Question

The rate constant of a reaction is $\mathbf{4}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}$at $\mathbf{25}{\mathbf{}}^{\mathbf{0}}\mathbf{C}$, and the activation energy is 33.6 kJ/mol. What is k at $\mathbf{75}{\mathbf{}}^{\mathbf{0}}\mathbf{C}$?

Short answer

The rate constant of a reaction is $4.7\times {10}^{-3}{\mathrm{s}}^{-1}$at, and the activation energy is 33.6 kJ/mol. Thenat 75°C, the rate constant is role="math" localid="1654921474710" $0.33{\mathrm{s}}^{-1}$.

Step-by-step solution

Step 1: The Arrhenius equation

TheArrhenius equationis written like this:

$$\begin{gathered} \mathrm{K}={\mathrm{Ae}}^{-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}} \\ \mathrm{In}\mathrm{K}=\mathrm{In}\mathrm{A}-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT} \end{gathered}$$

Where k is the rate constant, A is the frequency factor, ${\mathrm{E}}_{\mathrm{a}}$is the reaction's activation energy at a certain temperature T, and R is the gas constant.

Step 2: What is k at 75 0C ?

Create a new expression for the two rate constants, ${\mathrm{K}}_1$and ${\mathrm{K}}_2$, for temperatures ${\mathrm{T}}_1$and ${\mathrm{T}}_2$ , respectively:

$$\begin{gathered} {\mathrm{InK}}_1=\mathrm{In}\mathrm{A}-{\mathrm{E}}_{\mathrm{a}}/{\mathrm{RT}}_1 \\ {\mathrm{InK}}_2=\mathrm{In}\mathrm{A}-{\mathrm{E}}_{\mathrm{a}}/{\mathrm{RT}}_2 \\ {\mathrm{InK}}_2-{\mathrm{InK}}_1=-\frac{{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}\left(\frac{1}{{\mathrm{T}}_1}-\frac{1}{{\mathrm{T}}_2}\right) \\ \mathrm{In}\left(\frac{{\mathrm{K}}_2}{{\mathrm{K}}_1}\right)=-\frac{{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}\left(\frac{1}{{\mathrm{T}}_1}-\frac{1}{{\mathrm{T}}_2}\right) \end{gathered}$$

Replace the values and calculate the rate constant:

role="math" localid="1654922725958" $$\begin{gathered} \mathrm{In}\left(\frac{{\mathrm{K}}_2}{4.7\times {10}^{-3}{\mathrm{s}}^{-1}}\right)=\frac{33.6\mathrm{KJ}/\mathrm{mol}}{8.314\mathrm{J}/\mathrm{moLK}}\left(\frac{1}{(273+75)\mathrm{K}}-\frac{1}{(273+25)\mathrm{K}}\right)\times \left(\frac{1000\mathrm{J}}{1\mathrm{KJ}}\right) \\ \mathrm{In}\left(\frac{{\mathrm{K}}_2}{4.7\times {10}^{-3}{\mathrm{s}}^{-1}}\right)=1.948515 \\ {\mathrm{K}}_2=4.7\times {10}^{-3}{\mathrm{s}}^{-1}\times \left({\mathrm{e}}^{1.948515}\right) \\ {\mathrm{K}}_2=0.33{\mathrm{s}}^{-1} \end{gathered}$$

At 75°C, the rate constant is $0.33{\mathrm{s}}^{-1}$.