Question

Consider the reaction

$\mathrm{A}+\mathrm{B}\underset{}{\overset{}{\leftrightharpoons }}\mathrm{C}+\mathrm{D}$

with all reactants and products gaseous (for simplicity) and an equilibrium constant K.

(a) Assume that the elementary steps in the reaction are those indicated by the stoichiometric equation (in each direction), with specific rate constants for the forward reaction and the reverse reaction, respectively, k_f and k_r. Derive the relation between k_f, k_r, and K. Comment on the general validity of the assumptions made about the relation of elementary steps and the stoichiometric equation and also on the general validity of K.

(b) Assume that the reaction as written is exothermic. Explain what this implies about the change of K with temperature. Explain also what it implies about the relation of the activation energies of the forward and reverse reactions and how this relation is consistent with your statement about the variation of K with temperature.

Short answer

a.The above relation is$\mathrm{K}=\frac{{\mathrm{k}}_{\mathrm{r}}}{{\mathrm{k}}_{\mathrm{f}}}$ valid for gas phase reactions.

b. $\mathrm{InK}={\mathrm{InA}}_{\mathrm{E}}+\left(\frac{{{\mathrm{E}}_{\mathrm{a}}}_{\mathrm{r}}-{{\mathrm{E}}_{\mathrm{a}}}_{\mathrm{f}}}{\mathrm{RT}}\right)$

Step-by-step solution

Relation between equilbrium constant and  rate of forward and backward reactions:

a.For the reaction:

$\mathrm{A}+\mathrm{B}\underset{{\mathrm{k}}_{-1_{}}}{\overset{{\mathrm{k}}_1}{\leftrightharpoons }}\mathrm{C}+\mathrm{D}$

Where k_f is the rate of forward reaction and k_b is the rate constant of backward reaction.

$$\begin{gathered} \mathrm{Rate}\mathrm{of}\mathrm{forward}\mathrm{reaction}={\mathrm{k}}_{\mathrm{f}}\left[\mathrm{A}\right]\left[\mathrm{B}\right] \\ \mathrm{Rate}\mathrm{of}\mathrm{backward}\mathrm{reaction}={\mathrm{k}}_{\mathrm{r}}\left[\mathrm{C}\right]\left[\mathrm{D}\right] \end{gathered}$$

Atthestate of equilbrium Rate of forward reaction is equal to backward reaction

The expression of equilbrium constant is given by

localid="1661585908550" $$\begin{gathered} \mathrm{K}=\frac{\left[\mathrm{C}\right]\left[\mathrm{D}\right]}{\left[\mathrm{A}\right]\left[\mathrm{B}\right]}..............\left(1\right) \\ \frac{{\mathrm{rate}}_{\mathrm{f}}}{{\mathrm{rate}}_{\mathrm{r}}}=\frac{{\mathrm{k}}_{\mathrm{f}}\left[\mathrm{A}\right]\left[\mathrm{B}\right]}{{\mathrm{k}}_{\mathrm{r}}\left[\mathrm{C}\right]\left[\mathrm{D}\right]}.......\left(2\right) \\ 1=\frac{{\mathrm{k}}_{\mathrm{f}}\left[\mathrm{A}\right]\left[\mathrm{B}\right]}{{\mathrm{k}}_{\mathrm{r}}\left[\mathrm{C}\right]\left[\mathrm{D}\right]} \\ \frac{\left[\mathrm{C}\right]\left[\mathrm{D}\right]}{\left[\mathrm{A}\right]\left[\mathrm{B}\right]}=\frac{{\mathrm{k}}_{\mathrm{r}}}{{\mathrm{k}}_{\mathrm{f}}} \\ \mathrm{But} \\ \mathrm{K}=\frac{\left[\mathrm{C}\right]\left[\mathrm{D}\right]}{\left[\mathrm{A}\right]\left[\mathrm{B}\right]}..............\left(1\right) \\ \mathrm{K}=\frac{{\mathrm{k}}_{\mathrm{r}}}{{\mathrm{k}}_{\mathrm{f}}}.............\left(3\right) \end{gathered}$$

The above relation $\mathrm{K}=\frac{{\mathrm{k}}_{\mathrm{r}}}{{\mathrm{k}}_{\mathrm{f}}}$is valid for gas phase reactions.

Temparature variation with forward and reverse reactions:

b) if the forward reaction is exothermic

$\mathrm{A}+\mathrm{B}\underset{{\mathrm{k}}_{\mathrm{b}}}{\overset{{\mathrm{k}}_{\mathrm{f}}}{\leftrightharpoons }}\mathrm{C}+\mathrm{D}-{\mathrm{E}}_{\mathrm{heat}}$

If temperature increases reverse reaction is favourable, k_f>k_r,so K value decreases

According to arrehenius equation.

localid="1661586568776" $\mathrm{k}={\mathrm{Ae}}^{{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}}.........\left(4\right)$

Where k is rate constant of the reaction,A is the reaction constant,E is the activation energy,R is the universal gas constant,T is the absolute temparature.

localid="1661586579807" $$\begin{gathered} {\mathrm{k}}_{\mathrm{f}}={\mathrm{A}}_{\mathrm{f}}{\mathrm{e}}^{\frac{{\mathrm{E}}_{\mathrm{f}}}{\mathrm{RT}}}........\left(4\mathrm{a}\right) \\ {\mathrm{k}}_{\mathrm{r}}={\mathrm{A}}_{\mathrm{r}}{\mathrm{e}}^{\frac{{\mathrm{E}}_{\mathrm{r}}}{\mathrm{RT}}}..........\left(4\mathrm{b}\right) \end{gathered}$$

Where E_fand E_r are the the activation energy of forward and backward reactions.

Ratio of these equation has to be taken

localid="1661586617497" $$\begin{gathered} \mathrm{K}=\frac{{\mathrm{A}}_{\mathrm{f}}{\mathrm{e}}^{\frac{{\mathrm{E}}_{\mathrm{f}}}{\mathrm{RT}}}}{{\mathrm{A}}_{\mathrm{r}}{\mathrm{e}}^{\frac{\mathrm{Er}}{\mathrm{RT}}}}...................\left(5\right) \\ \mathrm{K}={\mathrm{A}}_{\mathrm{E}}{\mathrm{e}}^{-\left({\mathrm{E}}_{\mathrm{f}}-{\mathrm{E}}_{\mathrm{r}}\right)/\mathrm{RT}}...........................\left(6\right) \\ \mathrm{Taking}\mathrm{logartham}\mathrm{on}\mathrm{both}\mathrm{sides} \\ \mathrm{InK}={\mathrm{InA}}_{\mathrm{E}}+\left(\frac{{{\mathrm{E}}_{\mathrm{a}}}_{\mathrm{r}}-{{\mathrm{E}}_{\mathrm{a}}}_{\mathrm{f}}}{\mathrm{RT}}\right) \end{gathered}$$

The graph is drawn between Energy vs reaction co-ordinate, if the forward reaction is exothermic , the graph is

For exothermic reactionE_ar>E_{af .}

For the eqaution a, (E_ar-E_af) is positive value and hence Temparature increases and InK value decreases.