>. The products of the decay are nitrogen atoms and electrons (beta particles):[latex]{}_{6}^{14}\\text{C}\\rightarrow{}_{6}^{14}\\text{N}+{\\text{e}}^{-}[\/latex][latex]\\text{rate}=k\\left[{}_{6}^{14}\\text{C}\\right][\/latex]What is the instantaneous rate of production of N atoms in a sample with a carbon-14 content of 6.5 \u00d7 10<sup>\u22129<\/sup><em>M<\/em>?<\/li>\r\n \t<li>The decomposition of acetaldehyde is a second order reaction with a rate constant of 4.71 \u00d7 10<sup>\u22128<\/sup>\u00a0L\/mol\/s. What is the instantaneous rate of decomposition of acetaldehyde in a solution with a concentration of 5.55 \u00d7 10<sup>\u22124<\/sup><em>M<\/em>?<\/li>\r\n \t<li>Alcohol is removed from the bloodstream by a series of metabolic reactions. The first reaction produces acetaldehyde; then other products are formed. The following data have been determined for the rate at which alcohol is removed from the blood of an average male, although individual rates can vary by 25\u201330%. Women metabolize alcohol a little more slowly than men:\r\n<table>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[C<sub>2<\/sub>H<sub>5<\/sub>OH] (<em>M<\/em>)<\/td>\r\n<td>4.4 \u00d7 10<sup>\u22122<\/sup><\/td>\r\n<td>3.3 \u00d7 10<sup>\u22122<\/sup><\/td>\r\n<td>2.2 \u00d7 10<sup>\u22122<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rate (mol\/L\/h)<\/td>\r\n<td>2.0 \u00d7 10<sup>\u22122<\/sup><\/td>\r\n<td>2.0 \u00d7 10<sup>\u22122<\/sup><\/td>\r\n<td>2.0 \u00d7 10<sup>\u22122<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDetermine the rate equation, the rate constant, and the overall order for this reaction.<\/li>\r\n \t<li>Under certain conditions the decomposition of ammonia on a metal surface gives the following data:\r\n<table>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[NH<sub>3<\/sub>] (<em>M<\/em>)<\/td>\r\n<td>1.0 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<td>2.0 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<td>3.0 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rate (mol\/L\/h<sup>1<\/sup>)<\/td>\r\n<td>1.5 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>1.5 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<td>1.5 \u00d7 10<sup>\u22126<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDetermine the rate equation, the rate constant, and the overall order for this reaction.<\/li>\r\n \t<li>Nitrosyl chloride, NOCl, decomposes to NO and Cl<sub>2<\/sub>.[latex]\\text{2NOCl(}g\\text{)}\\rightarrow\\text{2NO(}g\\text{)}+{\\text{Cl}}_{2}\\text{(}g\\text{)}[\/latex]Determine the rate equation, the rate constant, and the overall order for this reaction from the following data:\r\n<table>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[NOCl] (<em>M<\/em>)<\/td>\r\n<td>0.10<\/td>\r\n<td>0.20<\/td>\r\n<td>0.30<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rate (mol\/L\/h)<\/td>\r\n<td>8.0 \u00d7 10<sup>-10<\/sup><\/td>\r\n<td>3.2 \u00d7 10<sup>\u22129<\/sup><\/td>\r\n<td>7.2 \u00d7 10<sup>\u22129<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>From the following data, determine the rate equation, the rate constant, and the order with respect to <em>A<\/em> for the reaction [latex]A\\rightarrow 2C.[\/latex]\r\n<table>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[<em>A<\/em>] (<em>M<\/em>)<\/td>\r\n<td>[latex]1.33\\times {10}^{-2}[\/latex]<\/td>\r\n<td>[latex]\\text{2.66}\\times {10}^{-2}[\/latex]<\/td>\r\n<td>[latex]\\text{3.99}\\times {10}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rate (mol\/L\/h)<\/td>\r\n<td>[latex]\\text{3.80}\\times {10}^{-7}[\/latex]<\/td>\r\n<td>[latex]\\text{1.52}\\times {10}^{-6}[\/latex]<\/td>\r\n<td>[latex]\\text{3.42}\\times {10}^{-6}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Nitrogen(II) oxide reacts with chlorine according to the equation:[latex]\\text{2NO(}g\\text{)}+{\\text{Cl}}_{2}\\text{(}g\\text{)}\\rightarrow\\text{2NOCl(}g\\text{)}[\/latex]The following initial rates of reaction have been observed for certain reactant concentrations:\r\n<table>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[NO] (mol\/L<sup>1<\/sup>)<\/th>\r\n<th>[Cl<sub>2<\/sub>] (mol\/L)<\/th>\r\n<th>Rate (mol\/L\/h)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>0.50<\/td>\r\n<td>0.50<\/td>\r\n<td>1.14<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1.00<\/td>\r\n<td>0.50<\/td>\r\n<td>4.56<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1.00<\/td>\r\n<td>1.00<\/td>\r\n<td>9.12<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat is the rate equation that describes the rate\u2019s dependence on the concentrations of NO and Cl<sub>2<\/sub>? What is the rate constant? What are the orders with respect to each reactant?<\/li>\r\n \t<li>Hydrogen reacts with nitrogen monoxide to form dinitrogen monoxide (laughing gas) according to the equation: [latex]{\\text{H}}_{2}\\text{(}g\\text{)}+\\text{2NO(}g\\text{)}\\rightarrow{\\text{N}}_{2}\\text{O(}g\\text{)}+{\\text{H}}_{2}\\text{O(}g\\text{)}[\/latex]Determine the rate equation, the rate constant, and the orders with respect to each reactant from the following data:\r\n<table>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[NO] (<em>M<\/em>)<\/td>\r\n<td>0.30<\/td>\r\n<td>0.60<\/td>\r\n<td>0.60<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[H<sub>2<\/sub>] (<em>M<\/em>)<\/td>\r\n<td>0.35<\/td>\r\n<td>0.35<\/td>\r\n<td>0.70<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rate (mol\/L\/s)<\/td>\r\n<td>[latex]2.835\\times {\\text{10}}^{-3}[\/latex]<\/td>\r\n<td>[latex]1.134\\times {\\text{10}}^{-2}[\/latex]<\/td>\r\n<td>[latex]2.268\\times {\\text{10}}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>For the reaction [latex]A\\rightarrow B+C,[\/latex] the following data were obtained at 30 \u00b0C:\r\n<table>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[<em>A<\/em>] (<em>M<\/em>)<\/td>\r\n<td>0.230<\/td>\r\n<td>0.356<\/td>\r\n<td>0.557<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rate (mol\/L\/s)<\/td>\r\n<td>4.17\u00a0\u00d7 10<sup>\u22124<\/sup><\/td>\r\n<td>9.99 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<td>2.44 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>What is the order of the reaction with respect to [<em>A<\/em>], and what is the rate equation?<\/li>\r\n \t<li>What is the rate constant?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>For the reaction [latex]Q\\rightarrow W+X,[\/latex] the following data were obtained at 30 \u00b0C:\r\n<table>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[<em>Q<\/em>]<sub>initial<\/sub> (<em>M<\/em>)<\/td>\r\n<td>0.170<\/td>\r\n<td>0.212<\/td>\r\n<td>0.357<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rate (mol\/L\/s)<\/td>\r\n<td>6.68 \u00d7 10<sup>\u22123<\/sup><\/td>\r\n<td>1.04 \u00d7 10<sup>\u22122<\/sup><\/td>\r\n<td>2.94 \u00d7 10<sup>\u22122<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>What is the order of the reaction with respect to [<em>Q<\/em>], and what is the rate equation?<\/li>\r\n \t<li>What is the rate constant?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>The rate constant for the first-order decomposition at 45 \u00b0C of dinitrogen pentoxide, N<sub>2<\/sub>O<sub>5<\/sub>, dissolved in chloroform, CHCl<sub>3<\/sub>, is 6.2 \u00d7 10<sup>\u22124<\/sup> min<sup>\u22121<\/sup>.[latex]{\\text{2N}}_{2}{\\text{O}}_{5}\\rightarrow{\\text{4NO}}_{2}+{\\text{O}}_{2}[\/latex]What is the rate of the reaction when [N<sub>2<\/sub>O<sub>5<\/sub>] = 0.40 <em>M<\/em>?<\/li>\r\n \t<li>The annual production of HNO<sub>3<\/sub> in 2013 was 60 million metric tons Most of that was prepared by the following sequence of reactions, each run in a separate reaction vessel.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{\\text{4NH}}_{3}\\text{(}g\\text{)}+{\\text{5O}}_{2}\\text{(}g\\text{)}\\rightarrow\\text{4NO(}g\\text{)}+{\\text{6H}}_{2}\\text{O(}g\\text{)}[\/latex]<\/li>\r\n \t<li>[latex]\\text{2NO(}g\\text{)}+{\\text{O}}_{2}\\text{(}g\\text{)}\\rightarrow{\\text{2NO}}_{2}\\text{(}g\\text{)}[\/latex]<\/li>\r\n \t<li>[latex]{\\text{3NO}}_{2}\\text{(}g\\text{)}+{\\text{H}}_{2}\\text{O(}l\\text{)}\\rightarrow{\\text{2HNO}}_{3}\\text{(}aq\\text{)}+\\text{NO(}g\\text{)}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>The first reaction is run by burning ammonia in air over a platinum catalyst. This reaction is fast. The reaction in equation (c) is also fast. The second reaction limits the rate at which nitric acid can be prepared from ammonia. If equation (b) is second order in NO and first order in O<sub>2<\/sub>, what is the rate of formation of NO<sub>2<\/sub> when the oxygen concentration is 0.50 M and the nitric oxide concentration is 0.75 <em>M<\/em>? The rate constant for the reaction is 5.8 \u00d7 10<sup>\u22126<\/sup> L<sup>2<\/sup>\/mol<sup>2<\/sup>\/s.<\/li>\r\n \t<li>The following data have been determined for the reaction:[latex]{\\text{I}}^{-}+{\\text{OCl}}^{-}\\rightarrow{\\text{IO}}^{-}+{\\text{Cl}}^{-}[\/latex]\r\n<table id=\"fs-idm189410736\" class=\"medium unnumbered\" summary=\"This table has four columns and four rows. The first column labels each row. The first cell in the first column is blank, and then each row is labeled, \u201c[ I superscript negative sign ] subscript initial ( M ),\u201d \u201c[ O C l superscript negative sign ] subscript initial ( M ),\u201d and, \u201cRate ( mol \/ L \/ s ).