Chapter 16, Problem 16.81P

(a)

Interpretation Introduction

Interpretation:

If $\left[{\text{NO}}_2\right]$ is doubled, then rate quadruples, then the rate law for the given reaction (1) has to be predicted.

Concept introduction:

Rate law or rate equation: The relationship between the reactant concentrations and reaction rate is expressed by an equation.

$\begin{array}{l}\text{aA + bB}\to x\text{X}\\ \\ {\text{Rate of reaction = k [A]}}^{\text{m}}{\text{[B]}}^{\text{n}}\\ \\ \text{Total}\text{order}\text{of reaction = }\left(\text{m + n}\right)\end{array}$

Order of a reaction: The order of a reaction with respect to a particular reactant is the exponent of its concentration term in the rate law expression, and the overall reaction order is the sum of the exponents on all concentration terms.

Rate constant, k: It is a proportionality constant that relates rate and concentration at a given temperature.

Explanation of Solution

The overall rate law of reaction (1) is,

$\text{Rate}\text{ = k }{\left[\text{A}\right]}^{\text{m}}{\left[\text{B}\right]}^{\text{n}}\text{= k }{\left[{\text{NO}}_{\text{2}}\right]}^{\text{m}}{\left[\text{CO}\right]}^{\text{n}}$

When $\left[{\text{NO}}_2\right]$ increases by a factor of $2$, the rate increases by a factor of $4.$

While considering only the concentration of $\left[{\text{NO}}_{\text{2}}\right]$, the reaction rate becomes,

$\begin{array}{l}{\text{Rate}}_{\text{1}}\text{ = k }{\left[{\text{NO}}_{\text{2}}\right]}_{\text{1}}^{\text{m}}\text{;}{\text{Rate}}_{\text{2}}\text{ = k }{\left[{\text{NO}}_{\text{2}}\right]}_{\text{2}}^{\text{m}}\\ \\ \frac{{\text{Rate}}_{\text{2}}}{{\text{Rate}}_{\text{1}}}\text{ = }\frac{\text{ k }{\left[{\text{NO}}_{\text{2}}\right]}_{\text{2}}^{\text{m}}}{\text{k }{\left[{\text{NO}}_{\text{2}}\right]}_{\text{1}}^{\text{m}}}\\ \\ \text{ratio}\text{of}\text{rate}\text{is}\text{4;}\left[{\text{NO}}_{\text{2}}\right]\text{increases}\text{by}\text{a}\text{factor}\text{of}\text{2,thus}\\ \\ \text{4 = }{\left[2\right]}^m\\ \\ \boxed{\text{m}\text{=}\text{2}}\end{array}$

So, $\text{Rate}\text{ = k }{\left[{\text{NO}}_{\text{2}}\right]}^{\text{m}}{\left[\text{CO}\right]}^{\text{n}}$

$\begin{array}{l}\text{Total}\text{order}\text{of}\text{reaction}\left(Secondorder\right)\text{:}\\ \\ \text{m + n = 2}\Rightarrow 2\text{ + n = 2}\\ \\ \text{n}=2-2=0.\end{array}$

The overall order of the reaction is SECOND; but the order with respect to $\left[\text{CO}\right]$ is ZERO ORDER.

Therefore, the overall reaction rate law is $\text{Rate}\text{ = k }{\left[{\text{NO}}_{\text{2}}\right]}^{\text{2}}{\left[\text{CO}\right]}^{\text{0}}\left(\text{or}\right)k{\left[N{O}_2\right]}^2.$

(b)

Interpretation Introduction

Interpretation:

If $\left[\text{NO}\right]$ is doubled, then rate doubles, then the rate law for the given reaction (2) has to be predicted.

Concept introduction:

Rate law or rate equation: The relationship between the reactant concentrations and reaction rate is expressed by an equation.

$\begin{array}{l}\text{aA + bB}\to x\text{X}\\ \\ {\text{Rate of reaction = k [A]}}^{\text{m}}{\text{[B]}}^{\text{n}}\\ \\ \text{Total}\text{order}\text{of reaction = }\left(\text{m + n}\right)\end{array}$

Order of a reaction: The order of a reaction with respect to a particular reactant is the exponent of its concentration term in the rate law expression, and the overall reaction order is the sum of the exponents on all concentration terms.

Rate constant, k: It is a proportionality constant that relates rate and concentration at a given temperature.

