Chapter 16, Problem 85A

Apply the method of initial rates to determine the order of a chemical reaction with respect to Reactant X. Create a set of hypothetical experimental data that would lead you to conclude that the reaction is second order in X.

Interpretation Introduction

Interpretation: The order of a chemical reaction with respect to reactant X needs to be determined.

Concept Introduction: A rate law expresses the relationship between the rate of a chemical reaction and reactants concentration. For example, the reaction $\text{A }\to \text{ B}$ is a one-step reaction. The rate law for this reaction is expressed as follows.

$\text{rate}=\text{k}\left[\text{A}\right]$

Here, k and [A] represent the rate constant and concentration of the reactant A.

Answer to Problem 85A

From the table, it is clear that the initial concentration of X is doubled and the initial rate of the reaction is increased by four times. Therefore, the order of the reaction with respect to X is 2.

Experiment	Initial $\left[\text{X}\right]\left(\text{mol/L}\right)$	Initial rate $\left(\text{mol/L}\text{.s}\right)$
1	$1.0$	$2.5\times {10}^{-2}$
2	$2.0$	$1.0\times {10}^{-1}$

Explanation of Solution

Consider the general reaction

$\text{aX}\to \text{Products}$

In order to determine the rate law experimentally for a reaction, the method of initial rates is used. In this method, at least two trials with different initial amounts of the reactants are taken to determine the effect that the reactant has on the initial reaction rate.

Experiment	Initial $\left[\text{X}\right]\left(\text{mol/L}\right)$	Initial rate $\left(\text{mol/L}\text{.s}\right)$
1	$1.0$	$2.5\times {10}^{-2}$
2	$2.0$	$1.0\times {10}^{-1}$

When the initial concentration of X is doubled, the initial rate of the reaction is increased by 4 times.

$\text{rate}=\left(2.5\times {10}^{-2}\right)\frac{\text{L}}{\text{mol}\text{.s}}{\left[\text{X}\right]}^2$ when X is $\text{1}\text{.0}\text{mol/L}$

When X is $\text{2}\text{.0}\text{mol/L}$ , $\text{rate}=\text{k}\left[\text{A}\right]$ which is shown in the table. Therefore, the above data shows that the order of the reaction is 2.

Conclusion

The method of initial rates specifies the order of reaction by comparing the initial rates of a reaction conducted to different concentrations of reactants.