\u201d To the right of the blank cell are the numbers: 1, 2, and 3. To the right of the header, \u201c[ I superscript negative sign ] subscript initial ( M ),\u201d are the numbers: 0.10, 0.20, and 0.30. To the right of the header, \u201c[ O C l superscript negative sign ] subscript initial ( M ),\u201d are the numbers: 0.050, 0.050, and 0.010. To the right of the header, \u201cRate ( mol \/ L \/ s ),\u201d are the numbers: 3.05 times ten to the negative 4; 6.20 times ten to the negative 4; and 1.83 times ten to the negative 4.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th><\/th>\r\n<th>1<\/th>\r\n<th>2<\/th>\r\n<th>3<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\left[{\\text{I}}^{-}\\right]}_{\\text{initial}}[\/latex] (<em>M<\/em>)<\/td>\r\n<td>0.10<\/td>\r\n<td>0.20<\/td>\r\n<td>0.30<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\left[{\\text{OCl}}^{-}\\right]}_{\\text{initial}}[\/latex] (<em>M<\/em>)<\/td>\r\n<td>0.050<\/td>\r\n<td>0.050<\/td>\r\n<td>0.010<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Rate (mol\/L\/s)<\/td>\r\n<td>3.05 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<td>6.20 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<td>1.83 \u00d7 10<sup>\u22124<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDetermine the rate equation and the rate constant for this reaction.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"453158\"]Show Selected Answers[\/reveal-answer]\r\n[hidden-answer a=\"453158\"]\r\n\r\n2.\u00a0(a) Since the concentration of the reactant doubled and the rate quadrupled, we can conclude that the order with respect to the reactant is 2, since 2<sup>2<\/sup> = 4.\r\n\r\n[latex]\\begin{array}{l}\\text{rate}=k{\\left[\\text{reactant}\\right]}^{m}\\\\ \\text{4}\\left(\\text{rate}\\right)=k{\\left[\\text{2}\\left(\\text{reactant}\\right)\\right]}^{2}\\end{array}[\/latex]\r\n\r\n(b) Since the concentration of the reactant and the rate both tripled, we can conclude that [latex]m=1,[\/latex] and the order with respect to this reactant is 1.\r\n\r\n[latex]\\begin{array}{l}\\text{rate}=k{\\left[\\text{reactant}\\right]}^{m}\\\\ \\text{3}\\left(\\text{rate}\\right)=k{\\left[\\text{3}\\left(\\text{reactant}\\right)\\right]}^{1}\\end{array}[\/latex]\r\n\r\n4.\u00a0(a) [latex]\\frac{{\\text{rate}}_{2}}{{\\text{rate}}_{1}}=\\frac{\\text{k}{\\text{[}0.25{\\text{NO}}_{2}\\text{]}}^{2}}{\\text{k}{\\text{[}0.50{\\text{NO}}_{2}\\text{]}}^{2}}=\\frac{0.0675}{0.25}=\\frac{1}{4}[\/latex]\r\n\r\nSince Rate<sub>1<\/sub> is four times as large as Rate<sub>2<\/sub>, the process reduces the rate by a factor of 4.\r\n\r\n(b) Since CO does not appear in the rate law, the rate is not affected.\r\n\r\n6.\u00a0Rate = <em>k<\/em> [NO][O<sub>3<\/sub>] = 2.20 \u00d7 10<sup>7<\/sup> L\/mol\/s[3.3 \u00d7 10<sup>\u22126<\/sup><em>M<\/em>][5.9 \u00d7 10<sup>-7<\/sup><em>M<\/em>] = 4.3 \u00d7 10<sup>\u22125<\/sup> mol\/L\/s\r\n\r\n8.\u00a0rate = 1.21 \u00d7 10<sup>\u22124<\/sup> year<sup>\u22121<\/sup> [6.5 \u00d7 10<sup>-9<\/sup><em>M<\/em>] = 7.9 \u00d7 10<sup>\u221213<\/sup> mol\/L\/year\r\n\r\n10.\u00a0The rate is independent of the concentration. Therefore, rate = <em>k<\/em> ; <em>k<\/em> = 2.0 \u00d7 10<sup>-2<\/sup> mol\/L\/h (about 0.9 g\/L\/h for the average male); The reaction is zero order\u2014that is, it does not depend on the concentration of any reagent.\r\n\r\n12.\u00a0The object of this problem is to use the general rate expression: rate = <em>k<\/em>[NOC]<sup>m<\/sup>, first to determine the value of <em>m<\/em> and then, by substituting data from one experiment into the equation, to find the value of <em>k<\/em>. The data listed as substituted into the rate equation give:\r\n\r\nExperiment 1: 8.0 \u00d7 10<sup>\u221210<\/sup> mol\/L\/s = <em>k<\/em>[0.10 mol\/L]<sup><em>m<\/em><\/sup>\r\n\r\nExperiment 2: 3.20 \u00d7 10<sup>\u22129<\/sup> mol\/L\/s = <em>k<\/em>[0.20 mol\/L]<sup><em>m<\/em><\/sup>\r\n\r\nExperiment 3: 7.2 \u00d7 10<sup>\u22129<\/sup> mol\/L\/s = <em>k<\/em>[0.30 mol\/L]<sup><em>m<\/em><\/sup>\r\n\r\nThe value of <em>m<\/em> can be found by inspection. Examining Experiments 1 and 2, it is found that the rate increases by a factor of four as the concentration increases by a factor of two; from Experiments 1 and 3, the rate increases by a factor of nine while the concentration increases by a factor of three. This can happen only if <em>m<\/em> is 2. The value of <em>k<\/em> as calculated from the first set of data is:\r\n<p style=\"text-align: center;\">[latex]k=\\frac{8.0\\times {\\text{10}}^{-10}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}{{\\left[0.10\\text{mol}{\\text{L}}^{-1}\\right]}^{2}}=8.0\\times {\\text{10}}^{-8}\\text{L}{\\text{mol}}^{-1}{\\text{s}}^{-1}[\/latex]<\/p>\r\nrate = <em>k<\/em>[NOC]<sup>2<\/sup>; <em>k<\/em> = 8.0 \u00d7 10<sup>\u22128<\/sup> L\/mol\/s; second order\r\n\r\n14.\u00a0The rate law has the general form:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{m}{\\left[{\\text{Cl}}_{2}\\right]}^{n}[\/latex]<\/p>\r\nComparing the data in rows 1 and 2, [Cl<sub>2<\/sub>] remains constant, [NO] doubles, and the rate becomes four times as large, so <em>m<\/em> = 2. Comparing data in rows 2 and 3, [NP] remains constant, [Cl<sub>2<\/sub>] doubles, and the rate doubles, so <em>n<\/em> = 1. The rate equation is:\r\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{2}\\left[{\\text{Cl}}_{2}\\right][\/latex]<\/p>\r\nData from row 1 are used to determine <em>k<\/em>.\r\n<p style=\"text-align: center;\">[latex]k=\\frac{\\text{rate}}{{\\text{[NO]}}^{2}{\\text{[Cl]}}_{2}}=\\frac{1.14\\text{mol}{\\text{L}}^{-1}{\\text{h}}^{-1}}{{\\left(0.50\\text{mol}{\\text{L}}^{-1}\\right)}^{2}\\left(0.50\\text{mol}{\\text{L}}^{-1}\\right)}=9.12{\\text{L}}^{2}{\\text{mol}}^{-2}{\\text{h}}^{-1}[\/latex]<\/p>\r\n[latex]\\text{rate}=k{\\text{[NO]}}^{2}{\\text{[Cl]}}_{2}\\text{;}[\/latex] second order in NO; first order in Cl<sub>2<\/sub>\r\n\r\n16.\u00a0(a) The rate equation will be of the form rate = <em>k<\/em>[A]<sup><em>m<\/em><\/sup> and <em>m<\/em> will be the same for all three sets of experimental data. Therefore, we can write:\r\n\r\nExperiment 1: 4.17 \u00d7 10<sup>\u22124<\/sup> mol\/L\/s = <em>k<\/em>[0.230 <em>M<\/em>]<sup><em>m<\/em><\/sup>\r\n\r\nExperiment 2: 9.99 \u00d7 10<sup>\u22124<\/sup> mol\/L\/s = <em>k<\/em>[0.356 <em>M<\/em>]<sup><em>m<\/em><\/sup>\r\n\r\nExperiment 3: 2.44 \u00d7 10<sup>\u22123<\/sup> mol\/L\/s = <em>k<\/em>[0.557 <em>M<\/em>]<sup><em>m<\/em><\/sup>\r\n\r\nThe first two experiments can be set up so as to cancel one of the unknowns (that is, <em>k<\/em>) and solve for the other unknown (that is, <em>m<\/em>):\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\frac{4.17\\times 10^{-4}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}{9.99\\times 10^{-4}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}&amp; =&amp; \\frac{{\\text{(0.230 M)}}^{m}}{{\\text{(0.356 M)}}^{m}}\\hfill \\\\ \\hfill 0.4174&amp; =&amp; \\frac{{\\text{(0.230 M)}}^{m}}{{\\text{(0.356 M)}}^{m}}\\hfill \\end{array}[\/latex]<\/p>\r\nTaking the natural log of each side gives:\r\n\r\nln 0.4174 = <em>m<\/em>(ln 0.230) \u2212 <em>m<\/em>(ln 0.356)\r\n\r\n\u22120.8737 = \u22121.4697<em>m<\/em> + 1.0328<em>m<\/em>\r\n\r\n\u22120.8737 = \u22120.4369<em>m<\/em>\r\n\r\n[latex]m=\\frac{-0.8737}{-0.4369}=2.00[\/latex]\r\n\r\nTherefore, the rate equation is second order in A and is written as rate = <em>k<\/em>[A]<sup>2<\/sup>.\r\n\r\n(b) The rate constant can be calculated from any of the three sets of data by using the rate equation in conjunction with data found by substituting any of the three sets of data into the rate equation. Using the data from Equation 1 gives:\r\n\r\n4.17 \u00d7 10<sup>\u221214<\/sup> mol\/L\/s = <em>k<\/em>[0.230 <em>M<\/em>]<sup>2<\/sup>\r\n\r\n[latex]k=\\frac{4.17\\times {\\text{10}}^{-14}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}{\\left(0.059{\\text{mol}}^{2}{\\text{L}}^{-2}\\right)}=7.88\\times {\\text{10}}^{-13}\\text{L}{\\text{mol}}^{-1}{\\text{s}}^{-1}[\/latex]\r\n\r\n18. The rate of reaction for a first-order reaction in N<sub>2<\/sub>O<sub>5<\/sub> is written as rate = <em>k<\/em>[N<sub>2<\/sub>O<sub>5<\/sub>] where <em>k<\/em>, the rate constant at 45 \u00b0C, is 6.2 \u00d7 10<sup>\u22124<\/sup> min<sup>\u22121<\/sup>. When [N<sub>2<\/sub>O<sub>5<\/sub>] = 0.40 <em>M<\/em>, rate = 6.2 \u00d7 10<sup>\u22124<\/sup> min<sup>\u22121<\/sup> (0.40 mol\/L) = 2.5 \u00d7 10<sup>\u22124<\/sup> mol\/L\/min\r\n\r\n20.\u00a0The rate equation has the form rate = <em>k<\/em>[I<sup>\u2212<\/sup>]<sup><em>m<\/em><\/sup>[OCl<sup>\u2212<\/sup>]<sup><em>n<\/em><\/sup> and the values for <em>m<\/em> and <em>n<\/em> must be determined. Comparing data from columns 1 and 2, [OCl<sup>-<\/sup>] remains constant and [I<sup>\u2212<\/sup>] doubles. As [I<sup>\u2212<\/sup>] doubles, the rate doubles, so <em>m<\/em> = 1.