## Nernst Equilibrium Potential

We discussed the Nernst equation in class. I had foolishly assumed that the textbook describes it, forgetting that it is not mentioned (at least not prominently) in this book. Here’s an overview.

The equation was developed by Walther Nernst, a German who won the Nobel Prize for Chemistry in 1920. Its full form, for general use in any situation that meets the assumptions (see below) is presented here, pulled from a Wikipedia page:

From en.wikipedia.org/wiki/Resting_potential:

where

*E*_{eq,K+}is the equilibrium potential for potassium, measured in volts*R*is the universal gas constant, equal to 8.314 joules·K^{−1}·mol^{−1}*T*is the absolute temperature, measured in kelvins (= K = degrees Celsius + 273.15)*z*is the number of elementary charges of the ion in question involved in the reaction*F*is the Faraday constant, equal to 96,485 coulombs·mol^{−1}or J·V^{−1}·mol^{−1}- [K
^{+}]_{o}is the extracellular concentration of potassium, measured in mol·m^{−3}or mmol·l^{−1} - [K
^{+}]_{i}is likewise the intracellular concentration of potassium

This can be simplified for neurons, because the temperature essentially becomes a constant (body temperature) so *RT/zF* reduces to +/- 61.54 for ions with a single charge (sign is the charge of the ion), twice 61.54 for ions with two charges. For potassium,

and depending on the exact concentrations, this will reduce to approximately 5 – 10 mV more negative than the resting potential. For sodium, it will reduce to a positive voltage.

What the equation tells us is the electrical energy necessary to balance a given concentration gradient,

assuming:

- the only forces at work on the ion are the electrostatic and the osmotic force,
- the ion is free to move across the membrane, and
- the concentrations are unchanging.

K and Na are both free to move across the membrane, and the concentrations are stable (so long as the cell is not active).

Typical Nernst equilibrium potentials (Nep) for these ions:

**Nep-K = -80 mV**

**Nep-Na = +50 mV**

The equilibrium potential for K is close to the electrical potential of the cell at rest; the equilibrium potential for Na is very far away from the resting potential.

The fact that the Nep-Na is so far from the resting potential means that one of the assumptions of the Nernst equation is being violated. Another force is at work – cells spend LOTS of energy driving the **Na-K pump**, a protein embedded in the cell membrane. The Na-K pump moves Na out of the cell, and in the process pulls in a little extra K. Without the Na-K pump, Na would enter the cell in large amounts.