Why is the resting membrane potential of excitable tissue not affected by the extracellular sodium concentration?

Question

I know that the resting membrane potential for excitable tissue (eg, nerve) is primarily determined by the electronegative difference between the inside and the outside of the membrane for potassium ions, as they are most permeate ion by far. And since the membrane is largely impermeable to sodium, it's diffusion potential is insignificant, as such it does not contribute much to the resting membrane potential. However, is it not correct to think that if the concentration of sodium ions in the extracellular fluid drops, outward diffusion of the positive potassium ions will increase, because the outside of the membrane is now less positive (more negative), subsequently making the diffusion potential of potassium and the resting membrane potential more on the negative side and further from the action potential threshold. I am asking this because my professor was mentioning factors that affect the resting membrane potential, and he mentioned that the ECF concentration of sodium has no effect on the resting membrane potential.

Answer 1

I cannot imagine anyone being able to explain it more clearly than this webpage. Your question is addressed in the middle of the page.

It's easier to split the idea into two separate, but parallel ideas: electrical potential and chemical potential. Both sodium and potassium contribute to resting potential. Sodium affects the voltage (electric potential) across the membrane. However, their chemical potentials (difference-across-membrane) are separate. The key idea is that there is a difference between resting membrane potential and potassium equilibrium potential.

I quote a helpful thing to keep in mind:

In a neuron, the resting membrane potential is closer to the potassium equilibrium potential than it is to the sodium equilibrium potential. That's because the resting membrane is much more permeable to K+ than to Na+.

    If more potassium channels were to open up—making it even easier for K+ to cross the cell membrane—the membrane would hyperpolarize, getting even closer to the potassium equilibrium potential.
    If, on the other hand, additional sodium channels were to open up—making it easier for Na+ to cross the membrane—the cell membrane would depolarize toward the sodium equilibrium potential.

Answer 2

This is a common misconception where ion concentrations and charge concentrations get confused.

Although the resting potential makes it sound like there is more "positive charge" outside than inside a cell, that difference is really really tiny, we are talking about a tiny tiny fraction of the number of ions in a cell (see for example Why is it possible to calculate the equilibrium potential of an ion using the Nernst equation from empirical measurements in the cell at rest?). For all intents and purposes, the sum charge of positive and negative ions is neutral both inside and outside the cell, to several decimal points.

If someone says the sodium concentration outside the cell changes they don't really mean just sodium ions, they mean sodium ions plus some equivalent number of negative ions. We can ignore those negative ions if they don't have any membrane permeability (see the Goldman equation: ions with no permeability don't count at all for membrane potential).

So in your question you already identified why extracellular sodium concentration doesn't matter much: its permeability is low and the extracelullar sodium concentration is already high. You should not think about adding extracellular sodium changing the charge of ions outside, you should only think about it as changing the driving force for sodium. The resting membrane potential is then a function of a sum of all those driving forces weighted by their permeabilities and calculated with the Goldman equation.

I won't do the math here, but if you actually were able to dump a bucket of just sodium ions, all with positive charge, in the environment outside the cell without a corresponding negative ion in solution, you would create something like lightning traveling at the speed of light and release enough heat to boil your apparatus.