\r\n\r\nComparing data from columns 1 and 3,\r\n<p style=\"text-align: center;\">[latex]3.05\\times {10}^{-4}=k{\\text{[0.10]}}^{1}{\\text{[0.05]}}^{n}\\rightarrow 3.05\\times {10}^{-3}=k{\\text{[0.05]}}^{n}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]1.83\\times {10}^{-4}=k{\\text{[0.30]}}^{1}{\\text{[0.01]}}^{n}\\rightarrow 6.1\\times {10}^{-4}=k{\\text{[0.01]}}^{n}[\/latex]<\/p>\r\nThe first numerical value is five times larger than the second, corresponding to a fivefold increase in concentration. Therefore, <em>n<\/em> = 1; rate = <em>k<\/em>[I<sup>\u2212<\/sup>][OCl<sup>\u22121<\/sup>]\r\n\r\nThe rate constant is determined by putting the data from column 2 into the rate equation:\r\n<p style=\"text-align: center;\">[latex]k=\\frac{\\text{rate}}{\\left[{\\text{I}}^{-}\\right]\\left[{\\text{OCl}}^{-}\\right]}=\\frac{6.10\\times {\\text{10}}^{-4}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}{\\left(0.20\\text{mol}{\\text{L}}^{-1}\\right)\\left(0.050\\text{mol}{\\text{L}}^{-1}\\right)}=6.1\\times {\\text{10}}^{-2}\\text{L}{\\text{mol}}^{-1}{\\text{s}}^{-1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<strong>method of initial rates: <\/strong>use of a more explicit algebraic method to determine the orders in a rate law\r\n\r\n<strong>overall reaction order: <\/strong>sum of the reaction orders for each substance represented in the rate law\r\n\r\n<strong>rate constant (<em>k<\/em>): <\/strong>proportionality constant in the relationship between reaction rate and concentrations of reactants\r\n\r\n<strong>rate law (also, rate equation): <\/strong>mathematical equation showing the dependence of reaction rate on the rate constant and the concentration of one or more reactants\r\n\r\n<strong>reaction order: <\/strong>value of an exponent in a rate law, expressed as an ordinal number (for example, zero order for 0, first order for 1, second order for 2, and so on)","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this module, you will be able to:<\/p>\n<ul>\n<li>Explain the form and function of a rate law<\/li>\n<li>Use rate laws to calculate reaction rates<\/li>\n<li>Use rate and concentration data to identify reaction orders and derive rate laws<\/li>\n<\/ul>\n<\/div>\n<p>As described in the previous module, the rate of a reaction is affected by the concentrations of reactants. Rate laws or rate equations are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants. In general, a rate law (or differential rate law, as it is sometimes called) takes this form:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[A\\right]}^{m}{\\left[B\\right]}^{n}{\\left[C\\right]}^{p}\\dots[\/latex]<\/p>\n<p>in which [<em>A<\/em>], [<em>B<\/em>], and [<em>C<\/em>] represent the molar concentrations of reactants, and <em>k<\/em> is the rate constant, which is specific for a particular reaction at a particular temperature. The exponents <em>m<\/em>, <em>n<\/em>, and <em>p<\/em> are usually positive integers (although it is possible for them to be fractions or negative numbers). The rate constant <em>k<\/em> and the exponents <em>m<\/em>, <em>n<\/em>, and <em>p<\/em> must be determined experimentally by observing how the rate of a reaction changes as the concentrations of the reactants are changed. The rate constant <em>k<\/em> is independent of the concentration of <em>A<\/em>, <em>B<\/em>, or <em>C<\/em>, but it does vary with temperature and surface area.<\/p>\n<p>The exponents in a rate law describe the effects of the reactant concentrations on the reaction rate and define the reaction order. Consider a reaction for which the rate law is:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[A\\right]}^{m}{\\left[B\\right]}^{n}[\/latex]<\/p>\n<p>If the exponent <em>m<\/em> is 1, the reaction is first order with respect to <em>A<\/em>. If <em>m<\/em> is 2, the reaction is second order with respect to <em>A<\/em>. If <em>n<\/em> is 1, the reaction is first order in <em>B<\/em>. If <em>n<\/em> is 2, the reaction is second order in <em>B<\/em>. If <em>m<\/em> or <em>n<\/em> is zero, the reaction is zero order in <em>A<\/em> or <em>B<\/em>, respectively, and the rate of the reaction is not affected by the concentration of that reactant. The overall reaction order is the sum of the orders with respect to each reactant. If <em>m<\/em> = 1 and <em>n<\/em> = 1, the overall order of the reaction is second order (<em>m<\/em> + <em>n<\/em> = 1 + 1 = 2).<\/p>\n<p>The rate law:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k\\left[{\\text{H}}_{2}{\\text{O}}_{2}\\right][\/latex]<\/p>\n<p>describes a reaction that is first order in hydrogen peroxide and first order overall. The rate law:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[{\\text{C}}_{4}{\\text{H}}_{6}\\right]}^{2}[\/latex]<\/p>\n<p>describes a reaction that is second order in C<sub>4<\/sub>H<sub>6<\/sub> and second order overall. The rate law:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k\\left[{\\text{H}}^{+}\\right]\\left[{\\text{OH}}^{-}\\right][\/latex]<\/p>\n<p>describes a reaction that is first order in H<sup>+<\/sup>, first order in OH<sup>&#8211;<\/sup>, and second order overall.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1:\u00a0Writing Rate Laws from Reaction Orders<\/h3>\n<p>An experiment shows that the reaction of nitrogen dioxide with carbon monoxide:<\/p>\n<p style=\"text-align: center;\">[latex]{\\text{NO}}_{2}\\text{(}g\\text{)}+\\text{CO(}g\\text{)}\\rightarrow\\text{NO(}g\\text{)}+{\\text{CO}}_{2}\\left(g\\right)[\/latex]<\/p>\n<p>is second order in NO<sub>2<\/sub> and zero order in CO at 100 \u00b0C. What is the rate law for the reaction?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q341650\">Show Answer<\/span><\/p>\n<div id=\"q341650\" class=\"hidden-answer\" style=\"display: none\">\n<p>The reaction will have the form:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[{\\text{NO}}_{2}\\right]}^{m}{\\left[\\text{CO}\\right]}^{n}[\/latex]<\/p>\n<p>The reaction is second order in NO<sub>2<\/sub>; thus <em>m<\/em> = 2. The reaction is zero order in CO; thus <em>n<\/em> = 0. The rate law is:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[{\\text{NO}}_{2}\\right]}^{2}{\\left[\\text{CO}\\right]}^{0}=k{\\left[{\\text{NO}}_{2}\\right]}^{2}[\/latex]<\/p>\n<p>Remember that a number raised to the zero power is equal to 1, thus [CO]<sup>0<\/sup> = 1, which is why we can simply drop the concentration of CO from the rate equation: the rate of reaction is solely dependent on the concentration of NO<sub>2<\/sub>. When we consider rate mechanisms later in this chapter, we will explain how a reactant\u2019s concentration can have no effect on a reaction despite being involved in the reaction.<\/p>\n<\/div>\n<\/div>\n<h4>Check Your Learning<\/h4>\n<p>The rate law for the reaction:<\/p>\n<p style=\"text-align: center;\">[latex]{\\text{H}}_{2}\\text{(}g\\text{)}+2\\text{NO(}g\\text{)}\\rightarrow{\\text{N}}_{2}\\text{O(}g\\text{)}+{\\text{H}}_{2}\\text{O(}g\\text{)}[\/latex]<\/p>\n<p>has been determined to be rate = <em>k<\/em>[NO]<sup>2<\/sup>[H<sub>2<\/sub>]. What are the orders with respect to each reactant, and what is the overall order of the reaction?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q126680\">Show Answer<\/span><\/p>\n<div id=\"q126680\" class=\"hidden-answer\" style=\"display: none\">order in NO = 2; order in H<sub>2<\/sub> = 1; overall order = 3<\/div>\n<\/div>\n<h4>Check Your Learning<\/h4>\n<p>In a transesterification reaction, a triglyceride reacts with an alcohol to form an ester and glycerol. Many students learn about the reaction between methanol (CH<sub>3<\/sub>OH) and ethyl acetate (CH<sub>3<\/sub>CH<sub>2<\/sub>OCOCH<sub>3<\/sub>) as a sample reaction before studying the chemical reactions that produce biodiesel:<\/p>\n<p>[latex]{\\text{CH}}_{3}\\text{OH}+{\\text{CH}}_{3}{\\text{CH}}_{2}{\\text{OCOCH}}_{3}\\rightarrow{\\text{CH}}_{3}{\\text{OCOCH}}_{3}+{\\text{CH}}_{3}{\\text{CH}}_{2}\\text{OH}[\/latex]<\/p>\n<p>The rate law for the reaction between methanol and ethyl acetate is, under certain conditions, determined to be:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k\\left[{\\text{CH}}_{3}\\text{OH}\\right][\/latex]<\/p>\n<p>What is the order of reaction with respect to methanol and ethyl acetate, and what is the overall order of reaction?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q36335\">Show Answer<\/span><\/p>\n<div id=\"q36335\" class=\"hidden-answer\" style=\"display: none\">order in CH<sub>3<\/sub>OH = 1; order in CH<sub>3<\/sub>CH<sub>2<\/sub>OCOCH<sub>3<\/sub> = 0; overall order = 1<\/div>\n<\/div>\n<\/div>\n<p>It is sometimes helpful to use a more explicit algebraic method, often referred to as the method of initial rates, to determine the orders in rate laws. To use this method, we select two sets of rate data that differ in the concentration of only one reactant and set up a ratio of the two rates and the two rate laws. After canceling terms that are equal, we are left with an equation that contains only one unknown, the coefficient of the concentration that varies. We then solve this equation for the coefficient.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2:\u00a0Determining a Rate Law from Initial Rates<\/h3>\n<p>Ozone in the upper atmosphere is depleted when it reacts with nitrogen oxides. The rates of the reactions of nitrogen oxides with ozone are important factors in deciding how significant these reactions are in the formation of the ozone hole over Antarctica.