/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 94 The following kinetic data are c... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 94

The following kinetic data are collected for the initial rates of a reaction \(2 \mathrm{X}+\mathrm{Z} \longrightarrow\) products: $$ \begin{array}{llll} \hline \text { Experiment } & {[\mathrm{X}]_{0}(M)} & {[\mathrm{Z}]_{0}(M)} & \text { Rate }(M / \mathrm{s}) \\ \hline 1 & 0.25 & 0.25 & 4.0 \times 10^{1} \\ 2 & 0.50 & 0.50 & 3.2 \times 10^{2} \\ 3 & 0.50 & 0.75 & 7.2 \times 10^{2} \\ \hline \end{array} $$ (a) What is the rate law for this reaction? (b) What is the value of the rate constant with proper units? (c) What is the reaction rate when the initial concentration of \(X\) is \(0.75 M\) and that of \(Z\) is \(1.25 M ?\)

Short Answer

Expert verified
(a) The rate law for this reaction is: Rate = \( k [\mathrm{X}]^2 [\mathrm{Z}] \) (b) The value of the rate constant is approximately \( 2.56 \times 10^3 \, \mathrm{M^{-3} s^{-1}} \) (c) The reaction rate when the initial concentration of X is 0.75 M and that of Z is 1.25 M is approximately \( 7.2 \times 10^3 \, \mathrm{M \, s^{-1}} \).

Step by step solution

01

Determine the order of the reaction with respect to reactant X

Compare Experiment 1 and Experiment 2, as both have a change in concentration for X but not for Z. The rate and concentration ratio can help us determine the order of the reaction with respect to X. For the change between Experiment 1 and Experiment 2: Change in concentration of X: \( \cfrac{[\mathrm{X}]_{0}\text{ (Experiment 2)}}{[\mathrm{X}]_{0}\text{ (Experiment 1)}} = \cfrac{0.50}{0.25} = 2 \) Change in rate: \( \cfrac{\mathrm{Rate} \cdot \mathrm{(Experiment 2)}}{\mathrm{Rate} \cdot \mathrm{(Experiment 1)}} = \cfrac{3.2 \times 10^2}{4.0 \times 10^1} = 8 \) Since the rate changes in 8 times while the concentration of X changes in 2 times, we can conclude that the reaction is second order with respect to X.
02

Determine the order of the reaction with respect to reactant Z

Compare Experiment 2 and Experiment 3, as both have a change in concentration for Z but not for X. The rate and concentration ratio can help us determine the order of the reaction with respect to Z. For the change between Experiment 2 and Experiment 3: Change in concentration of Z: \( \cfrac{[\mathrm{Z}]_{0}\text{ (Experiment 3)}}{[\mathrm{Z}]_{0}\text{ (Experiment 2)}} = \cfrac{0.75}{0.50} = 1.5 \) Change in rate: \( \cfrac{\mathrm{Rate} \cdot \mathrm{(Experiment 3)}}{\mathrm{Rate} \cdot \mathrm{(Experiment 2)}} = \cfrac{7.2 \times 10^2}{3.2 \times 10^2} = 2.25 \) Since the rate changes in 2.25 times while the concentration of Z changes in 1.5 times, we can conclude that the reaction is first order with respect to Z.
03

Determine the rate law and calculate the rate constant

From Steps 1 and 2, we know that the reaction is second order with respect to X and first order with respect to Z. Therefore, the rate law can be written as: Rate \( = k[\mathrm{X}]^2[\mathrm{Z}] \) Now let's use the data from Experiment 1 to calculate the rate constant k: \( 4.0 \times 10^1 = k (0.25)^2(0.25) \) \( k = \cfrac{4.0 \times 10^1}{(0.25)^2(0.25)} \) Calculate k: \( k \approx 2.56 \times 10^3 \, \mathrm{M^{-3} s^{-1}} \)
04

Calculate the reaction rate for the given initial concentrations

Now that we have the rate law and the rate constant, we can calculate the reaction rate for the given initial concentrations of X = 0.75 M and Z = 1.25 M: Rate \( = k [\mathrm{X}]^2 [\mathrm{Z}] \) Rate \( = (2.56 \times 10^3 \, \mathrm{M^{-3} s^{-1}}) (0.75 \, \mathrm{M} )^2 (1.25 \, \mathrm{M}) \) Calculate the reaction rate: Rate \( \approx 7.2 \times 10^3 \, \mathrm{M \, s^{-1}} \) #Summary# (a) The rate law for this reaction is: Rate = \( k [\mathrm{X}]^2 [\mathrm{Z}] \) (b) The value of the rate constant is approximately \( 2.56 \times 10^3 \, \mathrm{M^{-3} s^{-1}} \) (c) The reaction rate when the initial concentration of X is 0.75 M and that of Z is 1.25 M is approximately \( 7.2 \times 10^3 \, \mathrm{M \, s^{-1}} \).

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Most popular questions from this chapter

Heterogeneous catalysts that perform hydrogenation reactions, as illustrated in Figure 14.24, are subject to "poisoning," which shuts down their catalytic ability. Compounds of sulfur are often poisons. Suggest a mechanism by which such compounds might act as poisons.

(a) The activation energy for the isomerization of methyl isonitrile (Figure 14.7) is \(160 \mathrm{~kJ} / \mathrm{mol}\). Calculate the fraction of methyl isonitrile molecules that has an energy of \(160.0 \mathrm{~kJ}\) or greater at \(500 \mathrm{~K}\). (b) Calculate this fraction for a temperature of \(520 \mathrm{~K}\). What is the ratio of the fraction at \(520 \mathrm{~K}\) to that at \(500 \mathrm{~K} ?\)

As described in Exercise 14.41, the decomposition of sulfuryl chloride \(\left(\mathrm{SO}_{2} \mathrm{Cl}_{2}\right)\) is a first-order process. The rate constant for the decomposition at \(660 \mathrm{~K}\) is \(4.5 \times 10^{-2} \mathrm{~s}^{-1}\). (a) If we begin with an initial \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) pressure of 450 torr, what is the partial pressure of this substance after \(60 \mathrm{~s}\) ? (b) At what time will the partial pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) decline to one-tenth its initial value?

Which of the following linear plots do you expect for a reaction \(A \longrightarrow\) products if the kinetics are (a) zero order, (b) first order, or (c) second order? [Section 14.4]

(a) A certain first-order reaction has a rate constant of \(2.75 \times 10^{-2} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\). What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if \(E_{a}=75.5 \mathrm{~kJ} / \mathrm{mol}\) ? (b) Another first-order reaction also has a rate constant of \(2.75 \times 10^{-2} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\) What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if \(E_{a}=125 \mathrm{~kJ} / \mathrm{mol}\) ? (c) What assumptions do you need to make in order to calculate answers for parts (a) and (b)?
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