Chapter 12, Problem 105CP

The reaction

$\text{NO}\left(g\right)+{\text{O}}_{\text{3}}\to {\text{NO}}_{\text{2}}\left(g\right)+{\text{O}}_{\text{2}}\left(g\right)$

was studied by performing two experiments. In the first experiment the rate of disappearance of NO was followed in the presence of a large excess of O₃. The results were as follows ([O₃] remains effectively constant at 1.0 × 10¹⁴ molecules/cm³):

Time(ms)	[NO] (molecules/cm3)
0	6.0 × 108
100 ± 1	5.0 × 108
500 ± 1	2.4 × 108
700 ± 1	1.7 × 108
1000 ± 1	9.9 × 107

In the second experiment [NO] was held constant at2.0 × 10¹⁴ molecules/cm³. The data for the disappearance of O₃ are as follows:

Time(ms)	[O3] (molecules/cm3)
0	1.0 × 1010
50± 1	8.4 × 109
100 ± 1	7.0 × 109
200 ± 1	4.9 × 109
300 ± 1	3.4 × 109

a. What is the order with respect to each reactant?

b. What is the overall rate law?

c. What is the value of the rate constant from each set of experiments?

$\text{Rate }=k\text{'}{\left[\text{NO}\right]}^x\text{Rate }=k"{\left[{\text{O}}_3\right]}^y$

d. What is the value of the rate constant for the overall rate law?

$\text{Rate}=k{\left[\text{NO}\right]}^x{\left[{\text{O}}_3\right]}^y$

Interpretation Introduction

Interpretation: The order of the reaction with respect to each reactant is to be calculated. The overall rate law is to be stated. From each set of the given experiments the value of rate constants are to be calculated. The overall rate constant is to be stated.

Concept introduction: The order of a reaction is the number or the index raise to the concentration term with respect to each component.

The order of reaction may or may not be equal to the stoichiometric coefficient.

Stoichiometric coefficient is the number which tells the number of the reactant or products in a chemical reaction.

The first order reaction always gives the straight line plot of $\ln \left[\text{reactant}\right]$ Vs time.

The rate law or rate equation is a mathematical representation that relates the rate of the reaction with the concentration or pressure.

Rate constant is a proportionality coefficient that relates the rate of chemical reaction at a specific temperature to the concentration of the reactant or the product.

To determine: The order of the reaction with respect to each reactant.

Answer to Problem 105CP

Answer

The order of the reaction with respect to both the reactant is $\underset{\_}{1^{st}}$.

Explanation of Solution

Explanation

Given

The given reaction is stated as,

$\text{NO}\left(g\right)+{\text{O}}_{\text{3}}\left(g\right)\to {\text{NO}}_{\text{2}}\left(g\right)+{\text{O}}_{\text{2}}\left(g\right)$

There are two reactants are present in the given reaction. Therefore to calculate the order of the reaction with respect to each reactant the following steps are observed.

With the help of the given data the value of $\ln \left[\text{NO}\right]$ is determined and are given by the following expression,

Time (ms)	$\left[\text{NO}\right]$ $\left({\text{molecules/cm}}^{\text{3}}\right)$	$\text{ln}\left[\text{NO}\right]$
$0$	$6.0\times {10}^8$	$20.21$
$100$	$5.0\times {10}^8$	$20.03$
$500$	$2.4\times {10}^8$	$19.29$
$700$	$1.7\times {10}^8$	$18.95$
$1000$	$9.9\times {10}^7$	$18.41$

Table 1

With the help of the above determined values the plot of $\ln \left[\text{NO}\right]$ Vs time is drawn as,

Figure 1

The above figure clearly indicates that in the $\ln \left[\text{NO}\right]$ Vs time plot the straight line is observed therefore, the given reaction is first order with respect to reactant $\left[\text{NO}\right]$.