<\/p>\n<div style=\"width: 362px\" 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\/23212248\/CNX_Chem_12_03_OzoneHole1.jpg\" alt=\"A view of Earth\u2019s southern hemisphere is shown. A nearly circular region of approximately half the diameter of the image is shown in shades of purple, with Antarctica appearing in a slightly lighter color than the surrounding ocean areas. Immediately outside this region is a narrow bright blue zone followed by a bright green zone. In the top half of the figure, the purple region extends slightly outward from the circle and the blue zone extends more outward to the right of the center as compared to the lower half of the image. In the upper half of the image, the majority of the space outside the purple region is shaded green, with a few small strips of interspersed blue regions. The lower half however shows the majority of the space outside the central purple zone in yellow, orange, and red. The red zones appear in the lower central and left regions outside the purple zone. To the lower right of this image is a color scale that is labeled \u201cTotal Ozone (Dobsone units).\u201d This scale begins at 0 and increases by 100\u2019s up to 700. At the left end of the scale, the value 0 shows a very deep purple color, 100 is indigo, 200 is blue, 300 is green, 400 is a yellow-orange, 500 is red, 600 is pink, and 700 is white.\" width=\"352\" height=\"360\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure\u00a01. Over the past several years, the atmospheric ozone concentration over Antarctica has decreased during the winter. This map shows the decreased concentration as a purple area. (credit: NASA)<\/p>\n<\/div>\n<p>One such reaction is the combination of nitric oxide, NO, with ozone, O<sub>3<\/sub>:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{NO(}g\\text{)}+{\\text{O}}_{3}\\left(g\\right)\\rightarrow{\\text{NO}}_{2}\\text{(}g\\text{)}+{\\text{O}}_{2}\\left(g\\right)[\/latex]<\/p>\n<p>This reaction has been studied in the laboratory, and the following rate data were determined at 25 \u00b0C.<\/p>\n<table id=\"fs-idm205685856\" class=\"medium unnumbered\" summary=\"This table has four columns and six rows. The first row is a header row, and it labels each column: \u201cTrial,\u201d \u201c[ N O ] ( mol \/ L),\u201d \u201c[ O subscript 3 ] ( mol \/ L ),\u201d and \u201ccapital delta [ N O subscript 2 ] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 ).\u201d Under the \u201cTrial\u201d column are the numbers: 1, 2, 3, 4, and 5. Under the \u201c[ N O ] ( mol \/ L)\u201d column are the numbers: 1.00 times ten to the negative six power; 1.00 times ten to the negative six power; 1.00 times ten to the negative six power; 2.00 times ten to the negative six power; and 3.00 times ten to the negative six power. Under the \u201c[ O subscript 3 ] ( mol \/ L )\u201d column are the numbers: 3.00 times ten to the negative six; 6.00 times ten to the negative six; 9.00 times ten to the negative six; 9.00 times ten to the negative six; and 9.00 times ten to the negative six. Under the column \u201ccapital delta [ N O subscript 2 ] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 )\u201d are the numbers: 6.60 times ten to the negative five; 1.32 times ten to the negative four; 1.98 times ten to the negative four; 3.96 times ten to the negative four; and 5.94 times ten to the negative four.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th>Trial<\/th>\n<th>[NO] (mol\/L)<\/th>\n<th>[O<sub>3<\/sub>] (mol\/L)<\/th>\n<th>[latex]\\frac{{\\text{Delta [NO}}_{\\text{2}}\\text{]}}{\\Delta t}\\left(\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>1<\/td>\n<td>1.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>3.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>6.60 \u00d7 10<sup>\u22125<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2<\/td>\n<td>1.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>6.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>1.32 \u00d7 10<sup>\u22124<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3<\/td>\n<td>1.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>9.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>1.98 \u00d7 10<sup>\u22124<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>4<\/td>\n<td>2.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>9.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>3.96 \u00d7 10<sup>\u22124<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>5<\/td>\n<td>3.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>9.00 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>5.94 \u00d7 10<sup>\u22124<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Determine the rate law and the rate constant for the reaction at 25 \u00b0C.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q646205\">Show Answer<\/span><\/p>\n<div id=\"q646205\" class=\"hidden-answer\" style=\"display: none\">\n<p>The rate law will have the form:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{m}{\\left[{\\text{O}}_{3}\\right]}^{n}[\/latex]<\/p>\n<p>We can determine the values of <em>m<\/em>, <em>n<\/em>, and <em>k<\/em> from the experimental data using the following three-part process:<\/p>\n<ol>\n<li>Determine the value of <em>m<\/em> from the data in which [NO] varies and [O<sub>3<\/sub>] is constant:\u00a0 In the last three experiments, [NO] varies while [O<sub>3<\/sub>] remains constant. When [NO] doubles from trial 3 to 4, the rate doubles, and when [NO] triples from trial 3 to 5, the rate also triples. Thus, the rate is also directly proportional to [NO], and <em>m<\/em> in the rate law is equal to 1.<\/li>\n<li>Determine the value of <em>n<\/em> from data in which [O<sub>3<\/sub>] varies and [NO] is constant: In the first three experiments, [NO] is constant and [O<sub>3<\/sub>] varies. The reaction rate changes in direct proportion to the change in [O<sub>3<\/sub>]. When [O<sub>3<\/sub>] doubles from trial 1 to 2, the rate doubles; when [O<sub>3<\/sub>] triples from trial 1 to 3, the rate increases also triples. Thus, the rate is directly proportional to [O<sub>3<\/sub>], and <em>n<\/em> is equal to 1.The rate law is thus:[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{1}{\\left[{\\text{O}}_{3}\\right]}^{1}=k\\left[\\text{NO}\\right]\\left[{\\text{O}}_{3}\\right][\/latex]<\/li>\n<li>Determine the value of <em>k<\/em> from one set of concentrations and the corresponding rate:[latex]\\begin{array}{cc}\\hfill k& =\\frac{\\text{rate}}{\\left[\\text{NO}\\right]\\left[{\\text{O}}_{3}\\right]}\\hfill \\\\ & =\\frac{0.660\\times {10}^{-4}\\cancel{{\\text{mol L}}^{-1}}{\\text{s}}^{-1}}{\\left(1.00\\times {10}^{-6}\\cancel{{\\text{mol L}}^{-1}}\\right)\\left(3.00\\times {10}^{-6}\\text{mol}{\\text{L}}^{-1}\\right)}\\hfill \\\\ & =\\text{2}.\\text{2}0\\times {10}^{7}\\text{L}{\\text{mol}}^{-1}{\\text{s}}^{-1}\\hfill \\end{array}[\/latex]The large value of <em>k<\/em> tells us that this is a fast reaction that could play an important role in ozone depletion if [NO] is large enough.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h4>Check Your Learning<\/h4>\n<p>Acetaldehyde decomposes when heated to yield methane and carbon monoxide according to the equation:<\/p>\n<p style=\"text-align: center;\">[latex]{\\text{CH}}_{3}\\text{CHO(}g\\text{)}\\rightarrow{\\text{CH}}_{4}\\text{(}g\\text{)}+\\text{CO(}g\\text{)}[\/latex]<\/p>\n<p>Determine the rate law and the rate constant for the reaction from the following experimental data:<\/p>\n<table id=\"fs-idm276791216\" class=\"medium unnumbered\" summary=\"This table has three columns and four rows. The first row is a header row, and it labels each column, \u201cTrial,\u201d \u201c[ C H subscript 3 C H O ] ( mol \/ L),\u201d and \u201cnegative capital delta [ C H subscript 3 C H O ] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 ).\u201d Under the \u201cTrial\u201d column are the numbers: 1, 2, and 3. Under the \u201c[ C H subscript 3 C H O ] ( mol \/ L)\u201d are the numbers: 1.75 times ten to the negative three; 3.50 times ten to the negative three; and 7.00 times ten to the negative three. Under the column \u201cnegative capital delta [ C H subscript 3 C H O ] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 )\u201d are the numbers: 2.06 times ten to the negative 11; 8.24 times ten to the negative 11; and 3.30 times ten to the negative ten.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th>Trial<\/th>\n<th>[CH<sub>3<\/sub>CHO] (mol\/L)<\/th>\n<th>[latex]-\\frac{{\\text{Delta [CH}}_{3}\\text{CHO]}}{\\Delta t}\\text{(mol}{\\text{L}}^{-1}{\\text{s}}^{-1}\\text{)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>1<\/td>\n<td>1.75 \u00d7 10<sup>\u22123<\/sup><\/td>\n<td>2.06 \u00d7 10<sup>\u221211<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2<\/td>\n<td>3.50 \u00d7 10<sup>\u22123<\/sup><\/td>\n<td>8.24 \u00d7 10<sup>\u221211<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3<\/td>\n<td>7.00 \u00d7 10<sup>\u22123<\/sup><\/td>\n<td>3.30 \u00d7 10<sup>\u221210<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q950345\">Show Answer<\/span><\/p>\n<div id=\"q950345\" class=\"hidden-answer\" style=\"display: none\">[latex]\\text{rate}=k{\\left[{\\text{CH}}_{3}\\text{CHO}\\right]}^{2}[\/latex] with <em>k<\/em> = 6.73 \u00d7 10<sup>-6<\/sup> L\/mol\/s<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3:\u00a0Determining Rate Laws from Initial Rates<\/h3>\n<p>Using the initial rates method and the experimental data, determine the rate law and the value of the rate constant for this reaction:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{2NO(}g\\text{)}+{\\text{Cl}}_{2}\\left(g\\right)\\rightarrow\\text{2NOCl(}g\\text{)}[\/latex]<\/p>\n<table id=\"fs-idm285249664\" class=\"medium unnumbered\" summary=\"This table has four columns and four rows. The first row is a header row, and it labels the columns, \u201cTrial,\u201d \u201c[ N O ] ( mol \/ L),\u201d \u201c[ C l subscript 2 ] ( mol \/ L ),\u201d and \u201cnegative capital delta [ N O] divided by capital delta t ( mol L superscript negative 1 s superscript negative 1 ).