Chapter 16, Problem 16.72P

(a)

Interpretation Introduction

Interpretation:

An overall equation for the given reaction has to be written.

Concept introduction:

Reaction: Substances which are mutually involved each other in a chemical process and changed into different substances.

Mechanism of a reaction: The representation of step by step process involved in the chemical process is said to be mechanism of a chemical reaction.

Elementary step: The first step in a reaction mechanism is said to be elementary step.

Intermediate: Sometimes, in between the reaction some separable amount and useful amount of substances are formed during the reaction and which are said to be intermediates

Explanation of Solution

The given reaction involves multiple mechanism steps, by adding the entire individual steps gives rise to an overall reaction equation. Hence, the reaction equation as follows,

$\underset{\_}{\begin{array}{l}\underset{\_}{\begin{array}{l}A+B\rightleftharpoons \cancel{X}\left[fast\right]\\ \cancel{X}+C\stackrel{}{\to }\cancel{Y}\left[slow\right]\\ \cancel{Y}\stackrel{}{\to }D\left[fast\right]\end{array}}\\ A+B+C\stackrel{}{\to }DOverallreaction\end{array}}$

In the reaction, the slow step is the rate determining step; and its rate law is the overall rate law. The overall equation becomes,

$A+B+C\stackrel{}{\to }D$

(b)

Interpretation Introduction

Interpretation:

The intermediate has to be identified.

Concept introduction:

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

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

$\text{Reaction Rate}\text{ = k }{\left[\text{A}\right]}^{\text{m}}{\left[\text{B}\right]}^{\text{n}}{\left[\text{C}\right]}^{\text{p}}\text{,where 'm, n and p' are orders of the reactants}\text{.}$

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 given reaction involves multiple mechanism steps, by adding the entire individual steps gives rise to an overall reaction equation. Hence, the reaction equation as follows,

$\underset{\_}{\begin{array}{l}\underset{\_}{\begin{array}{l}A+B\rightleftharpoons \cancel{X}\left[fast\right]\\ \cancel{X}+C\stackrel{}{\to }\cancel{Y}\left[slow\right]\\ \cancel{Y}\stackrel{}{\to }D\left[fast\right]\end{array}}\\ A+B+C\stackrel{}{\to }DOverallreaction\end{array}}$

The intermediates are not involved in the overall reaction; thus, the components ‘X and Y’ are intermediates and produced and consumed in the reaction.

(c)

Interpretation Introduction

Interpretation:

The molecularity and the rate law for each step has to be identified.

Concept introduction:

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

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

$\text{Reaction Rate}\text{ = k }{\left[\text{A}\right]}^{\text{m}}{\left[\text{B}\right]}^{\text{n}}{\left[\text{C}\right]}^{\text{p}}\text{,where 'm, n and p' are orders of the reactants}\text{.}$

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

Step 1: $A_{\text{(g)}}+B_{\text{(g)}}\rightleftharpoons X_{\text{(g)}}$

Molecularity (collision) of a reaction step (1) is the number of reactant involved in that elementary step. Here, two molecules A and B are involved; thus, molecularity is BIMOLECULAR. The rate law of this step is ${\text{Rate}}_1\text{ = k }\left[\text{A}\right]\left[\text{B}\right].$

Step 2: $X_{\text{(g)}}+C_{\text{(g)}}\to Y_{\text{(g)}}$

Molecularity (collision) of a reaction step is the number of reactant involved in that elementary step. Here, two molecules X and C are involved; thus, molecularity is BIMOLECULAR. The rate law of this step is ${\text{Rate}}_{\text{2}}\text{ = k }\left[\text{X}\right]\left[\text{C}\right]\text{.}$

Step 3: $Y_{\text{(g)}}\to D_{\text{(g)}}$

Molecularity (collision) of a reaction step is the number of reactant involved in that elementary step. Here, only one molecule Y is involved; thus, molecularity is UNIMOLECULAR. The rate law of this step is ${\text{Rate}}_3\text{ = k }\left[\text{Y}\right].$

(d)

Interpretation Introduction

Interpretation:

The mechanism whether consistent with the given rate law 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.

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

$\text{Reaction Rate}\text{ = k }{\left[\text{A}\right]}^{\text{m}}{\left[\text{B}\right]}^{\text{n}}{\left[\text{C}\right]}^{\text{p}}\text{,where 'm, n and p' are orders of the reactants}\text{.}$

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

In the reaction, the slow step is the rate determining step; and its rate law is the overall rate law. The overall equation becomes,

$A+B+C\stackrel{}{\to }D$

The overall reaction rate is same as the rate of slowest reaction step.

The slowest step is, $X+C\stackrel{}{\to }Y\left[slow\right]$

The rate law for the above reaction is, $\text{Rate}{\text{ = k}}^{\text{'}}\left[\text{X}\right]\left[C\right]$        (1)

The concentration $\left[X\right]$ is the intermediate that cannot be involved in the overall rate law. Hence, substitution is needed for the $\left[X\right]$ intermediate.

The rate law for the given mechanism steps are,

The rate of mechanism (1), ${\text{Rate}}_{\text{1}}{\text{ = k}}_1\left[\text{A}\right]\left[\text{B}\right]\text{and}{\text{Rate}}_{\text{1}}{\text{ = k}}_{-1}\left[\text{X}\right]$, thus the step is in equilibrium, the rate of forward and backward reaction rate are equal,

$\begin{array}{l}{\text{k}}_1\left[\text{A}\right]\left[\text{B}\right]{\text{ = k}}_{-1}\left[\text{X}\right]\\ \\ \left[\text{X}\right]=\frac{{\text{k}}_1}{{\text{ k}}_{-1}}\left[\text{A}\right]\left[\text{B}\right]-----------------\left(2\right)\end{array}$

Thus, by substituting above relation into the equation (1), the rate law becomes,

$\begin{array}{l}\text{Rate}{\text{ = k}}^{\text{'}}\left[\text{X}\right]\left[C\right]\\ \\ =\frac{{\text{ k}}^{\text{'}}{\text{k}}_1}{{\text{ k}}_{-1}}\left[\text{A}\right]\left[\text{B}\right]\left[C\right]\\ \text{=}\text{k}\left[\text{A}\right]\left[\text{B}\right]\left[\text{C}\right]\text{,}\text{where}\frac{{\text{ k}}^{\text{'}}{\text{k}}_1}{{\text{ k}}_{-1}}\text{=}\text{k}\\ \\ \boxed{Rate=k\left[A\right]\left[B\right]\left[C\right]}\end{array}$

Therefore, the given rate law is consistent with the rate law of overall reaction $Rate=k\left[A\right]\left[B\right]\left[C\right].$

(e)

Interpretation Introduction

Interpretation:

The given one-step mechanism whether is equally valid 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.

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

$\text{Reaction Rate}\text{ = k }{\left[\text{A}\right]}^{\text{m}}{\left[\text{B}\right]}^{\text{n}}{\left[\text{C}\right]}^{\text{p}}\text{,where 'm, n and p' are orders of the reactants}\text{.}$

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 given one-step mechanism is,

$A+B+C\stackrel{}{\to }D$

The rate law for the above reaction is, $\text{Rate}\text{ = k}\left[\text{A}\right]\left[\text{B}\right]\left[\text{C}\right]$        (1)

Yes, the rate law of one-step reaction mechanism is equal to the actual rate law.