Chapter 12, Problem 29E

What are the units for each of the following if the concentrations are expressed in moles per liter and the time in seconds?

a. rate of a chemical reaction

b. rate constant for a zero-order rate law

c. rate constant for a first-order rate law

d. rate constant for a second-order rate law

e. rate constant for a third-order rate law

Interpretation Introduction

Interpretation: The units for each of the given terms are to be calculated taking concentration in terms of moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

Concept introduction: The rate of a reaction is directly proportional to the concentration of the reactant. For a reaction, $\text{X}\to \text{Products}$,

$\begin{array}{l}\text{Rate}\propto {\left[\text{X}\right]}^n\\ \text{Rate}=k{\left[\text{X}\right]}^n\end{array}$

The constant $k$ in the above expression is the rate constant.

Answer to Problem 29E

The unit of the rate of a chemical reaction is $\text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}$.

Explanation of Solution

Given

Concentration is expressed in moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

The rate of a reaction is calculated by the formula,

$\text{Rate}\text{of}\text{reaction}=\frac{\text{Change}\text{in}\text{the}\text{concentration}}{\text{Change}\text{in}\text{time}}$

Substitute the given units of concentration and time in the above expression.

$\begin{array}{c}\text{Rate}\text{of}\text{reaction}=\frac{\text{mol}/\text{L}}{\text{s}}\\ =\text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}\end{array}$

Conclusion

The unit of the rate of a chemical reaction is $\text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}$.

Interpretation Introduction

Interpretation: The units for each of the given terms are to be calculated taking concentration in terms of moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

Concept introduction: The rate of a reaction is directly proportional to the concentration of the reactant. For a reaction, $\text{X}\to \text{Products}$,

$\begin{array}{l}\text{Rate}\propto {\left[\text{X}\right]}^n\\ \text{Rate}=k{\left[\text{X}\right]}^n\end{array}$

The constant $k$ in the above expression is the rate constant.

Answer to Problem 29E

The unit of the rate constant for a zero order reaction is $\text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}$.

Explanation of Solution

Given

The reaction is a zero order reaction.

Concentration is expressed in moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

For a reaction, $\text{X}\to \text{Products}$,

$\text{Rate}=k{\left[\text{X}\right]}^n$

Where,

• $k$ is the rate constant.
• $n$ is the order of a reaction.
• $\left[\text{X}\right]$ is the concentration of the reactant $\text{X}$.

The reaction is of zero order.

Substitute the order and the units in the above expression.

$\begin{array}{c}\text{Rate}=k{\left[\text{X}\right]}^n\\ \text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}=k{\left[\text{X}\right]}^0\\ k=\text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}\end{array}$

Conclusion

The unit of the rate constant for a zero order reaction is $\text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}$.

Interpretation Introduction

Interpretation: The units for each of the given terms are to be calculated taking concentration in terms of moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

Concept introduction: The rate of a reaction is directly proportional to the concentration of the reactant. For a reaction, $\text{X}\to \text{Products}$,

$\begin{array}{l}\text{Rate}\propto {\left[\text{X}\right]}^n\\ \text{Rate}=k{\left[\text{X}\right]}^n\end{array}$

The constant $k$ in the above expression is the rate constant.

To determine: The unit of the rate constant for a first order reaction.

Answer to Problem 29E

The unit of the rate constant for a first order reaction is ${\text{s}}^{-1}$.

Given

The reaction is a first order reaction.

Concentration is expressed in moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

For a reaction, $\text{X}\to \text{Products}$,

$\text{Rate}=k{\left[\text{X}\right]}^n$

Where,

• $k$ is the rate constant.
• $n$ is the order of a reaction.
• $\left[\text{X}\right]$ is the concentration of the reactant $\text{X}$.

The reaction is of first order.

Substitute the order and the units in the above expression.

$\begin{array}{c}\text{Rate}=k{\left[\text{X}\right]}^1\\ \text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}=k{\left[\text{mol}\cdot {\text{L}}^{-1}\right]}^1\\ k={\text{s}}^{-1}\end{array}$

Explanation of Solution

Conclusion

The unit of the rate constant for a first order reaction is ${\text{s}}^{-1}$.