\u201d Under the column \u201cTrial\u201d are the numbers: 1, 2, and 3. Under the column \u201c[ N O ] ( mol \/ L)\u201d are the numbers: 0.10, 0.10, and 0.15. Under the column \u201c[ C l subscript 2 ] ( mol \/ L )\u201d are the numbers: 0.10, 0.15, and 0.10. Under the column \u201cnegative delta [ N O] divided by delta t ( mol L superscript negative 1 s superscript negative 1\u201d are the numbers: 0.00300, 0.00450, 0.00675.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th>Trial<\/th>\n<th>[NO] (mol\/L)<\/th>\n<th>[Cl<sub>2<\/sub>] (mol\/L)<\/th>\n<th>[latex]-\\frac{\\Delta\\left[\\text{NO}\\right]}{\\Delta t}\\left(\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>1<\/td>\n<td>0.10<\/td>\n<td>0.10<\/td>\n<td>0.00300<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2<\/td>\n<td>0.10<\/td>\n<td>0.15<\/td>\n<td>0.00450<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3<\/td>\n<td>0.15<\/td>\n<td>0.10<\/td>\n<td>0.00675<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q64050\">Show Answer<\/span><\/p>\n<div id=\"q64050\" class=\"hidden-answer\" style=\"display: none\">\n<p>The rate law for this reaction will have the form:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{m}{\\left[{\\text{Cl}}_{2}\\right]}^{n}[\/latex]<\/p>\n<p>As in Example 2, we can approach this problem in a stepwise fashion, determining the values of <em>m<\/em> and <em>n<\/em> from the experimental data and then using these values to determine the value of <em>k<\/em>. In this example, however, we will use a different approach to determine the values of <em>m<\/em> and <em>n<\/em>:<\/p>\n<ol>\n<li>Determine the value of <em>m<\/em> from the data in which [NO] varies and [Cl<sub>2<\/sub>] is constant: We can write the ratios with the subscripts <em>x<\/em> and <em>y<\/em> to indicate data from two different trials:[latex]\\frac{{\\text{rate}}_{x}}{{\\text{rate}}_{y}}=\\frac{k{\\text{[}\\text{NO}\\text{]}}_{x}^{m}{\\text{[}{\\text{Cl}}_{2}\\text{]}}_{x}^{n}}{k{\\text{[}\\text{NO}\\text{]}}_{y}^{m}{\\text{[}{\\text{Cl}}_{2}\\text{]}}_{y}^{n}}[\/latex]Using the third trial and the first trial, in which [Cl<sub>2<\/sub>] does not vary, gives:[latex]\\frac{\\text{rate 3}}{\\text{rate 1}}=\\frac{0.00675}{0.00300}=\\frac{k{\\left(0.15\\right)}^{m}{\\left(0.10\\right)}^{n}}{k{\\left(0.10\\right)}^{m}{\\left(0.10\\right)}^{n}}[\/latex]\n<p>After canceling equivalent terms in the numerator and denominator, we are left with:<\/p>\n<p>[latex]\\frac{0.00675}{0.00300}=\\frac{{\\left(0.15\\right)}^{m}}{{\\left(0.10\\right)}^{m}}[\/latex]<\/p>\n<p>which simplifies to:<\/p>\n<p>[latex]2.25={\\left(1.5\\right)}^{m}[\/latex]<\/p>\n<p>We can use natural logs to determine the value of the exponent <em>m<\/em>:<\/p>\n<p>[latex]\\begin{array}{ccc}\\hfill \\text{ln}\\left(2.25\\right)& =& m\\text{ln}\\left(1.5\\right)\\hfill \\\\ \\hfill \\frac{\\text{ln}\\left(2.25\\right)}{\\text{ln}\\left(1.5\\right)}& =& m\\hfill \\\\ \\hfill 2& =& m\\hfill \\end{array}[\/latex]<\/p>\n<p>We can confirm the result easily, since:<\/p>\n<p>1.5<sup>2<\/sup> = 2.25<\/li>\n<li>Determine the value of <em>n<\/em> from data in which [Cl<sub>2<\/sub>] varies and [NO] is constant:[latex]\\frac{\\text{rate 2}}{\\text{rate 1}}=\\frac{0.00450}{0.00300}=\\frac{k{\\left(0.10\\right)}^{m}{\\left(0.15\\right)}^{n}}{k{\\left(0.10\\right)}^{m}{\\left(0.10\\right)}^{n}}[\/latex]Cancelation gives:[latex]\\frac{0.0045}{0.0030}=\\frac{{\\left(0.15\\right)}^{n}}{{\\left(0.10\\right)}^{n}}[\/latex]\n<p>which simplifies to:<\/p>\n<p>1.5 = (1.5)<sup><em>n<\/em><\/sup><\/p>\n<p>Thus <em>n<\/em> must be 1, and the form of the rate law is:<\/p>\n<p>[latex]\\text{Rate}=k{\\left[\\text{NO}\\right]}^{m}{\\left[{\\text{Cl}}_{2}\\right]}^{n}=k{\\left[\\text{NO}\\right]}^{2}\\left[{\\text{Cl}}_{2}\\right][\/latex]<\/li>\n<li>Determine the numerical value of the rate constant <em>k<\/em> with appropriate units:\u00a0The units for the rate of a reaction are mol\/L\/ s. The units for <em>k<\/em> are whatever is needed so that substituting into the rate law expression affords the appropriate units for the rate. In this example, the concentration units are mol<sup>3<\/sup>\/L<sup>3<\/sup>. The units for <em>k<\/em> should be mol<sup>-2<\/sup> L<sup>2\/<\/sup>s so that the rate is in terms of mol\/L\/ s.To determine the value of <em>k<\/em> once the rate law expression has been solved, simply plug in values from the first experimental trial and solve for <em>k<\/em>:<br \/>\n[latex]\\begin{array}{ccc}\\hfill 0.00300\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}& =& k{\\left(0.10\\text{mol}{\\text{L}}^{-1}\\right)}^{2}{\\left(0.10\\text{mol}{\\text{L}}^{-1}\\right)}^{1}\\hfill \\\\ \\hfill k& =& 3.0{\\text{mol}}^{-2}{\\text{L}}^{2}{\\text{s}}^{-1}\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h4>Check Your Learning<\/h4>\n<p>Use the provided initial rate data to derive the rate law for the reaction whose equation is:<\/p>\n<p style=\"text-align: center;\">[latex]{\\text{OCl}}^{-}\\left(aq\\right)+{\\text{I}}^{-}\\left(aq\\right)\\rightarrow{\\text{OI}}^{-}\\left(aq\\right)+{\\text{Cl}}^{-}\\left(aq\\right)[\/latex]<\/p>\n<table id=\"fs-idm275492048\" class=\"medium unnumbered\" summary=\"This table has four columns and four rows. The first row is a header row and it labels each column, \u201cTrial,\u201d \u201c[ O C l superscript negative sign ] ( mol \/ L),\u201d \u201c[ I superscript negative sign ] ( mol \/ l ),\u201d and \u201cInitial Rate ( mol \/ L \/ s).\u201d Under the column \u201cTrial\u201d are the numbers: 1, 2, and 3. Under the column \u201c[ O C l superscript negative sign ] ( mol \/ L)\u201d are the numbers 0.0040, 0.0020, and 0.0020. Under the column \u201c[ I superscript negative sign ] ( mol \/ l )\u201d are the numbers: 0.0020, 0.0040, and 0.0020. Under the column \u201cInitial Rate ( mol \/ L \/ s)\u201d are the numbers: 0.00184, 0.00092, and 0.00046.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th>Trial<\/th>\n<th>[OCl<sup>&#8211;<\/sup>] (mol\/L)<\/th>\n<th>[I<sup>&#8211;<\/sup>] (mol\/L)<\/th>\n<th>Initial Rate (mol\/L\/s)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>1<\/td>\n<td>0.0040<\/td>\n<td>0.0020<\/td>\n<td>0.00184<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2<\/td>\n<td>0.0020<\/td>\n<td>0.0040<\/td>\n<td>0.00092<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3<\/td>\n<td>0.0020<\/td>\n<td>0.0020<\/td>\n<td>0.00046<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Determine the rate law expression and the value of the rate constant <em>k<\/em> with appropriate units for this reaction.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q858130\">Show Answer<\/span><\/p>\n<div id=\"q858130\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{\\text{rate 2}}{\\text{rate 3}}=\\frac{0.00092}{0.00046}=\\frac{k{\\left(0.0020\\right)}^{x}{\\left(0.0040\\right)}^{y}}{k{\\left(0.0020\\right)}^{x}{\\left(0.0020\\right)}^{y}}[\/latex]<\/p>\n<p>2.00 = 2.00<sup><em>y<\/em><\/sup><\/p>\n<p>y = 1<\/p>\n<p>[latex]\\frac{\\text{rate 1}}{\\text{rate 2}}=\\frac{0.00184}{0.00092}=\\frac{k{\\left(0.0040\\right)}^{x}{\\left(0.0020\\right)}^{y}}{k{\\left(0.0020\\right)}^{x}{\\left(0.0040\\right)}^{y}}[\/latex]<\/p>\n<p>[latex]\\begin{array}{ccc}\\hfill 2.00& =& \\frac{{2}^{x}}{{2}^{y}}\\hfill \\\\ \\hfill 2.00& =& \\frac{{2}^{x}}{{2}^{1}}\\hfill \\\\ \\hfill 4.00& =& {2}^{x}\\hfill \\\\ x& =& 2\\hfill \\end{array}[\/latex]<\/p>\n<p>Substituting the concentration data from trial 1 and solving for <em>k<\/em> yields:<\/p>\n<p>[latex]\\begin{array}{ccc}\\hfill \\text{rate}& =& k{\\left[{\\text{OCl}}^{-}\\right]}^{2}{\\left[{\\text{I}}^{-}\\right]}^{1}\\hfill \\\\ 0.00184\\hfill & =& k{\\left(0.0040\\right)}^{2}{\\left(0.0020\\right)}^{1}\\hfill \\\\ \\hfill k& =& 5.75{10}^{4}{\\text{mol}}^{-2}{\\text{L}}^{2}{\\text{s}}^{-1}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Reaction Order and Rate Constant Units<\/h2>\n<p>In some of our examples, the reaction orders in the rate law happen to be the same as the coefficients in the chemical equation for the reaction. This is merely a coincidence and very often not the case.<\/p>\n<p>Rate laws may exhibit fractional orders for some reactants, and negative reaction orders are sometimes observed when an increase in the concentration of one reactant causes a decrease in reaction rate. A few examples illustrating these points are provided:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\\\ {\\text{NO}}_{2}+\\text{CO}\\rightarrow\\text{NO}+{\\text{CO}}_{\\text{2}}\\text{rate}=\\text{k}{\\left[{\\text{NO}}_{2}\\right]}^{2}\\\\ {\\text{CH}}_{3}\\text{CHO}\\rightarrow{\\text{CH}}_{4}+\\text{CO}\\text{rate}=\\text{k}{\\left[{\\text{CH}}_{3}\\text{CHO}\\right]}^{2}\\\\ {\\text{2N}}_{2}{\\text{O}}_{5}\\rightarrow{\\text{2NO}}_{2}+{\\text{O}}_{\\text{2}}\\text{rate}=\\text{k}\\left[{\\text{N}}_{2}{\\text{O}}_{5}\\right]\\\\ {\\text{2NO}}_{2}+{\\text{F}}_{2}\\rightarrow{\\text{2NO}}_{2}\\text{F}\\text{rate}=\\text{k}\\left[{\\text{NO}}_{2}\\right]\\left[{\\text{F}}_{2}\\right]\\\\ {\\text{2NO}}_{2}\\text{Cl}\\rightarrow{\\text{2NO}}_{2}+{\\text{Cl}}_{2}\\text{rate}=\\text{k}\\left[{\\text{NO}}_{2}\\text{Cl}\\right]\\end{array}[\/latex]<\/p>\n<p>It is important to note that <em>rate laws are determined by experiment only and are not reliably predicted by reaction stoichiometry.