A membrane can store free energy by maintaining both a proton concentration difference and an electrical potential difference across it. For protons, these two contributions are combined into the proton-motive force.

For movement of one mole of $\mathrm{H^+}$ from side 1 to side 2,

$$\Delta G_{H^+}=RT\ln!\left(\frac{a_{H^+,2}}{a_{H^+,1}}\right)+F(\psi_2-\psi_1).$$

Because

$$\mathrm{pH}=-\log_{10}a_{H^+},$$

the chemical term can also be written in terms of a pH difference.

Define

$$\Delta\psi=\psi_2-\psi_1$$

and

$$\Delta\mathrm{pH}=\mathrm{pH}_2-\mathrm{pH}_1.$$

Then

$$\boxed{\Delta G_{H^+}=F\Delta p}$$

with

$$\boxed{\Delta p=\Delta\psi-\frac{2.303RT}{F}\Delta\mathrm{pH}},$$

for this chosen direction and sign convention.

$\Delta p$ has units of volts and is called the proton-motive force.

Electrical and chemical contributions can cooperate or oppose

Suppose side 1 has a higher proton activity than side 2. The concentration contribution favors proton movement from side 1 to side 2.

If side 2 is also electrically negative relative to side 1, the electrical contribution likewise favors movement of positively charged protons into side 2. The two terms then reinforce one another.

If instead side 2 is electrically positive, the electrical contribution opposes the chemical contribution. The total free-energy change, not either term alone, determines the favored passive direction.

A proton gradient is a stored nonequilibrium state

Maintaining unequal proton electrochemical potentials across a membrane requires the membrane to restrict spontaneous proton equilibration. Pumps or redox-driven transport processes can create the gradient by moving protons in the thermodynamically uphill direction.

When protons later return downhill through a selective molecular pathway, the decrease in their electrochemical free energy can be coupled to another process.

Biological systems use this principle for ATP synthesis, transport and other work.

The reusable idea is therefore:

energy input → move H+ uphill across membrane
                     ↓
             proton-motive force
                     ↓
H+ returns downhill through coupled machinery → useful work

The proton-motive force is not a new kind of force separate from thermodynamics. It is a compact way to express the electrochemical free-energy difference of protons across a membrane.