Past exam of the biological physics course of the University of Oxford / 2022 / 3 / a / Solution

In the continuum cylindrical-pore estimate, the electrical resistance is $R_p=\rho \ell /(\pi a^2)$, with radius $a=0.075\mathrm{nm}$, length $\ell =0.5\mathrm{nm}$ and electrical resistivity $\rho =80\Omega \mathrm{cm}=0.80\Omega \mathrm{m}$. ThereforeUnrestricted approach from the reservoirs removes access barriers, but still permits a continuum spreading or access resistance of a circular pore. Including $\rho /(4a)$ on each side givesThus the usual estimate neglecting the access contribution is about $44\mathrm{pS}$, or about $36\mathrm{pS}$ with spreading resistance. A $1.5$-ångström diameter is at the molecular scale, smaller than hydrated ions; bulk electrical resistivity cannot quantitatively describe dehydration, selectivity or discrete-ion transport. These values are the requested continuum estimates, not evidence that such a pore freely passes hydrated potassium ions.