<\/em><\/p>\n<p>Reaction orders also play a role in determining the units for the rate constant <em>k<\/em>. In Example 2, a second-order reaction, we found the units for <em>k<\/em> to be [latex]\\text{L}{\\text{mol}}^{-4}{\\text{s}}^{-1},[\/latex] whereas in Example 3, a third order reaction, we found the units for <em>k<\/em> to be mol<sup>\u22122<\/sup> L<sup>2<\/sup>\/s. More generally speaking, the units for the rate constant for a reaction of order [latex]\\left(m+n\\right)[\/latex] are [latex]{\\text{mol}}^{1-\\left(m+n\\right)}{\\text{L}}^{\\left(m+n\\right)-1}{\\text{s}}^{-1}[\/latex]. Table\u00a01\u00a0summarizes the rate constant units for common reaction orders.<\/p>\n<table id=\"fs-idm233742304\" summary=\"This table has two columns and four rows. The first row is a header row and it labels each column, \u201cReaction order,\u201d and \u201cUnits of k.\u201d Under the \u201cReaction Order\u201d column are: \u201c( m + n ),\u201d \u201czero,\u201d \u201cfirst,\u201d \u201csecond,\u201d and, \u201cthird.\u201d Under the \u201cUnits of k\u201d column are: \u201cmol superscript 1 minus ( m + n) L superscript ( m + n) minus 1 s superscript negative one,\u201d \u201cmol \/ L \/ s,\u201d \u201cs superscript negative 1,\u201d \u201cL \/ mol \/ s,\u201d and \u201cmol superscript negative 2 L superscript 2 s superscript negative 1.\u201d\">\n<thead>\n<tr>\n<th colspan=\"2\">Table\u00a01. Rate Constants for Common Reaction Orders<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<th>Reaction Order<\/th>\n<th>Units of <em>k<\/em><\/th>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\left(m+n\\right)[\/latex]<\/td>\n<td>[latex]{\\text{mol}}^{1-\\left(m+n\\right)}{\\text{L}}^{\\left(m+n\\right)-1}{\\text{s}}^{-1}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>zero<\/td>\n<td>mol\/L\/s<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>first<\/td>\n<td>s<sup>\u22121<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>second<\/td>\n<td>L\/mol\/s<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>third<\/td>\n<td>mol<sup>\u22122<\/sup> L<sup>2<\/sup> s<sup>\u22121<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note that the units in the table can also be expressed in terms of molarity (<em>M<\/em>) instead of mol\/L. Also, units of time other than the second (such as minutes, hours, days) may be used, depending on the situation.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts and Summary<\/h3>\n<p>Rate laws provide a mathematical description of how changes in the amount of a substance affect the rate of a chemical reaction. Rate laws are determined experimentally and cannot be predicted by reaction stoichiometry. The order of reaction describes how much a change in the amount of each substance affects the overall rate, and the overall order of a reaction is the sum of the orders for each substance present in the reaction. Reaction orders are typically first order, second order, or zero order, but fractional and even negative orders are possible.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<ol>\n<li>How do the rate of a reaction and its rate constant differ?<\/li>\n<li>Doubling the concentration of a reactant increases the rate of a reaction four times. With this knowledge, answer the following questions:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>What is the order of the reaction with respect to that reactant?<\/li>\n<li>Tripling the concentration of a different reactant increases the rate of a reaction three times. What is the order of the reaction with respect to that reactant?<\/li>\n<\/ol>\n<\/li>\n<li>Tripling the concentration of a reactant increases the rate of a reaction nine times. With this knowledge, answer the following questions:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>What is the order of the reaction with respect to that reactant?<\/li>\n<li>Increasing the concentration of a reactant by a factor of four increases the rate of a reaction four times. What is the order of the reaction with respect to that reactant?<\/li>\n<\/ol>\n<\/li>\n<li>How much and in what direction will each of the following affect the rate of the reaction: [latex]\\text{CO(}g\\text{)}+{\\text{NO}}_{2}\\text{(}g\\text{)}\\rightarrow{\\text{CO}}_{2}\\text{(}g\\text{)}+\\text{NO(}g\\text{)}[\/latex] if the rate law for the reaction is [latex]\\text{rate}=k{\\left[{\\text{NO}}_{2}\\right]}^{2}?[\/latex]\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Decreasing the pressure of NO<sub>2<\/sub> from 0.50 atm to 0.250 atm.<\/li>\n<li>Increasing the concentration of CO from 0.01 <em>M<\/em> to 0.03 <em>M<\/em>.<\/li>\n<\/ol>\n<\/li>\n<li>How will each of the following affect the rate of the reaction: [latex]\\text{CO(}g\\text{)}+{\\text{NO}}_{2}\\text{(}g\\text{)}\\rightarrow{\\text{CO}}_{2}\\text{(}g\\text{)}+\\text{NO(}g\\text{)}[\/latex] if the rate law for the reaction is [latex]\\text{rate}=k\\left[{\\text{NO}}_{2}\\right]\\left[\\text{CO}\\right][\/latex] ?\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Increasing the pressure of NO<sub>2<\/sub> from 0.1 atm to 0.3 atm<\/li>\n<li>Increasing the concentration of CO from 0.02 <em>M<\/em> to 0.06 <em>M<\/em>.<\/li>\n<\/ol>\n<\/li>\n<li>Regular flights of supersonic aircraft in the stratosphere are of concern because such aircraft produce nitric oxide, NO, as a byproduct in the exhaust of their engines. Nitric oxide reacts with ozone, and it has been suggested that this could contribute to depletion of the ozone layer. The reaction [latex]\\text{NO}+{\\text{O}}_{3}\\rightarrow{\\text{NO}}_{2}+{\\text{O}}_{2}[\/latex] is first order with respect to both NO and O<sub>3<\/sub> with a rate constant of 2.20 \u00d7 10<sup>7<\/sup> L\/mol\/s. What is the instantaneous rate of disappearance of NO when [NO] = 3.3 \u00d7 10<sup>\u22126<\/sup><em>M<\/em> and [O<sub>3<\/sub>] = 5.9 \u00d7 10<sup>\u22127<\/sup><em>M<\/em>?<\/li>\n<li>Radioactive phosphorus is used in the study of biochemical reaction mechanisms because phosphorus atoms are components of many biochemical molecules. The location of the phosphorus (and the location of the molecule it is bound in) can be detected from the electrons (beta particles) it produces:[latex]{}_{15}^{32}\\text{P}\\rightarrow{}_{16}^{32}\\text{S}+{\\text{e}}^{-}[\/latex]Rate = 4.85 \u00d7 10<sup>\u22122<\/sup> day<sup>-1<\/sup>[<sup>32<\/sup>P]What is the instantaneous rate of production of electrons in a sample with a phosphorus concentration of 0.0033 <em>M<\/em>?<\/li>\n<li>The rate constant for the radioactive decay of <sup>14<\/sup>C is 1.21 \u00d7 10<sup>\u22124<\/sup> year<sup>\u22121<\/sup>. The products of the decay are nitrogen atoms and electrons (beta particles):[latex]{}_{6}^{14}\\text{C}\\rightarrow{}_{6}^{14}\\text{N}+{\\text{e}}^{-}[\/latex][latex]\\text{rate}=k\\left[{}_{6}^{14}\\text{C}\\right][\/latex]What is the instantaneous rate of production of N atoms in a sample with a carbon-14 content of 6.5 \u00d7 10<sup>\u22129<\/sup><em>M<\/em>?<\/li>\n<li>The decomposition of acetaldehyde is a second order reaction with a rate constant of 4.71 \u00d7 10<sup>\u22128<\/sup>\u00a0L\/mol\/s. What is the instantaneous rate of decomposition of acetaldehyde in a solution with a concentration of 5.55 \u00d7 10<sup>\u22124<\/sup><em>M<\/em>?<\/li>\n<li>Alcohol is removed from the bloodstream by a series of metabolic reactions. The first reaction produces acetaldehyde; then other products are formed. The following data have been determined for the rate at which alcohol is removed from the blood of an average male, although individual rates can vary by 25\u201330%. Women metabolize alcohol a little more slowly than men:<br \/>\n<table>\n<tbody>\n<tr valign=\"top\">\n<td>[C<sub>2<\/sub>H<sub>5<\/sub>OH] (<em>M<\/em>)<\/td>\n<td>4.4 \u00d7 10<sup>\u22122<\/sup><\/td>\n<td>3.3 \u00d7 10<sup>\u22122<\/sup><\/td>\n<td>2.2 \u00d7 10<sup>\u22122<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rate (mol\/L\/h)<\/td>\n<td>2.0 \u00d7 10<sup>\u22122<\/sup><\/td>\n<td>2.0 \u00d7 10<sup>\u22122<\/sup><\/td>\n<td>2.0 \u00d7 10<sup>\u22122<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Determine the rate equation, the rate constant, and the overall order for this reaction.<\/li>\n<li>Under certain conditions the decomposition of ammonia on a metal surface gives the following data:<br \/>\n<table>\n<tbody>\n<tr valign=\"top\">\n<td>[NH<sub>3<\/sub>] (<em>M<\/em>)<\/td>\n<td>1.0 \u00d7 10<sup>\u22123<\/sup><\/td>\n<td>2.0 \u00d7 10<sup>\u22123<\/sup><\/td>\n<td>3.0 \u00d7 10<sup>\u22123<\/sup><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rate (mol\/L\/h<sup>1<\/sup>)<\/td>\n<td>1.5 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>1.5 \u00d7 10<sup>\u22126<\/sup><\/td>\n<td>1.5 \u00d7 10<sup>\u22126<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Determine the rate equation, the rate constant, and the overall order for this reaction.<\/li>\n<li>Nitrosyl chloride, NOCl, decomposes to NO and Cl<sub>2<\/sub>.