Interpretation Introduction

Interpretation: The units for each of the given terms are to be calculated taking concentration in terms of moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

Concept introduction: The rate of a reaction is directly proportional to the concentration of the reactant. For a reaction, $\text{X}\to \text{Products}$,

$\begin{array}{l}\text{Rate}\propto {\left[\text{X}\right]}^n\\ \text{Rate}=k{\left[\text{X}\right]}^n\end{array}$

The constant $k$ in the above expression is the rate constant.

Answer to Problem 29E

The unit of the rate constant for a second order reaction is ${\text{mol}}^{-1}{\text{L}}^{}{\text{s}}^{-1}$.

Explanation of Solution

Given

The reaction is a second order reaction.

Concentration is expressed in moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

For a reaction, $\text{X}\to \text{Products}$,

$\text{Rate}=k{\left[\text{X}\right]}^n$

Where,

• $k$ is the rate constant.
• $n$ is the order of a reaction.
• $\left[\text{X}\right]$ is the concentration of the reactant $\text{X}$.

The reaction is of second order.

Substitute the order and the units in the above expression.

$\begin{array}{c}\text{Rate}=k{\left[\text{X}\right]}^2\\ \text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}=k{\left[\text{mol}\cdot {\text{L}}^{-1}\right]}^2\\ \text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}=k\left[{\text{mol}}^2\cdot {\text{L}}^{-2}\right]\end{array}$

Simplify the above expression.

$\begin{array}{c}k=\frac{\text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}}{\left[{\text{mol}}^2\cdot {\text{L}}^{-2}\right]}\\ ={\text{mol}}^{-1}\cdot \text{L}\cdot {\text{s}}^{-1}\end{array}$

Conclusion

The unit of the rate constant for a second order reaction is ${\text{mol}}^{-1}{\text{L}}^{}{\text{s}}^{-1}$.

Interpretation Introduction

Interpretation: The units for each of the given terms are to be calculated taking concentration in terms of moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

Concept introduction: The rate of a reaction is directly proportional to the concentration of the reactant. For a reaction, $\text{X}\to \text{Products}$,

$\begin{array}{l}\text{Rate}\propto {\left[\text{X}\right]}^n\\ \text{Rate}=k{\left[\text{X}\right]}^n\end{array}$

The constant $k$ in the above expression is the rate constant.

Answer to Problem 29E

The unit of the rate constant for a third order reaction is ${\text{mol}}^{\text{-2}}{\text{L}}^{\text{2}}{\text{s}}^{\text{-1}}$.

To determine: The unit of the rate constant for a third order reaction.

Explanation of Solution

Given:

The reaction is a third order reaction.

Concentration is expressed in moles per liter $\left(\text{mol}/\text{L}\right)$ and time in seconds $\left(\text{s}\right)$.

For a reaction, $\text{X}\to \text{Products}$,

$\text{Rate}=k{\left[\text{X}\right]}^n$

Where,

• $k$ is the rate constant.
• $n$ is the order of a reaction.
• $\left[\text{X}\right]$ is the concentration of the reactant $\text{X}$.

The reaction is of third order.

Substitute the order and the units in the above expression.

$\begin{array}{c}\text{Rate}=k{\left[\text{X}\right]}^3\\ \text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}=k{\left[\text{mol}\cdot {\text{L}}^{-1}\right]}^3\\ \text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}=k\left[{\text{mol}}^3\cdot {\text{L}}^{-3}\right]\end{array}$

Simplify the above expression.

$\begin{array}{c}k=\frac{\text{mol}\cdot {\text{L}}^{-1}\cdot {\text{s}}^{-1}}{\left[{\text{mol}}^3\cdot {\text{L}}^{-3}\right]}\\ ={\text{mol}}^{-2}\cdot {\text{L}}^2\cdot {\text{s}}^{-1}\end{array}$

Conclusion

The unit of the rate constant for a third order reaction is ${\text{mol}}^{\text{-2}}{\text{L}}^{\text{2}}{\text{s}}^{\text{-1}}$.