With the help of the given data the value of $\ln \left[{\text{O}}_3\right]$ is determined and are given by the following expression,

Time (ms)	$\left[{\text{O}}_3\right]$ $\left({\text{molecules/cm}}^{\text{3}}\right)$	$\text{ln}\left[{\text{O}}_3\right]$
$0$	$1.0\times {10}^{10}$	$23.02$
$50$	$8.4\times {10}^9$	$22.85$
$100$	$7.0\times {10}^9$	$22.66$
$200$	$4.9\times {10}^9$	$22.31$
$300$	$3.4\times {10}^9$	$21.94$

Table 2

With the help of the above determined values the plot of $\ln \left[{\text{O}}_3\right]$ Vs time is drawn as,

Figure 2

The above figure clearly indicates that in the $\ln \left[{\text{O}}_3\right]$ Vs time plot the straight line is observed therefore, the given reaction is first order with respect to reactant $\left[{\text{O}}_3\right]$.

Interpretation Introduction

Interpretation: The order of the reaction with respect to each reactant is to be calculated. The overall rate law is to be stated. From each set of the given experiments the value of rate constants are to be calculated. The overall rate constant is to be stated.

Concept introduction: The order of a reaction is the number or the index raise to the concentration term with respect to each component.

The order of reaction may or may not be equal to the stoichiometric coefficient.

Stoichiometric coefficient is the number which tells the number of the reactant or products in a chemical reaction.

The first order reaction always gives the straight line plot of $\ln \left[\text{reactant}\right]$ Vs time.

The rate law or rate equation is a mathematical representation that relates the rate of the reaction with the concentration or pressure.

Rate constant is a proportionality coefficient that relates the rate of chemical reaction at a specific temperature to the concentration of the reactant or the product.

To determine: The overall rate law for the given reaction.

Answer to Problem 105CP

Answer

The overall rate law is given as, $\underset{\_}{Rate=k\left[NO\right]\left[O_3\right]}$

Explanation of Solution

Explanation

In the step 1 it is determined that the reaction is first order with respect to both the reactants. Rate of the reaction is a mathematical representation that relates the rate to the concentration of the reactants.

Therefore, the overall rate law is given as,

Interpretation Introduction

Interpretation: The order of the reaction with respect to each reactant is to be calculated. The overall rate law is to be stated. From each set of the given experiments the value of rate constants are to be calculated. The overall rate constant is to be stated.

Concept introduction: The order of a reaction is the number or the index raise to the concentration term with respect to each component.

The order of reaction may or may not be equal to the stoichiometric coefficient.

Stoichiometric coefficient is the number which tells the number of the reactant or products in a chemical reaction.

The first order reaction always gives the straight line plot of $\ln \left[\text{reactant}\right]$ Vs time.

The rate law or rate equation is a mathematical representation that relates the rate of the reaction with the concentration or pressure.

Rate constant is a proportionality coefficient that relates the rate of chemical reaction at a specific temperature to the concentration of the reactant or the product.

To determine: The value of rate constant from each set of experiments.

Answer to Problem 105CP

Answer

The value of rate constant for first experiment is $\underset{\_}{k=0.0018m{s}^{-1}}$

The value of rate constant for first experiment is $\underset{\_}{k=0.0036m{s}^{-1}}$

Explanation of Solution

Explanation

As in both the experiments the reaction is first order with respect to both the reactant.

Therefore, the rate constant is calculated by the formula,

$\ln \left[\text{R}\right]=-k\text{t}+\ln {\left[\text{R}\right]}_0$

Where,

• $\ln \left[\text{R}\right]$ is the final concentration of the reactant.
• $\ln {\left[\text{R}\right]}_0$ is the initial concentration of the reactant.
• $k$ is the rate constant.
• $\text{t}$ is the time difference.