[latex]\\text{2NOCl(}g\\text{)}\\rightarrow\\text{2NO(}g\\text{)}+{\\text{Cl}}_{2}\\text{(}g\\text{)}[\/latex]Determine the rate equation, the rate constant, and the overall order for this reaction from the following data:<br \/>\n<table>\n<tbody>\n<tr valign=\"top\">\n<td>[NOCl] (<em>M<\/em>)<\/td>\n<td>0.10<\/td>\n<td>0.20<\/td>\n<td>0.30<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rate (mol\/L\/h)<\/td>\n<td>8.0 \u00d7 10<sup>-10<\/sup><\/td>\n<td>3.2 \u00d7 10<sup>\u22129<\/sup><\/td>\n<td>7.2 \u00d7 10<sup>\u22129<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>From the following data, determine the rate equation, the rate constant, and the order with respect to <em>A<\/em> for the reaction [latex]A\\rightarrow 2C.[\/latex]<br \/>\n<table>\n<tbody>\n<tr valign=\"top\">\n<td>[<em>A<\/em>] (<em>M<\/em>)<\/td>\n<td>[latex]1.33\\times {10}^{-2}[\/latex]<\/td>\n<td>[latex]\\text{2.66}\\times {10}^{-2}[\/latex]<\/td>\n<td>[latex]\\text{3.99}\\times {10}^{-2}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rate (mol\/L\/h)<\/td>\n<td>[latex]\\text{3.80}\\times {10}^{-7}[\/latex]<\/td>\n<td>[latex]\\text{1.52}\\times {10}^{-6}[\/latex]<\/td>\n<td>[latex]\\text{3.42}\\times {10}^{-6}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Nitrogen(II) oxide reacts with chlorine according to the equation:[latex]\\text{2NO(}g\\text{)}+{\\text{Cl}}_{2}\\text{(}g\\text{)}\\rightarrow\\text{2NOCl(}g\\text{)}[\/latex]The following initial rates of reaction have been observed for certain reactant concentrations:<br \/>\n<table>\n<thead>\n<tr valign=\"top\">\n<th>[NO] (mol\/L<sup>1<\/sup>)<\/th>\n<th>[Cl<sub>2<\/sub>] (mol\/L)<\/th>\n<th>Rate (mol\/L\/h)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>0.50<\/td>\n<td>0.50<\/td>\n<td>1.14<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1.00<\/td>\n<td>0.50<\/td>\n<td>4.56<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1.00<\/td>\n<td>1.00<\/td>\n<td>9.12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What is the rate equation that describes the rate\u2019s dependence on the concentrations of NO and Cl<sub>2<\/sub>? What is the rate constant? What are the orders with respect to each reactant?<\/li>\n<li>Hydrogen reacts with nitrogen monoxide to form dinitrogen monoxide (laughing gas) according to the equation: [latex]{\\text{H}}_{2}\\text{(}g\\text{)}+\\text{2NO(}g\\text{)}\\rightarrow{\\text{N}}_{2}\\text{O(}g\\text{)}+{\\text{H}}_{2}\\text{O(}g\\text{)}[\/latex]Determine the rate equation, the rate constant, and the orders with respect to each reactant from the following data:<br \/>\n<table>\n<tbody>\n<tr valign=\"top\">\n<td>[NO] (<em>M<\/em>)<\/td>\n<td>0.30<\/td>\n<td>0.60<\/td>\n<td>0.60<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[H<sub>2<\/sub>] (<em>M<\/em>)<\/td>\n<td>0.35<\/td>\n<td>0.35<\/td>\n<td>0.70<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rate (mol\/L\/s)<\/td>\n<td>[latex]2.835\\times {\\text{10}}^{-3}[\/latex]<\/td>\n<td>[latex]1.134\\times {\\text{10}}^{-2}[\/latex]<\/td>\n<td>[latex]2.268\\times {\\text{10}}^{-2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>For the reaction [latex]A\\rightarrow B+C,[\/latex] the following data were obtained at 30 \u00b0C:<br \/>\n<table>\n<tbody>\n<tr valign=\"top\">\n<td>[<em>A<\/em>] (<em>M<\/em>)<\/td>\n<td>0.230<\/td>\n<td>0.356<\/td>\n<td>0.557<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rate (mol\/L\/s)<\/td>\n<td>4.17\u00a0\u00d7 10<sup>\u22124<\/sup><\/td>\n<td>9.99 \u00d7 10<sup>\u22124<\/sup><\/td>\n<td>2.44 \u00d7 10<sup>\u22123<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>What is the order of the reaction with respect to [<em>A<\/em>], and what is the rate equation?<\/li>\n<li>What is the rate constant?<\/li>\n<\/ol>\n<\/li>\n<li>For the reaction [latex]Q\\rightarrow W+X,[\/latex] the following data were obtained at 30 \u00b0C:<br \/>\n<table>\n<tbody>\n<tr valign=\"top\">\n<td>[<em>Q<\/em>]<sub>initial<\/sub> (<em>M<\/em>)<\/td>\n<td>0.170<\/td>\n<td>0.212<\/td>\n<td>0.357<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rate (mol\/L\/s)<\/td>\n<td>6.68 \u00d7 10<sup>\u22123<\/sup><\/td>\n<td>1.04 \u00d7 10<sup>\u22122<\/sup><\/td>\n<td>2.94 \u00d7 10<sup>\u22122<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>What is the order of the reaction with respect to [<em>Q<\/em>], and what is the rate equation?<\/li>\n<li>What is the rate constant?<\/li>\n<\/ol>\n<\/li>\n<li>The rate constant for the first-order decomposition at 45 \u00b0C of dinitrogen pentoxide, N<sub>2<\/sub>O<sub>5<\/sub>, dissolved in chloroform, CHCl<sub>3<\/sub>, is 6.2 \u00d7 10<sup>\u22124<\/sup> min<sup>\u22121<\/sup>.[latex]{\\text{2N}}_{2}{\\text{O}}_{5}\\rightarrow{\\text{4NO}}_{2}+{\\text{O}}_{2}[\/latex]What is the rate of the reaction when [N<sub>2<\/sub>O<sub>5<\/sub>] = 0.40 <em>M<\/em>?<\/li>\n<li>The annual production of HNO<sub>3<\/sub> in 2013 was 60 million metric tons Most of that was prepared by the following sequence of reactions, each run in a separate reaction vessel.\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{\\text{4NH}}_{3}\\text{(}g\\text{)}+{\\text{5O}}_{2}\\text{(}g\\text{)}\\rightarrow\\text{4NO(}g\\text{)}+{\\text{6H}}_{2}\\text{O(}g\\text{)}[\/latex]<\/li>\n<li>[latex]\\text{2NO(}g\\text{)}+{\\text{O}}_{2}\\text{(}g\\text{)}\\rightarrow{\\text{2NO}}_{2}\\text{(}g\\text{)}[\/latex]<\/li>\n<li>[latex]{\\text{3NO}}_{2}\\text{(}g\\text{)}+{\\text{H}}_{2}\\text{O(}l\\text{)}\\rightarrow{\\text{2HNO}}_{3}\\text{(}aq\\text{)}+\\text{NO(}g\\text{)}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>The first reaction is run by burning ammonia in air over a platinum catalyst. This reaction is fast. The reaction in equation (c) is also fast. The second reaction limits the rate at which nitric acid can be prepared from ammonia. If equation (b) is second order in NO and first order in O<sub>2<\/sub>, what is the rate of formation of NO<sub>2<\/sub> when the oxygen concentration is 0.50 M and the nitric oxide concentration is 0.75 <em>M<\/em>? The rate constant for the reaction is 5.8 \u00d7 10<sup>\u22126<\/sup> L<sup>2<\/sup>\/mol<sup>2<\/sup>\/s.<\/li>\n<li>The following data have been determined for the reaction:[latex]{\\text{I}}^{-}+{\\text{OCl}}^{-}\\rightarrow{\\text{IO}}^{-}+{\\text{Cl}}^{-}[\/latex]<br \/>\n<table id=\"fs-idm189410736\" class=\"medium unnumbered\" summary=\"This table has four columns and four rows. The first column labels each row. The first cell in the first column is blank, and then each row is labeled, \u201c[ I superscript negative sign ] subscript initial ( M ),\u201d \u201c[ O C l superscript negative sign ] subscript initial ( M ),\u201d and, \u201cRate ( mol \/ L \/ s ).\u201d To the right of the blank cell are the numbers: 1, 2, and 3. To the right of the header, \u201c[ I superscript negative sign ] subscript initial ( M ),\u201d are the numbers: 0.10, 0.20, and 0.30. To the right of the header, \u201c[ O C l superscript negative sign ] subscript initial ( M ),\u201d are the numbers: 0.050, 0.050, and 0.010. To the right of the header, \u201cRate ( mol \/ L \/ s ),\u201d are the numbers: 3.05 times ten to the negative 4; 6.20 times ten to the negative 4; and 1.83 times ten to the negative 4.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th><\/th>\n<th>1<\/th>\n<th>2<\/th>\n<th>3<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]{\\left[{\\text{I}}^{-}\\right]}_{\\text{initial}}[\/latex] (<em>M<\/em>)<\/td>\n<td>0.10<\/td>\n<td>0.20<\/td>\n<td>0.30<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\left[{\\text{OCl}}^{-}\\right]}_{\\text{initial}}[\/latex] (<em>M<\/em>)<\/td>\n<td>0.050<\/td>\n<td>0.050<\/td>\n<td>0.010<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Rate (mol\/L\/s)<\/td>\n<td>3.05 \u00d7 10<sup>\u22124<\/sup><\/td>\n<td>6.20 \u00d7 10<sup>\u22124<\/sup><\/td>\n<td>1.83 \u00d7 10<sup>\u22124<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Determine the rate equation and the rate constant for this reaction.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q453158\">Show Selected Answers<\/span><\/p>\n<div id=\"q453158\" class=\"hidden-answer\" style=\"display: none\">\n<p>2.\u00a0(a) Since the concentration of the reactant doubled and the rate quadrupled, we can conclude that the order with respect to the reactant is 2, since 2<sup>2<\/sup> = 4.<\/p>\n<p>[latex]\\begin{array}{l}\\text{rate}=k{\\left[\\text{reactant}\\right]}^{m}\\\\ \\text{4}\\left(\\text{rate}\\right)=k{\\left[\\text{2}\\left(\\text{reactant}\\right)\\right]}^{2}\\end{array}[\/latex]<\/p>\n<p>(b) Since the concentration of the reactant and the rate both tripled, we can conclude that [latex]m=1,[\/latex] and the order with respect to this reactant is 1.<\/p>\n<p>[latex]\\begin{array}{l}\\text{rate}=k{\\left[\\text{reactant}\\right]}^{m}\\\\ \\text{3}\\left(\\text{rate}\\right)=k{\\left[\\text{3}\\left(\\text{reactant}\\right)\\right]}^{1}\\end{array}[\/latex]<\/p>\n<p>4.\u00a0(a) [latex]\\frac{{\\text{rate}}_{2}}{{\\text{rate}}_{1}}=\\frac{\\text{k}{\\text{[}0.25{\\text{NO}}_{2}\\text{]}}^{2}}{\\text{k}{\\text{[}0.50{\\text{NO}}_{2}\\text{]}}^{2}}=\\frac{0.0675}{0.25}=\\frac{1}{4}[\/latex]<\/p>\n<p>Since Rate<sub>1<\/sub> is four times as large as Rate<sub>2<\/sub>, the process reduces the rate by a factor of 4.