The Nernst equation is derived as follows:
A molecule in solution (a solute) tends to move from a region of high concentration to a region of low concentration simply due to the random movement of molecules, which results in their equilibrium. Consequently, movement down a concentration gradient is accompanied by a favorable free-energy change (°G < 0), whereas movement up a concentration gradient is accompanied by an unfavorable free-energy change (°G > 0). (Free energy is introduced in Chapter 2 and discussed in the context of redox reactions in Panel 14–1, p. 765.)
The free-energy change per mole of solute moved across the plasma membrane (°Gconc) is equal to –RT In Co / Ci.
If the solute is an ion, moving it into a cell across a membrane whose inside is at a voltage V relative to the outside will cause an additional free-energy change (per mole of solute moved) of °Gvolt = zFV.
At the point where the concentration and voltage gradients just balance, and the ion distribution is at equilibrium across the membrane.
Thus, and, therefore, or, using the constant that converts natural logarithms to base 10,
V = 2.3 log10
For a univalent cation, 2.3 = 58 mV at 20oC and 61.5 mV at 37oC.
Thus, for such an ion at 37ºC,
V = + 61.5 mV for Co/ Ci = 10, whereas V = 0 for Co / Ci= 1.
The K+ equilibrium potential (VK), for example, is 61.5 log10([K+]o / [K+]i) millivolts (–89 mV for a typical cell, where [K+]o = 5 mM and [K+]i = 140 mM).
At VK, there is no net fow of K+ across the membrane.
Similarly, when the membrane potential has a value of 61.5 log10([Na+]o /[Na+]i), the Na+ equilibrium potential (VNa), there is no net fow of Na+.
For any particular membrane potential, VM, the net force tending to drive a particular type of ion out of the cell, is proportional to the difference between VM and the equilibrium potential for the ion: hence, for K+ it is VM – VK and for Na+ it is VM – VNa.
When there is a voltage gradient across the membrane, the ions responsible for it—the positive ions on one side and the negative ions on the other—are concentrated in thin layers on either side of the membrane because of the attraction between positive and negative electric charges. The number of ions that go to form the layer of charge adjacent to the membrane is minute compared with the total number inside the cell. For example, the movement of 6000 Na+ ions across 1 µm2 of membrane will carry suffcient charge to shift the membrane potential by about 100 mV.
Because there are about 3 × 107 Na+ ions in a typical cell (1 µm3 of bulk cytoplasm), such a movement of charge will generally have a negligible effect on the ion concentration gradients across the membrane.