Can you get a potential difference if the membrane is not permeable to the ion?

Question

If the outside compartment has 10 mM Na+ and the inside is 1 mM Na+, but only permeable to K+, is Vm=0?

Answer 1

A potential difference exists because you have a separation of charge.  Just because one ion is impermeable does not mean no potential difference can exist.

In fact, if all ions were permeable and there were no system to maintain the difference, such as an ATP-dependent ion pump, then no membrane voltage difference would exist.

The free energy of ion movement is a function of both the chemical (i.e. concentration difference) and electrical potential.  K+ ion will move across the membrane (a membrane current) in response to any chemical and electrical potential difference so that equilibrium is obtained.  The electric potential will reflect the difference between charges between the membrance without respect to the identity of the ions, whereas the chemical potential takes into account the identity of the ions.  It is quite possible for the electric potential (i.e. membrane voltage) to be zero, but if there is an imbalance of Na and K on the inside and outside, a chemical potential will exist, and may force some K+ ion in or out, restoring at equlibrium a nonzero membrane voltage.

Answer 2

So long as there's an electrochemical gradient, you can expect a corresponding transmembrane potential.