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Consider the following ion concentrations on either side of a cell membrane (in = inside cell, out = outside cell):

$[\text{Na}^+]_{\text{in}} = 10mM$,
$[\text{Na}^+]_{\text{out}} = 142mM$,
$[\text{K}^+]_{\text{in}} = 148mM$,
$[\text{K}^+]_{\text{out}} = 5mM$,
$[\text{Cl}^-]_{\text{in}} = 4mM$,
$[\text{Cl}^-]_{\text{out}} = 103mM$.

Let us further say that there are no leak channels or voltage-gated channels in the cell membrane, and also that ion pumps stop working once they achieve these concentration gradients (i.e. the state is at equilibrium with respect to the ion pump.

There is a greater net positive charge on the inside compared to the outside. This should give me a positive membrane potential when I think about the cell membrane as a capacitor.

I know, Nernst and Goldman equations say otherwise. (For my example it would be zero membrane potential due to zero permeability).

Can someone explain why there is or is not a positive membrane potential, with appropriate sources?

There are already multiple questions with a similar background but none of the answers are satisfactory.

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    $\begingroup$ can you say anything about why the other answers are not satisfactory, and link to those questions? $\endgroup$ Nov 17 '20 at 19:48
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Goldman equation is not applicable in this case. If you have a non-permeable membrane, all the permeabilities of the ionic species ($P_x$) are zero, which leads to an indeterminate form of the argument of the logarithm in the Goldman equation. The logarithm does not equate to $0$. Consequently, Goldman equation cannot be used to find the membrane potential ($E_m$) in this case.

Nernst potential is the potential when the ionic species are in thermodynamic equilibrium (i.e. the diffusive force and the electrostatic force are balanced) and there would be no net flow of the ionic species across a membrane even when the membrane is permeable to the ion. This again is not the case for your impermeable membrane since there is an "unbalanced" force which will cause a net ionic flux given some permeability.

The usual equations of a capacitor, however, are perfectly usable given that you know the spatial arrangement of the charge inside and outside the membrane (i.e. if the ions are closely pressed to the membrane or distributed in the cell and outside).

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