Domain: Biochemical redox chemistry

In aqueous \(0.10\ \mathrm{M}\) phosphate buffer at \(\mathrm{pH}=7.0\) and \(T=298.15\ \mathrm{K}\), the disulfide exchange \(\mathrm{BSSB}+\mathrm{MSSM}\rightleftharpoons 2\,\mathrm{BSSM}\) is to be analyzed. Potentials are reported vs the standard hydrogen electrode at the biochemical standard state \(c^\circ=1\ \mathrm{mM}\). The notation \(E^{0\prime}\) denotes an apparent standard reduction potential at \(\mathrm{pH}=7.0\), and the starred value \(E^{0\prime *}\) denotes the same potential with the heterodisulfide statistical factor removed. The couples are two-electron \(n=2\). Values: \(E^{0\prime *}(\mathrm{BSSM})=0.272\ \mathrm{V}\), \(E^{0\prime}(\mathrm{BSSM})=0.281\ \mathrm{V}\), \(E^{0\prime}(\mathrm{BSSB})=0.270\ \mathrm{V}\), \(E^{0\prime}(\mathrm{MSSM})=0.271\ \mathrm{V}\). An equimolar mixture of the two homodisulfides is prepared at a common initial concentration \(c_0\), with no heterodisulfide initially; activities are approximated by concentrations, and the equilibrium constant is dimensionless under \(c^\circ=1\ \mathrm{mM}\).

Calculate the equilibrium mole fraction of \(\mathrm{BSSM}\) in the disulfide pool at these conditions, round to three decimals.

Grading rubric:
criterion: The solution correctly formulates the structural exchange Gibbs energy using starred potentials as ΔG°_ex,struct = − n F [ 2E0'* (BSSM) − E0'(BSSB) − E0'(MSSM) ]
weight: 20
criterion: The solution correctly evaluates the potential difference inside the brackets as +0.003 V using the provided E0' values
weight: 10
criterion: The solution correctly computes ΔG°_ex,struct numerically as −579 J mol−1 without unit inconsistency
weight: 10
criterion: The solution correctly computes the statistical potential offset as ΔE_stat = 0.009 V from the difference between E0' and E0'* for BSSM
weight: 10
criterion: The solution correctly applies the statistical correction to obtain ΔG°_ex = −4.05 kJ mol−1
weight: 10
criterion: The solution correctly converts ΔG°_ex to the equilibrium constant using K = exp(−ΔG° / RT) at 298.15 K
weight: 10
criterion: The solution correctly evaluates the equilibrium constant numerically as K = 5.13
weight: 10
criterion: The solution correctly derives the extent ratio x/c0 = sqrt(K) / (2 + sqrt(K)) for the reaction stoichiometry A + B ⇌ 2 C
weight: 10
criterion: The solution correctly reports the equilibrium mole fraction of BSSM as 0.531 rounded to three decimals
weight: 10

Answer:
0.531