Substitute the values for the first experiment from the Table 1 for reactant $\text{NO}$,

$\begin{array}{c}\ln \left[\text{R}\right]=-k\text{t}+\ln {\left[\text{R}\right]}_0\\ \ln \left[\text{NO}\right]=-k\text{t}+\ln {\left[\text{NO}\right]}_0\\ -k=\frac{\ln \left[\text{NO}\right]-\ln {\left[\text{NO}\right]}_0}{\text{t}}\\ -k=\frac{18.41-20.21}{1000\text{ms}}\end{array}$

Simplify the above equation, the value of rate constant for the first experiment is,

$\underset{\_}{k=0.0018m{s}^{-1}}$

Substitute the values for the second experiment from the Table 2 for reactant ${\text{O}}_3$,

$\begin{array}{c}\ln \left[\text{R}\right]=-k\text{t}+\ln {\left[\text{R}\right]}_0\\ \ln \left[{\text{O}}_3\right]=-k\text{t}+\ln {\left[{\text{O}}_3\right]}_0\\ -k=\frac{\ln \left[{\text{O}}_3\right]-\ln {\left[{\text{O}}_3\right]}_0}{\text{t}}\\ -k=\frac{21.94-23.02}{300\text{ms}}\end{array}$

Simplify the above equation, the value of rate constant for the second experiment is,

$\underset{\_}{k=0.0036m{s}^{-1}}$

Interpretation Introduction

Interpretation: The order of the reaction with respect to each reactant is to be calculated. The overall rate law is to be stated. From each set of the given experiments the value of rate constants are to be calculated. The overall rate constant is to be stated.

Concept introduction: The order of a reaction is the number or the index raise to the concentration term with respect to each component.

The order of reaction may or may not be equal to the stoichiometric coefficient.

Stoichiometric coefficient is the number which tells the number of the reactant or products in a chemical reaction.

The first order reaction always gives the straight line plot of $\ln \left[\text{reactant}\right]$ Vs time.

The rate law or rate equation is a mathematical representation that relates the rate of the reaction with the concentration or pressure.

Rate constant is a proportionality coefficient that relates the rate of chemical reaction at a specific temperature to the concentration of the reactant or the product.

To determine: The value of rate constant for overall rate law.

Answer to Problem 105CP

Answer

The value of overall rate constant is

Explanation of Solution

Explanation

The rate is determined by the formula,

$\text{Rate}=-\frac{\Delta \left[\text{NO}\right]}{\Delta \text{t}}$

Where,

• $\Delta \left[\text{NO}\right]$ is difference in the concentration of the reactant.
• $\Delta \text{t}$ is the difference in time.

Substitute the value of $\Delta \left[\text{NO}\right]$ and $\Delta \text{t}$ the of the reaction is calculated as,

$\begin{array}{c}\text{Rate}=-\frac{\Delta \left[\text{NO}\right]}{\Delta \text{t}}\\ =-\frac{\left(5.0\times {10}^8-6.0\times {10}^8\right)}{100\text{ms}}\\ =1.0\times {10}^6{\text{molecules/cm}}^3\cdot \text{ms}\end{array}$

The rate constant for the given reaction is calculated by the formula,

$k=\frac{\text{Rate}}{\left[\text{NO}\right]\left[{\text{O}}_{\text{3}}\right]}$

Where,

• $\left[\text{NO}\right]$ is the concentration of the reactant $\text{NO}$.
• $\Delta \left[{\text{O}}_3\right]$ is the concentration of the reactant ${\text{O}}_{\text{3}}$.

Substitute the value of rate and concentrations of the reactants the rate constant is calculate as,

$\begin{array}{c}k=\frac{\text{Rate}}{\left[\text{NO}\right]\left[{\text{O}}_{\text{3}}\right]}\\ =\frac{1.0\times {10}^6{\text{molecules/cm}}^3\cdot \text{ms}}{\left(6.0\times {10}^8{\text{molecules/cm}}^3\right)\left(1.0\times {10}^{14}{\text{molecules/cm}}^3\right)}\\ =\underset{\_}{1.0\times 10^{6}molecules/c{m}^3\times ms}\end{array}$

Conclusion

The questions based on the given reaction have been rightfully determined.