For an ion crossing a membrane, two effects contribute to the thermodynamic driving force:

1. the chemical contribution, produced by unequal chemical potential on the two sides;
2. the electrical contribution, produced by a voltage difference across the membrane.

Together they define the ion's electrochemical gradient.

If an ion of charge number $z$ moves from side 1 to side 2, the molar free-energy change is

$$\boxed{\Delta G=RT\ln!\left(\frac{a_2}{a_1}\right)+zF(\psi_2-\psi_1)},$$

where

• $R$ is the universal gas constant;
• $T$ is absolute temperature;
• $a_1$ and $a_2$ are the ion activities on the two sides;
• $F$ is the Faraday constant, the electric charge per mole of unit positive charges;
• $\psi_2-\psi_1$ is the electric potential difference.

For an ideal dilute solution, activity ratios can be approximated by concentration ratios:

$$\frac{a_2}{a_1}\approx\frac{c_2}{c_1},$$

so

$$\Delta G\approx RT\ln!\left(\frac{c_2}{c_1}\right)+zF\Delta\psi.$$

Interpreting the two terms

Suppose $\mathrm{K^+}$ is more concentrated inside a cell than outside. Its chemical contribution favors outward movement.

If the inside of the cell is electrically negative relative to the outside, the electrical contribution favors inward movement of the positive ion.

The two effects can therefore oppose one another. Net passive movement is thermodynamically favored in the direction that makes the total $\Delta G$ negative.

At electrochemical equilibrium for that ion,

$$\Delta G=0,$$

so the chemical and electrical contributions exactly balance. This does not require equal concentrations on the two sides.

Membrane potential

The voltage difference across a membrane is called the membrane potential. Only a small separation of charge near the membrane is required to create a biologically important voltage; the bulk aqueous solutions on either side can remain close to electrically neutral overall.

Electrochemical gradients are stored thermodynamic disequilibria. Opening an ion channel can allow an existing gradient to drive rapid passive movement, while pumps can use another energy source to create or maintain the gradient.

This concept underlies secondary active transport, mitochondrial energy conversion and electrical signaling in excitable cells.