GATE 2021 CH – Question 34

Chemical Reaction Engineering · Reaction rates and kinetics of homogeneous reactions · 1 mark · Numerical answer

The following homogeneous liquid phase reactions are at equilibrium. The values of rate constants are given by: $k_1=0.1\ \mathrm{s^{-1}}$, $k_{-1}=0.2\ \mathrm{s^{-1}}$, $k_2=1\ \mathrm{s^{-1}}$, $k_{-2}=10\ \mathrm{s^{-1}}$, $k_3=10\ \mathrm{s^{-1}}$. The value of rate constant $k_{-3}$ is _____ $\mathrm{s^{-1}}$ (round off to 1 decimal place).

reversible cycle A → B → C → A has forward constants $k_1,k_2,k_3$ and reverse constants $k_{-1},k_{-2},k_{-3}$, respectively.

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Correct answer: 0.49 to 0.51

Explanation

At equilibrium each reversible elementary pair satisfies detailed balance. Thus $C_B/C_A=k_1/k_{-1}=0.1/0.2=0.5$ and $C_C/C_B=k_2/k_{-2}=1/10=0.1$.

Combining these gives $C_C/C_A=0.05$. On the C-to-A link, $k_3C_C=k_{-3}C_A$, so $k_{-3}=k_3(C_C/C_A)=10(0.05)=\mathbf{0.5\ \mathrm{s^{-1}}}$.

Equivalently, the product of rate constants around the equilibrium cycle must equal that around the reverse cycle.