Question #44743

A researcher attempts to determine the limiting molar conductivity ($\Lambda_m^\circ$) of the actinoid salt $\ce{UO2Cl}$ by preparing a $1.0 \times 10^{-3}\text{ M}$ solution in a $1.0\text{ M}$ $\ce{HClO4}$ buffer at $298\text{ K}$ (to prevent hydrolysis). However, the Debye-Hückel-Onsager extrapolation fails because $\ce{UO2+}$ undergoes the following disproportionation equilibrium:

$\ce{2UO2+_{(aq)} + 4H+_{(aq)} <=> U^{4+}_{(aq)} + UO2^{2+}_{(aq)} + 2H2O_{(l)}}$

Given the standard reduction potentials $E^\circ(\ce{UO2^{2+}}/\ce{UO2+}) = +0.162\text{ V}$ and $E^\circ(\ce{UO2+}/\ce{U^{4+}}) = +0.280\text{ V}$, calculate the percentage of $\ce{UO2+}$ that has disproportionated in this buffered solution, which fundamentally alters the charge carrier composition.

(Assume ideal behavior and $\frac{2.303 RT}{F} = 0.059\text{ V}$)