Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 4 / b

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- Solution b

Solution

For

A + B ⇌ C + D

, the forward and reverse rates are

r_{f} = k_{f} n_{A} n_{B}, r_{r} = k_{r} n_{C} n_{D} .

At thermochemical equilibrium,

r_{f} = r_{r}

by detailed balance, while

K_{c} = \frac{n _{C} n _{D}}{n _{A} n _{B}} = exp (- \frac{Δ _{r} G ^{\circ}}{RT})

after the appropriate standard-concentration factors are included. The standard reaction Gibbs free energy is

Δ_{r} G^{\circ} = G_{C}^{\circ} + G_{D}^{\circ} - G_{A}^{\circ} - G_{B}^{\circ} .

Consequently

k_{r} = \frac{k _{f}}{K _{c}} = k_{f} exp (\frac{Δ _{r} G ^{\circ}}{RT}) .

For other stoichiometries, the same argument uses the corresponding activity product and its standard-state factors.

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