Chapter 14, Problem 80AP

Interpretation Introduction

Interpretation:

The order of the given reaction andits rate constant is to be determined and the probable mechanism, which is consistent with the rate law, is to be suggested.

Concept introduction:

The branch of chemistry that deals with the relative rates of the chemical reactions and the factors affecting the rates of reactions is known as chemical kinetics.

The rate of a reaction is the measurement of concentration change of the reactants, within a given period of time.

Answer to Problem 80AP

Solution:

$3$

$0.38s^{-1}$

$\begin{array}{l}Step-I:H_2+2NO\to N_{2}O+H_{2}O\left(slow\right)\\ Step-II:N_{2}O+H_2\to N_2+H_{2}O\left(fast\right)\end{array}$

Explanation of Solution

Given information:

The reaction between hydrogen and nitric acid, at $700°C$ is as follows:

$2H_2\left(g\right)+2NO\left(g\right)\to 2H_{2}O\left(g\right)+N_2\left(g\right)$

Experiment	$\left[H_2\right]\left(M\right)$	$\left[NO\right]\left(M\right)$	Initial Rate $\left(M/s\right)$
1	$0.010$	$0.025$	$2.4\times {10}^{-6}$
2	$0.0050$	$0.025$	$1.2\times {10}^{-6}$
3	$0.010$	$0.0125$	$0.60\times {10}^{-6}$

a)The order of reaction

The order of the reaction is given by the following expression as:

$\frac{{\text{Rate}}_2}{{\text{Rate}}_1}={\left(\frac{{\left[\text{A}\right]}_2}{{\left[\text{A}\right]}_1}\right)}^x{\left(\frac{{\left[\text{B}\right]}_2}{{\left[\text{B}\right]}_1}\right)}^y$ …… (1)

Here, $x$ is the order with respect to A and $y$ is the order with respect to B.

Substitute the values of ${\text{Rate}}_1,{\text{Rate}}_2,{\left[\text{A}\right]}_1,{\left[\text{A}\right]}_2,{\left[\text{B}\right]}_1$, and ${\left[\text{B}\right]}_2$ from experiments 1 and 3 in equation (1),

$\begin{array}{c}\frac{2.4\times {10}^{-6}\cancel{\text{M}/\text{s}}}{0.60\times {10}^{-6}\cancel{\text{M}/\text{s}}}={\left(\frac{0.010\cancel{\text{M}}}{0.010\cancel{\text{M}}}\right)}^x{\left(\frac{0.025\cancel{\text{M}}}{0.0125\cancel{\text{M}}}\right)}^y\\ 4=2^y\\ y=2\end{array}$

Substitute the values of ${\text{Rate}}_1,{\text{Rate}}_2,{\left[\text{A}\right]}_1,{\left[\text{A}\right]}_2,{\left[\text{B}\right]}_1$, and ${\left[\text{B}\right]}_2$ from experiments 1 and 2 in the equation (1),

$\begin{array}{c}\frac{2.4\times {10}^{-6}\cancel{\text{M}/\text{s}}}{1.2\times {10}^{-6}\cancel{\text{M}/\text{s}}}={\left(\frac{0.010\cancel{\text{M}}}{0.0050\cancel{\text{M}}}\right)}^x{\left(\frac{0.025\cancel{\text{M}}}{0.025\cancel{\text{M}}}\right)}^y\\ 2=2^x\\ x=1\end{array}$

Hence, the overall order of the reaction is $3$.

b) The rate constant

The rate expression for the reaction is given as follows:

$\begin{array}{l}rate=k{\left[NO\right]}^2\left[H_2\right]\\ k=\frac{rate}{{\left[NO\right]}^2\left[H_2\right]}\end{array}$

Substitute the values of $\left[NO\right]$, $\left[H_2\right]$, and rate of reaction in the above equation:

$\begin{array}{c}rate=\frac{2.4\times {10}^{-6}M/s}{{\left[0.025M\right]}^2\left[0.010M\right]}\\ =\frac{2.4\times {10}^{-6}\cancel{M}/s}{3.125\times {10}^{-6}\cancel{M}}\\ =0.384s^{-1}\end{array}$

Hence, the rate constant for the reaction is $0.38s^{-1}$.

c) A plausible mechanism that is consistent to rate law.

The reaction between hydrogen with nitric oxide is as follows:

$2H_2\left(g\right)+2NO\left(g\right)\to 2H_{2}O\left(g\right)+N_2\left(g\right)$

The plausible mechanism of the reaction, which isconsistent with the rate law, is given as follows:

$\begin{array}{l}Step-I:H_2+2NO\to N_{2}O+H_{2}O\left(slow\right)\\ Step-II:N_{2}O+H_2\to N_2+H_{2}O\left(fast\right)\end{array}$

The overall reaction is obtained as follows:

$\begin{array}{l}{H}_2+2NO\to \cancel{N_{2}O}+H_{2}O\\ \underset{\_}{\cancel{N_{2}O}+H_2\to N_2+H_{2}O}\\ \underset{\_}{2H_2+2NO\to N_2+2H_{2}O}\end{array}$