text: The proton-motive force is derived from the Gibbs free energy. Let N denote the inside of a cell, and P denote the outside. Then
  where
  *  is the Gibbs free energy change per unit amount of cations transferred from P to N;
  *  is the charge number of the cation ;
  *  is the electric potential of N relative to P;
  *  and  are the cation concentrations at P and N, respectively;
  *  is the Faraday constant;
  *  is the gas constant; and
  *  is the temperature.
  The molar Gibbs free energy change  is frequently interpreted as a molar electrochemical ion potential .
  For an electrochemical proton gradient  and as a consequence:
  where
  Mitchell defined the proton-motive force (PMF) as
  For example,  implies . At  this equation takes the form:
  Note that for spontaneous proton import from the P side (relatively more positive and acidic) to the N side (relatively more negative and alkaline),  is negative (similar to ) whereas PMF is positive (similar to redox cell potential ).
  It is worth noting that, as with any transmembrane transport process, the PMF is directional.  The sign of the transmembrane electric potential difference  is chosen to represent the change in potential energy per unit charge flowing into the cell as above.  Furthermore, due to redox-driven proton pumping by coupling sites, the proton gradient is always inside-alkaline.  For both of these reasons, protons flow in spontaneously, from the P side to the N side; the available free energy is used to synthesize ATP (see below).  For this reason, PMF is defined for proton import, which is spontaneous.  PMF for proton export, i.e., proton pumping as catalyzed by the coupling sites, is simply the negative of PMF(import).
  The spontaneity of proton import (from the P to the N side) is universal in all bioenergetic membranes.  This fact was not recognized before the 1990s, because the chloroplast thylakoid lumen was interpreted as an interior phase, but in fact it is topologically equivalent to the exterior of the chloroplast.  Azzone et al. stressed that the inside phase (N side of the membrane) is the bacterial cytoplasm, mitochondrial matrix, or chloroplast stroma; the outside (P) side is the bacterial periplasmic space, mitochondrial intermembrane space, or chloroplast lumen. Furthermore, 3D tomography of the mitochondrial inner membrane shows its extensive invaginations to be stacked, similar to thylakoid disks; hence the mitochondrial intermembrane space is topologically quite similar to the chloroplast lumen.:
  The energy expressed here as Gibbs free energy, electrochemical proton gradient, or proton-motive force (PMF), is a combination of two gradients across the membrane:
  * the concentration gradient (via ) and
  * electric potential gradient .
  When a system reaches equilibrium, ; nevertheless, the concentrations on either side of the membrane need not be equal. Spontaneous movement across the potential membrane is determined by both concentration and electric potential gradients.
  The molar Gibbs free energy  of ATP synthesis
  is also called phosphorylation potential. The equilibrium concentration ratio  can be calculated by comparing  and , for example in case of the mammalian mitochondrion:
  H / ATP = &Delta;G / (&Delta;p / 10.4 kJ·mol/mV) = 40.2 kJ·mol / (173.5 mV / 10.4 kJ·mol/mV) = 40.2 / 16.7 = 2.4.  The actual ratio of the proton-binding c-subunit to the ATP-synthesizing beta-subunit copy numbers is 8/3 = 2.67, showing that under these conditions, the mitochondrion functions at 90% (2.4/2.67) efficiency.
  In fact, the thermodynamic efficiency is mostly lower in eukaryotic cells because ATP must be exported from the matrix to the cytoplasm, and ADP and phosphate must be imported from the cytoplasm. This "costs" one "extra" proton import per ATP, hence the actual efficiency is only 65% (= 2.4/3.67).