Chapter 13, Problem 13.49QA

Interpretation Introduction

To find:

The rate law and units of the rate constant (by using the units $M$ for concentration and $s$ for time) for the given reactions.

Answer to Problem 13.49QA

Solution:

a) The reaction of oxygen atoms with ${NO}_2$ is first order in both reactants.

Rate law is $Rate=k\left[O\right]\left[{NO}_2\right]$

Unit of rate constant is $M^{-1}{s}^{-1}$

b) The reaction between $NO$ and ${Cl}_2$ is second order in $NO$ and first order in ${Cl}_2$

Rate law is $Rate=k{\left[{NO}_2\right]}^2\left[{Cl}_2\right]$

Unit of rate constant is $M^{-2}{s}^{-1}$

c) The reaction between ${Cl}_2$ and chloroform $CH{Cl}_3$ is first order in $CH{Cl}_3$ and one half order in ${Cl}_2$.

Rate law is $Rate=k\left[{CHCl}_3\right][{{Cl}_2]}^{1/2}$

Unit of rate constant is $M^{-1/2}{s}^{-1}$

d) The decomposition of ozone ($O_3$) to $O_2$ is second order in $O_3$ and an order of -1 in $O$ atoms.

Rate law is $Rate=k{\left[O_3\right]}^2[{O]}^{-1}$

Unit of rate constant is $s^{-1}$

Explanation of Solution

1) Concept:

i. Rate law:

Rate of reaction is proportional to some power of the change in concentration of reactant.

Therefore,

$Rate \propto {\left[reactant\right]}^m$

Hence rate law will be,

$Rate=k{\left[reactant\right]}^m$….. (equation 1)

where $k$ is the rate constant, and $m$ is the order of reaction with respect to the reactant.

ii. Unit of reaction rate:

Rate is change in concentration per unit time, so the unit for rate is $M/s$(or $M s^{-1})$

Here unit for concentration is $M$(molarity) and time is $s$(second).

.

iii. Unit of rate constant $\mathit{k}$:

From equation 1,

$k=\frac{Rate}{{\left[reactant\right]}^m}$

Units of rate are $M s^{-1}$ and that of concentration is $M$. Therefore, units of rate constant are:

$k=\frac{M s^{-1}}{M^m}=M^{1-m}{s}^{-1}$

2) Calculations:

a. The reaction of oxygen atoms with ${NO}_2$ is first order in both reactants.

Rate law:

The reaction is first order in both reactants. Therefore, the exponents of concentration of both reactants is 1, and the rate law is:

$Rate=k\left[O\right]\left[{NO}_2\right]$

Units of rate constant $\mathit{k}$:

Rearranging the rate law for rate constant, we get:

$k=\frac{Rate}{\left[O\right]\left[{NO}_2\right]}$

$k=\frac{M s^{-1}}{M\times M}$

$k=\frac{s^{-1}}{M}$

$k=M^{-1}{s}^{-1}$

b. The reaction between $NO$ and ${Cl}_2$ is second order in $NO$ and first order in ${Cl}_2$.

Rate law:

The reaction is second order in $NO$ and first order in ${Cl}_2$. Therefore, the exponents of concentrations of $NO$ and of ${Cl}_2$ are 2 and 1 respectively.

Therefore, the rate law is:

$Rate=k{\left[{NO}_2\right]}^2\left[{Cl}_2\right]$

Unit of rate constant $\mathit{k}$:

Rearranging this rate law for rate constant, we get:

$k=\frac{Rate}{{\left[{NO}_2\right]}^2\left[{Cl}_2\right]}$

$k=\frac{M s^{-1}}{M^2\times M}$

$k=\frac{ s^{-1}}{M^2}$

$k=M^{-2}{s}^{-1}$

c. The reaction between ${Cl}_2$ and chloroform $CH{Cl}_3$ is first order in $CH{Cl}_3$ and one half order in ${Cl}_2$.

Rate law:

The reaction is first order in $CH{Cl}_3$ and one half order in ${Cl}_2$. Therefore, the exponents of concentrations of $CH{Cl}_3$ and ${Cl}_2$ are 1 and ½ respectively and the rate law is:

$Rate=k\left[{CHCl}_3\right][{{Cl}_2]}^{1/2}$

Units of rate constant $\mathit{k}$:

Rearranging the rate equation for rate constant, we get:

$k=\frac{Rate }{\left[{CHCl}_3\right][{{Cl}_2]}^{1/2}}$

$k=\frac{M s^{-1} }{M\times M^{1/2}}$

$k=\frac{ s^{-1} }{M^{1/2}}$

$k=M^{-1/2}{s}^{-1}$

d. The decomposition of ozone ($O_3$) to $O_2$ is second order in $O_3$ and an order of -1 in $O$ atoms.

Rate law:

The reaction is second order in $O_3$ and -1 order in $O$ atoms, i.e., the exponents of concentrations of $O_3$ and $O$ atoms are 2 and -1 respectively, and the rate law is

$Rate=k{\left[O_3\right]}^2[{O]}^{-1}$

Units of rate constant $\mathit{k}$:

Rearranging the rate equation for rate constant, we get:

$k= \frac{Rate }{{\left[O_3\right]}^2[{O]}^{-1}}$

$k=\frac{M s^{-1} }{M^2\times M^{-1}}$

$k=\frac{M\times M s^{-1} }{M\times M}$

$k=s^{-1}$

Conclusion:

Rate law determines the relation between the concentration of reactants and the rate of reaction. Order with respect to the reactants is the exponent of each reactant in the rate law.$k$, the constant of proportionality in the rate law, is called the rate constant. The units of $k$ can be found from the rate law.