Question

The reaction${\mathbf{A}}_{\mathbf{\left(}\mathbf{aq}\mathbf{\right)}}\mathbf{+}{\mathbf{B}}_{\mathbf{\left(}\mathbf{aq}\mathbf{\right)}}\mathbf{\to }{\mathbf{Products}}_{\mathbf{\left(}\mathbf{aq}\mathbf{\right)}}$ was studied, and the following data were obtained:

$$\begin{gathered} \mathbf{[}\mathbf{A}{\mathbf{]}}_{\mathbf{0}} \\ \mathbf{(}\mathbf{mol}\mathbf{/}\mathbf{L}\mathbf{)} \end{gathered}$$	$$\begin{gathered} \mathbf{[}\mathbf{B}{\mathbf{]}}_{\mathbf{0}} \\ \mathbf{(}\mathbf{mol}\mathbf{/}\mathbf{L}\mathbf{)} \end{gathered}$$	$$\begin{gathered} \mathbf{Initial}\mathbf{\text{Rate}} \\ \mathbf{\left(}{\mathbf{molL}}^{\mathbf{-}\mathbf{1}}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)} \end{gathered}$$
0.12	0.18	$\mathbf{3}\mathbf{.}\mathbf{46}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$
0.060	0.12	$\mathbf{1}\mathbf{.}\mathbf{15}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$
0.030	0.090	$\mathbf{4}\mathbf{.}\mathbf{32}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$
0.24	0.090	$\mathbf{4}\mathbf{.}\mathbf{32}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$

What is the order of the reaction with respect to A?

What is the order of the reaction with respect to B?

What is the value of the rate constant for the reaction?

Short answer

The order of the reaction with respect to A is 1.

The order of the reaction with respect to B is 1.

The value of the rate constant for the reaction is$1.60{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$

Step-by-step solution

Determining the order of the reaction with respect to A.

First, we can use last two experiments, inwhich concentration of reactant B is being held constant, to determine the order of the reaction with respect to reactant A:

$$\begin{gathered} \frac{{\mathrm{rate}}_4}{{\mathrm{rate}}_3}=\frac{\mathrm{k}{\left[\mathrm{A}\right]}^{\mathrm{x}}{\left[\mathrm{B}\right]}^{\mathrm{y}}}{\mathrm{k}{\left[\mathrm{A}\right]}^{\mathrm{x}}{\left[\mathrm{B}\right]}^{\mathrm{y}}}=\frac{{\left[\mathrm{A}\right]}^{\mathrm{x}}}{{\left[\mathrm{A}\right]}^{\mathrm{x}}} \\ \frac{3.46\times {10}^{-2}\mathrm{M}{\mathrm{s}}^{-1}}{4.32\times {10}^{-3}\mathrm{M}{\mathrm{s}}^{-1}}={\left(\frac{0.24\mathrm{M}}{0.030\mathrm{M}}\right)}^{\mathrm{x}} \\ 8=8^{\mathrm{x}} \\ \mathrm{x}=1 \end{gathered}$$

Determining the order of the reaction with respect to B.

Now that we know x, we can use any other two experiments to determine y the same way.

$$\begin{gathered} \frac{{\mathrm{rate}}_2}{{\mathrm{rate}}_1}=\frac{\mathrm{k}{\left[\mathrm{A}\right]}^{\mathrm{x}}{\left[\mathrm{B}\right]}^{\mathrm{y}}}{\mathrm{k}{\left[\mathrm{A}\right]}^{\mathrm{x}}{\left[\mathrm{B}\right]}^{\mathrm{y}}} \\ \mathrm{Now}\text{we can solve for y}: \\ \text{y =1} \end{gathered}$$

Determining the value of the rate constant for the reaction.

Now that we have both x and y, we can write out the rate law expression:

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

We can solve for the reaction rate constant:

role="math" localid="1663844189620" $$\begin{gathered} \mathrm{k}=\frac{\mathrm{rate}}{\left[\mathrm{A}\right]\left[\mathrm{B}\right]}=\frac{3.46\times {10}^{-2}\mathrm{M}{\mathrm{s}}^{-1}}{0.12\mathrm{M}\times 0.18\mathrm{M}} \\ \mathrm{k}=1.60{\mathrm{M}}^{-1}{\mathrm{s}}^{-1} \end{gathered}$$