Explanation of Solution

The overall rate law of reaction (2) is,

$\text{Rate}\text{ = k }{\left[\text{A}\right]}^{\text{m}}{\left[\text{B}\right]}^{\text{n}}\text{= k }{\left[\text{NO}\right]}^{\text{m}}{\left[{\text{O}}_3\right]}^{\text{n}}$

When $\left[\text{NO}\right]$ increases by a factor of $2$, the rate increases by a factor of $2.$

While considering only the concentration of $\left[\text{NO}\right]$, the reaction rate becomes,

$\begin{array}{l}{\text{Rate}}_{\text{1}}\text{ = k }{\left[\text{NO}\right]}_{\text{1}}^{\text{m}}\text{;}{\text{Rate}}_{\text{2}}\text{ = k }{\left[\text{NO}\right]}_{\text{2}}^{\text{m}}\\ \\ \frac{{\text{Rate}}_{\text{2}}}{{\text{Rate}}_{\text{1}}}\text{ = }\frac{\text{ k }{\left[\text{NO}\right]}_{\text{2}}^{\text{m}}}{\text{k }{\left[\text{NO}\right]}_{\text{1}}^{\text{m}}}\\ \\ \text{ratio}\text{of}\text{rate }\text{is }\text{2;}\left[\text{NO}\right]\text{increases}\text{by}\text{a}\text{factor}\text{of}\text{2,thus}\\ \\ \text{2 = }{\left[2\right]}^m\\ \\ \boxed{\text{m}\text{=}\text{1}}\end{array}$

So, $\text{Rate}\text{ = k }{\left[\text{A}\right]}^{\text{m}}{\left[\text{B}\right]}^{\text{n}}\text{= k }{\left[\text{NO}\right]}^{\text{m}}{\left[{\text{O}}_3\right]}^{\text{n}}$

$\begin{array}{l}\text{Total}\text{order}\text{of}\text{reaction}\left(Secondorder\right)\text{:}\\ \\ \text{m + n = 2}\Rightarrow 1\text{ + n = 2}\\ \\ \text{n}=2-1=1.\end{array}$

The overall order of the reaction is SECOND; but the order with respect to $\left[{\text{O}}_3\right]$ is FIRST ORDER.

Therefore, the overall reaction rate law is $Rate=k{\left[NO\right]}^1{\left[O_3\right]}^1.$

(c)

Interpretation Introduction

Interpretation:

The ratio of the initial rate to the rate when the reaction is $50\%$ complete has to be determined.

Concept introduction:

Rate law or rate equation: The relationship between the reactant concentrations and reaction rate is expressed by an equation.

$\begin{array}{l}\text{aA + bB}\to x\text{X}\\ \\ {\text{Rate of reaction = k [A]}}^{\text{m}}{\text{[B]}}^{\text{n}}\\ \\ \text{Total}\text{order}\text{of reaction = }\left(\text{m + n}\right)\end{array}$

Order of a reaction: The order of a reaction with respect to a particular reactant is the exponent of its concentration term in the rate law expression, and the overall reaction order is the sum of the exponents on all concentration terms.

Rate constant, k: It is a proportionality constant that relates rate and concentration at a given temperature.

Explanation of Solution

When the concentrations of the reactants are reduced to one-half of the initial concentration, then the reaction is considered as $50\%$ completion.

For reaction (1):

The concentration $\left[{\text{NO}}_2\right]$ decreases by one-half its initial during $50\%$ of reaction completion.

$\begin{array}{l}{\text{Rate}}_{initial}\text{ = k }{\left[{\text{NO}}_2\right]}^2\text{;}{\text{Rate}}_{50\%}\text{ = k }{\left[\text{0}\text{.5}\times {\text{NO}}_2\right]}^2\\ \\ \frac{{\text{Rate}}_{initial}}{{\text{Rate}}_{50\%}}\text{ = }\frac{\text{k }{\left[{\text{NO}}_2\right]}^2}{\text{k }{\left[\text{0}\text{.5}\times {\text{NO}}_2\right]}^2}=4\end{array}$

For reaction (2):

The concentration $\left[\text{NO}\right]\text{and}\left[{\text{O}}_{\text{3}}\right]$ decreases by one-half its initial during $50\%$ of reaction completion.

$\begin{array}{l}{\text{Rate}}_{\text{initial}}\text{ = k }{\left[\text{NO}\right]}^{\text{1}}{\left[{\text{O}}_{\text{3}}\right]}^{\text{1}}\text{;}{\text{Rate}}_{\text{50\%}}\text{ = k }{\left[\text{0}\text{.5}\text{×NO}\right]}^{\text{1}}{\left[\text{0}\text{.5}{\text{×O}}_{\text{3}}\right]}^{\text{1}}\\ \\ \frac{{\text{Rate}}_{\text{initial}}}{{\text{Rate}}_{\text{50\%}}}\text{ = }\frac{\text{k }{\left[\text{NO}\right]}^{\text{1}}{\left[{\text{O}}_{\text{3}}\right]}^{\text{1}}}{\text{ k }{\left[\text{0}\text{.5}\text{×NO}\right]}^{\text{1}}{\left[\text{0}\text{.5}{\text{×O}}_{\text{3}}\right]}^{\text{1}}}\text{=}4.\end{array}$

Therefore, the ratio of the initial rate to the rate when the reaction is $50\%$ complete was calculated as shown above.

