Chapter 11, Problem 11.36PAE

The reaction

NO(g) + O,(g) — NO,(g) + 0(g)

plays a role in the formation of nitrogen dioxide in automobile engines. Suppose that a series of experiments measured the rate of this reaction at 500 K and produced the following data;

[NO]

(mol L ’)[OJ

(mol L 1)Rate = -A[NO]/Af (mol L_1 s-1) 0.002 0.005 8.0 X 10"'⁷ 0.002 0.010 1.6 X 10-'⁶ 0.006 0.005 2.4 X IO-'⁶

Derive a rate law for the reaction and determine the value of the rate constant.

Interpretation Introduction

Interpretation: Given the experimental data obtained for a reaction at 500K, derive the rate law for the reaction and find the value of the rate constant.

Concept Introduction: Orders of reaction are constantly determined by doing experiments. Consequently without experimental information, we can't conclude anything about the order of a reaction just by having a look at the equation for the reaction. By doing experiments involving a reaction between A and B, the rate of the reaction is identified to be related to the concentrations of A and B as follows:

$rate=k{[A]}^a{[B]}^b$ ->

This is the Rate Equation.

Where,

Rate is in the units of mol dm-³s-1

k is the rate constant

A, B- concentrations in mol dm-3

a - Order of reaction with respect to A

b- Order of reaction with respect to B

If temperature is given, the rate is usually considered to be a function of the initial concentrations of the reactants A and B.

Answer to Problem 11.36PAE

Solution: The rate law of the reaction is $Rate=k\left[NO\right]\left[O_2\right]$ and the value of rate constant is 0.8

Explanation of Solution

Given information: Reaction: $NO(g)+O_2(g)\to N{O}_2(g)+O(g)$, Temperature =500K

Experimental Data

Step 1: For the reaction:

$NO(g)+O_2(g)\to N{O}_2(g)+O(g)$

The rate law can be determined using the rate equation as follows:

$Rate=k{\left[NO\right]}^a{\left[O_2\right]}^b$

Where,

a= Order of the reaction with respect to NO

b= Order of the reaction with respect to $O_2$

Step 2: From the first, second and third rows of the given experimental data,

$\begin{array}{l}k{\left(0.002\right)}^a{(0.005)}^b =\text{ 8}\text{.0}\times {\text{10}}^{-17}\to (1)\hfill \\ \begin{array}{l}k{\left(0.002\right)}^a{(}^{0.010} =1.6\times {10}^{-16}\to (2)\\ k{\left(0.006\right)}^a{(}^{0.005} =2.4\times {10}^{-16}\to (3)\end{array}\hfill \end{array}$

Step 3: Divide (2) by (1), we get

$\begin{array}{l}\begin{array}{l}\frac{1.6\times {10}^{-16}}{8\times {10}^{-17}}=\frac{k{\left(0.002\right)}^a{(0.010)}^b}{k{\left(0.002\right)}^a{(0.005)}^b}\\ \frac{16\times {10}^{-17}}{8\times {10}^{-17}}=\frac{k{\left(0.002\right)}^a{(0.010)}^b}{k{\left(0.002\right)}^a{(0.005)}^b}\\ 2={(}^2\end{array}\hfill \\ \begin{array}{l}{\left(2\right)}^1={\left(2\right)}^b\\ b=1\end{array}\hfill \end{array}$

Step 4: Divide (3) by (1), we get

$\begin{array}{l}\begin{array}{l}\frac{2.4\times {10}^{-16}}{8\times {10}^{-17}}=\frac{k{\left(0.006\right)}^a{(0.005)}^b}{k{\left(0.002\right)}^a{(0.005)}^b}\\ \frac{24\times {10}^{-17}}{8\times {10}^{-17}}=\frac{k{\left(0.006\right)}^a{(0.005)}^b}{k{\left(0.002\right)}^a{(0.005)}^b}\\ 3={(}^3\end{array}\hfill \\ \begin{array}{l}{\left(3\right)}^1={\left(3\right)}^a\\ a=1\end{array}\hfill \end{array}$

Step 5: Rate Equation = >$Rate=k{\left[A\right]}^a{\left[B\right]}^b \Rightarrow Rate=k{\left[NO\right]}^a{\left[O_2\right]}^b\text{= }k\left[NO\right]\left[O_2\right]$

Step 6: Substitute a=1, b=1 values in (1)

$\begin{array}{l}k{\left(0.002\right)}^a{(}^{0.005} =8\times {10}^{-7}\\ k \left(0.002\right)(0.005) =8\times {10}^{-7}\\ k\left(2\times {10}^{-3}\times 5\times {10}^{-3}\right)=8\times {10}^{-7}\\ k=\frac{8\times {10}^{-7}}{{10}^{-5}}=0.08\end{array}$

Conclusion

It does not make a difference what the number of reactants there are. The concentration of every reactant will be present in the rate equation, raised to some power. These powers resemble the individual orders with respect to each reactant. The sum of these powers results in the overall order of the reaction. The rate constant will be a constant value for a given reaction only if the concentration of the reactants is changed without changing any other factors.