(h) Reversible Reactions

Consider the following elementary first order reversible reaction. \[A\xrightleftharpoons[k_2]{k_1} R\] \[\begin{align*} \text{Thermodynamic equilibrium constant $K$ is given by} \\ K &= \frac{k_1}{k_2} \\ \text{Thermodynamic equilibrium constant based on concentration ($K_c$) is given by} \\ K_c &= \frac{C_R}{C_A}\end{align*}\] The relation between \(K\) and \(K_C\) is given by \[K = K_c(RT)^{\Delta \nu}\] where, \(\Delta \nu\) = difference in stoichiometric coefficients.

e.g.: For \(aA \rightarrow rR\),  \(\Delta \nu = r-a\).
For \(aA+bB\rightarrow rR + sS\),  \(\Delta \nu=(r+s)-(a+b)\).

So, here, \(\Delta \nu=1-1=0\). Hence, \(K=K_c\). \[K = \frac{C_{Re}}{C_{Ae}} \qquad \Longrightarrow \quad K = \frac{C_{A0}X_{Ae}}{C_{A0}(1-X_{Ae})} =\frac{X_{Ae}}{1-X_{Ae}}\] where, \(X_{Ae}\) = equilibrium conversion. \[X_{Ae} = \frac{K}{1+K}\] Note: the equilibrium conversion (\(X_{Ae}\)) is the maximum fractional conversion achievable at a given temperature. The integrated rate expression for this case is given by \[-\ln \left(1 - \frac{X_A}{X_{Ae}}\right) = \frac{M+1}{M+X_{Ae}} k_1 t\]