ΔG is the Gibbs free energy change during transfer of 1 mol of cations Xm+ from the phase A to B down the electrical potential, Δψ is the electrical potential difference (mV) between phases P and N (A and B), Xm+A and Xm+B are our cation concentrations on opposite sides of the membrane, F is the Faraday constant, R gas constant.
This energy is used to synthesize ATP when the protons flow across the membrane (down their concentration gradient) through the ATP synthase enzyme; this is known as chemiosmosis.