(d)

Interpretation Introduction

Interpretation:

The ratio of the initial rate to the rate when the reaction is $50\%$ complete when the initial $\left[{\text{NO}}_2\right]$ is twice the initial $\left[\text{CO}\right]$ has to be determined.

Concept introduction:

Rate law or rate equation: The relationship between the reactant concentrations and reaction rate is expressed by an equation.

$\begin{array}{l}\text{aA + bB}\to x\text{X}\\ \\ {\text{Rate of reaction = k [A]}}^{\text{m}}{\text{[B]}}^{\text{n}}\\ \\ \text{Total}\text{order}\text{of reaction = }\left(\text{m + n}\right)\end{array}$

Order of a reaction: The order of a reaction with respect to a particular reactant is the exponent of its concentration term in the rate law expression, and the overall reaction order is the sum of the exponents on all concentration terms.

Rate constant, k: It is a proportionality constant that relates rate and concentration at a given temperature.

Explanation of Solution

When the concentrations of the reactants are reduced to one-half of the initial concentration, then the reaction is considered as $50\%$ completion.

For reaction (1):

The ratio of the initial rate to the rate when the reaction is $50\%$ complete when the initial $\left[{\text{NO}}_2\right]$ is twice the initial $\left[\text{CO}\right]$

${\left[{\text{NO}}_{\text{2}}\right]}_{\text{initial}}\text{= 2}{\left[\text{CO}\right]}_{\text{initial}}$; $\left[\text{CO}\right]=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right){\left[\text{CO}\right]}_{initial};\left[{\text{NO}}_2\right]=\left(0.75\right){\left[{\text{NO}}_2\right]}_{initial}$

$\begin{array}{l}{\text{Rate}}_{initial}\text{ = k }{\left[{\text{NO}}_2\right]}^2\text{;}{\text{Rate}}_{50\%}\text{ = k }{\left[\text{0}\text{.5}\times {\text{NO}}_2\right]}^2\\ \\ \frac{{\text{Rate}}_{initial}}{{\text{Rate}}_{50\%}}\text{ = }\frac{\text{k }{\left[{\text{NO}}_2\right]}^2}{\text{k }{\left[\text{0}\text{.75}\times {\text{NO}}_2\right]}^2}=1.778=1.8\end{array}$

Therefore, the ratio of the initial rate to the rate when the reaction is $50\%$ complete was calculated as shown above.

(e)

Interpretation Introduction

Interpretation:

The ratio of the initial rate to the rate when the reaction is $50\%$ complete when the initial $\left[\text{NO}\right]$ is twice the initial $\left[{\text{O}}_3\right]$ has to be determined.

Concept introduction:

Rate law or rate equation: The relationship between the reactant concentrations and reaction rate is expressed by an equation.

$\begin{array}{l}\text{aA + bB}\to x\text{X}\\ \\ {\text{Rate of reaction = k [A]}}^{\text{m}}{\text{[B]}}^{\text{n}}\\ \\ \text{Total}\text{order}\text{of reaction = }\left(\text{m + n}\right)\end{array}$

Order of a reaction: The order of a reaction with respect to a particular reactant is the exponent of its concentration term in the rate law expression, and the overall reaction order is the sum of the exponents on all concentration terms.

Rate constant, k: It is a proportionality constant that relates rate and concentration at a given temperature.

Explanation of Solution

When the concentrations of the reactants are reduced to one-half of the initial concentration, then the reaction is considered as $50\%$ completion.

For reaction (2):

The ratio of the initial rate to the rate when the reaction is $50\%$ complete when the initial $\left[\text{NO}\right]$ is twice the initial $\left[{\text{O}}_3\right].$

${\left[\text{NO}\right]}_{\text{initial}}\text{= 2}{\left[{\text{O}}_3\right]}_{\text{initial}}$; $\left[{\text{O}}_3\right]=\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right){\left[{\text{O}}_3\right]}_{initial};\left[\text{NO}\right]=\left(0.75\right){\left[\text{NO}\right]}_{initial}$

$\begin{array}{l}{\text{Rate}}_{\text{initial}}\text{ = k }\left[\text{NO}\right]\left[{\text{O}}_{\text{3}}\right]\text{;}{\text{Rate}}_{\text{50\%}}\text{ = k }\left[\text{0}\text{.75}\text{×NO}\right]\left[\text{0}\text{.5}{\text{×O}}_{\text{3}}\right]\\ \\ \frac{{\text{Rate}}_{\text{initial}}}{{\text{Rate}}_{\text{50\%}}}\text{ = }\frac{\text{k }\left[\text{NO}\right]\left[{\text{O}}_{\text{3}}\right]}{\text{k }\left[\text{0}\text{.75}\text{×NO}\right]\left[\text{0}\text{.5}{\text{×O}}_{\text{3}}\right]}\text{=}\text{2}\text{.6667}\text{=}2.7.\end{array}$

Therefore, the ratio of the initial rate to the rate when the reaction is $50\%$ complete was calculated as shown above.