<\/p>\n<p>(b) Since CO does not appear in the rate law, the rate is not affected.<\/p>\n<p>6.\u00a0Rate = <em>k<\/em> [NO][O<sub>3<\/sub>] = 2.20 \u00d7 10<sup>7<\/sup> L\/mol\/s[3.3 \u00d7 10<sup>\u22126<\/sup><em>M<\/em>][5.9 \u00d7 10<sup>-7<\/sup><em>M<\/em>] = 4.3 \u00d7 10<sup>\u22125<\/sup> mol\/L\/s<\/p>\n<p>8.\u00a0rate = 1.21 \u00d7 10<sup>\u22124<\/sup> year<sup>\u22121<\/sup> [6.5 \u00d7 10<sup>-9<\/sup><em>M<\/em>] = 7.9 \u00d7 10<sup>\u221213<\/sup> mol\/L\/year<\/p>\n<p>10.\u00a0The rate is independent of the concentration. Therefore, rate = <em>k<\/em> ; <em>k<\/em> = 2.0 \u00d7 10<sup>-2<\/sup> mol\/L\/h (about 0.9 g\/L\/h for the average male); The reaction is zero order\u2014that is, it does not depend on the concentration of any reagent.<\/p>\n<p>12.\u00a0The object of this problem is to use the general rate expression: rate = <em>k<\/em>[NOC]<sup>m<\/sup>, first to determine the value of <em>m<\/em> and then, by substituting data from one experiment into the equation, to find the value of <em>k<\/em>. The data listed as substituted into the rate equation give:<\/p>\n<p>Experiment 1: 8.0 \u00d7 10<sup>\u221210<\/sup> mol\/L\/s = <em>k<\/em>[0.10 mol\/L]<sup><em>m<\/em><\/sup><\/p>\n<p>Experiment 2: 3.20 \u00d7 10<sup>\u22129<\/sup> mol\/L\/s = <em>k<\/em>[0.20 mol\/L]<sup><em>m<\/em><\/sup><\/p>\n<p>Experiment 3: 7.2 \u00d7 10<sup>\u22129<\/sup> mol\/L\/s = <em>k<\/em>[0.30 mol\/L]<sup><em>m<\/em><\/sup><\/p>\n<p>The value of <em>m<\/em> can be found by inspection. Examining Experiments 1 and 2, it is found that the rate increases by a factor of four as the concentration increases by a factor of two; from Experiments 1 and 3, the rate increases by a factor of nine while the concentration increases by a factor of three. This can happen only if <em>m<\/em> is 2. The value of <em>k<\/em> as calculated from the first set of data is:<\/p>\n<p style=\"text-align: center;\">[latex]k=\\frac{8.0\\times {\\text{10}}^{-10}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}{{\\left[0.10\\text{mol}{\\text{L}}^{-1}\\right]}^{2}}=8.0\\times {\\text{10}}^{-8}\\text{L}{\\text{mol}}^{-1}{\\text{s}}^{-1}[\/latex]<\/p>\n<p>rate = <em>k<\/em>[NOC]<sup>2<\/sup>; <em>k<\/em> = 8.0 \u00d7 10<sup>\u22128<\/sup> L\/mol\/s; second order<\/p>\n<p>14.\u00a0The rate law has the general form:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{m}{\\left[{\\text{Cl}}_{2}\\right]}^{n}[\/latex]<\/p>\n<p>Comparing the data in rows 1 and 2, [Cl<sub>2<\/sub>] remains constant, [NO] doubles, and the rate becomes four times as large, so <em>m<\/em> = 2. Comparing data in rows 2 and 3, [NP] remains constant, [Cl<sub>2<\/sub>] doubles, and the rate doubles, so <em>n<\/em> = 1. The rate equation is:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{rate}=k{\\left[\\text{NO}\\right]}^{2}\\left[{\\text{Cl}}_{2}\\right][\/latex]<\/p>\n<p>Data from row 1 are used to determine <em>k<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]k=\\frac{\\text{rate}}{{\\text{[NO]}}^{2}{\\text{[Cl]}}_{2}}=\\frac{1.14\\text{mol}{\\text{L}}^{-1}{\\text{h}}^{-1}}{{\\left(0.50\\text{mol}{\\text{L}}^{-1}\\right)}^{2}\\left(0.50\\text{mol}{\\text{L}}^{-1}\\right)}=9.12{\\text{L}}^{2}{\\text{mol}}^{-2}{\\text{h}}^{-1}[\/latex]<\/p>\n<p>[latex]\\text{rate}=k{\\text{[NO]}}^{2}{\\text{[Cl]}}_{2}\\text{;}[\/latex] second order in NO; first order in Cl<sub>2<\/sub><\/p>\n<p>16.\u00a0(a) The rate equation will be of the form rate = <em>k<\/em>[A]<sup><em>m<\/em><\/sup> and <em>m<\/em> will be the same for all three sets of experimental data. Therefore, we can write:<\/p>\n<p>Experiment 1: 4.17 \u00d7 10<sup>\u22124<\/sup> mol\/L\/s = <em>k<\/em>[0.230 <em>M<\/em>]<sup><em>m<\/em><\/sup><\/p>\n<p>Experiment 2: 9.99 \u00d7 10<sup>\u22124<\/sup> mol\/L\/s = <em>k<\/em>[0.356 <em>M<\/em>]<sup><em>m<\/em><\/sup><\/p>\n<p>Experiment 3: 2.44 \u00d7 10<sup>\u22123<\/sup> mol\/L\/s = <em>k<\/em>[0.557 <em>M<\/em>]<sup><em>m<\/em><\/sup><\/p>\n<p>The first two experiments can be set up so as to cancel one of the unknowns (that is, <em>k<\/em>) and solve for the other unknown (that is, <em>m<\/em>):<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\frac{4.17\\times 10^{-4}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}{9.99\\times 10^{-4}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}& =& \\frac{{\\text{(0.230 M)}}^{m}}{{\\text{(0.356 M)}}^{m}}\\hfill \\\\ \\hfill 0.4174& =& \\frac{{\\text{(0.230 M)}}^{m}}{{\\text{(0.356 M)}}^{m}}\\hfill \\end{array}[\/latex]<\/p>\n<p>Taking the natural log of each side gives:<\/p>\n<p>ln 0.4174 = <em>m<\/em>(ln 0.230) \u2212 <em>m<\/em>(ln 0.356)<\/p>\n<p>\u22120.8737 = \u22121.4697<em>m<\/em> + 1.0328<em>m<\/em><\/p>\n<p>\u22120.8737 = \u22120.4369<em>m<\/em><\/p>\n<p>[latex]m=\\frac{-0.8737}{-0.4369}=2.00[\/latex]<\/p>\n<p>Therefore, the rate equation is second order in A and is written as rate = <em>k<\/em>[A]<sup>2<\/sup>.<\/p>\n<p>(b) The rate constant can be calculated from any of the three sets of data by using the rate equation in conjunction with data found by substituting any of the three sets of data into the rate equation. Using the data from Equation 1 gives:<\/p>\n<p>4.17 \u00d7 10<sup>\u221214<\/sup> mol\/L\/s = <em>k<\/em>[0.230 <em>M<\/em>]<sup>2<\/sup><\/p>\n<p>[latex]k=\\frac{4.17\\times {\\text{10}}^{-14}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}{\\left(0.059{\\text{mol}}^{2}{\\text{L}}^{-2}\\right)}=7.88\\times {\\text{10}}^{-13}\\text{L}{\\text{mol}}^{-1}{\\text{s}}^{-1}[\/latex]<\/p>\n<p>18. The rate of reaction for a first-order reaction in N<sub>2<\/sub>O<sub>5<\/sub> is written as rate = <em>k<\/em>[N<sub>2<\/sub>O<sub>5<\/sub>] where <em>k<\/em>, the rate constant at 45 \u00b0C, is 6.2 \u00d7 10<sup>\u22124<\/sup> min<sup>\u22121<\/sup>. When [N<sub>2<\/sub>O<sub>5<\/sub>] = 0.40 <em>M<\/em>, rate = 6.2 \u00d7 10<sup>\u22124<\/sup> min<sup>\u22121<\/sup> (0.40 mol\/L) = 2.5 \u00d7 10<sup>\u22124<\/sup> mol\/L\/min<\/p>\n<p>20.\u00a0The rate equation has the form rate = <em>k<\/em>[I<sup>\u2212<\/sup>]<sup><em>m<\/em><\/sup>[OCl<sup>\u2212<\/sup>]<sup><em>n<\/em><\/sup> and the values for <em>m<\/em> and <em>n<\/em> must be determined. Comparing data from columns 1 and 2, [OCl<sup>&#8211;<\/sup>] remains constant and [I<sup>\u2212<\/sup>] doubles. As [I<sup>\u2212<\/sup>] doubles, the rate doubles, so <em>m<\/em> = 1.<\/p>\n<p>Comparing data from columns 1 and 3,<\/p>\n<p style=\"text-align: center;\">[latex]3.05\\times {10}^{-4}=k{\\text{[0.10]}}^{1}{\\text{[0.05]}}^{n}\\rightarrow 3.05\\times {10}^{-3}=k{\\text{[0.05]}}^{n}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]1.83\\times {10}^{-4}=k{\\text{[0.30]}}^{1}{\\text{[0.01]}}^{n}\\rightarrow 6.1\\times {10}^{-4}=k{\\text{[0.01]}}^{n}[\/latex]<\/p>\n<p>The first numerical value is five times larger than the second, corresponding to a fivefold increase in concentration. Therefore, <em>n<\/em> = 1; rate = <em>k<\/em>[I<sup>\u2212<\/sup>][OCl<sup>\u22121<\/sup>]<\/p>\n<p>The rate constant is determined by putting the data from column 2 into the rate equation:<\/p>\n<p style=\"text-align: center;\">[latex]k=\\frac{\\text{rate}}{\\left[{\\text{I}}^{-}\\right]\\left[{\\text{OCl}}^{-}\\right]}=\\frac{6.10\\times {\\text{10}}^{-4}\\text{mol}{\\text{L}}^{-1}{\\text{s}}^{-1}}{\\left(0.20\\text{mol}{\\text{L}}^{-1}\\right)\\left(0.050\\text{mol}{\\text{L}}^{-1}\\right)}=6.1\\times {\\text{10}}^{-2}\\text{L}{\\text{mol}}^{-1}{\\text{s}}^{-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Glossary<\/h2>\n<p><strong>method of initial rates: <\/strong>use of a more explicit algebraic method to determine the orders in a rate law<\/p>\n<p><strong>overall reaction order: <\/strong>sum of the reaction orders for each substance represented in the rate law<\/p>\n<p><strong>rate constant (<em>k<\/em>): <\/strong>proportionality constant in the relationship between reaction rate and concentrations of reactants<\/p>\n<p><strong>rate law (also, rate equation): <\/strong>mathematical equation showing the dependence of reaction rate on the rate constant and the concentration of one or more reactants<\/p>\n<p><strong>reaction order: <\/strong>value of an exponent in a rate law, expressed as an ordinal number (for example, zero order for 0, first order for 1, second order for 2, and so on)<\/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-2202\">\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":73,"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-2202","chapter","type-chapter","status-publish","hentry"],"part":2992,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters\/2202","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/users\/17"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters\/2202\/revisions"}],"predecessor-version":[{"id":6074,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters\/2202\/revisions\/6074"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/parts\/2992"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapters\/2202\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/media?parent=2202"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/pressbooks\/v2\/chapter-type?post=2202"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/contributor?post=2202"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-chem-atoms-first\/wp-json\/wp\/v2\/license?post=2202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}