Past exam of the chemistry course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 5 / a

With the forward direction defined as cis-to-trans, the elementary steps have rates $k_1[\mathrm{cis}]$ and $k_{-1}[\mathrm{trans}]$. One trans molecule is gained per forward event and lost per reverse event, soEach rate constant has units ${\mathrm{s}}^{-1}$, and the right side has units of concentration per unit time. This net rate can have either sign, even though the individual event rates are nonnegative.

At chemical equilibrium, the forward and reverse event rates are equal, rather than both reactions stopping:For the dilute-solution concentration model,The ratio is dimensionless because both concentrations and both rate constants have matching units. An exact thermodynamic equilibrium constant uses activities; here the common standard concentration and ideal-solution activity coefficients cancel.

Let $C_{\mathrm{tot}}=[\mathrm{cis}]+[\mathrm{trans}]$, which is conserved for this closed one-to-one isomerization at fixed volume. Substitute $[\mathrm{cis}{]}_{298}=C_{\mathrm{tot}}-[\mathrm{trans}{]}_{298}$ into the equilibrium ratio:ThereforeThe equilibrium split is independent of having started with pure cis, provided equilibration is complete and no side reaction changes the total concentration.