The British nobel laureate Peter Mitchell coined the term chemiosmosis to explain the coupling between gradient of electrochemical potentials of protons across selectively permeable membranes and the performance of cellular work (Harold 1986). In plants, proton gradients play a role in transport across the plasma membrane and the tonoplast and drive both transport and ATP synthesis on the inner membranes of chloroplasts and mitochondria. The passage of protons along their electric and chemical gradients can be coupled to cellular work because the change in free energy is negative. This process can be represented by the following equation:

where Δµ_H⁺ is the proton gradient of electrochemical potential, F is the Faraday constant, ΔE is the membrane potential, R is the gas constant, T is the absolute temperature, and the superscripts "i" and "o" refer to the inside and outside, respectively, of the cell or other membrane-enclosed compartment.

Mitchell introduced the term "proton motive force" (Δp) for the difference in electrochemical potential between protons inside and outside a cellular compartment, μ_H⁺. It is convenient to express Δp in units of electric potential, which we can do by dividing both sides of the foregoing equation by the Faraday constant, F, which changes the units to millivolts:

Since pH = –log[H⁺], the term log([H⁺]ⁱ/[H⁺]^o) simplifies to –(pHⁱ – pH^o). If ΔpH is defined as pHⁱ – pH^o, the general expression for proton motive force results:

At 25°C, 2.3RT/F = 59 mV, and substituting this value into the foregoing expression results in the most commonly used equation for proton motive force:

where Δp is expressed in millivolts.

Let’s consider the example of a cell bathed in a solution of 1 mM K Cl and 1 mM sucrose. Proton pumping by an H⁺-ATPase results in a membrane potential of –120 mV and a pH difference between the inside and the outside of the cell of 2 pH units. Thus, from the previous equation,

How much K⁺ can the cell take up by using this Δp? Because potassium is positively charged and the membrane potential differece across the cell membrane is inside negative, potassium will be taken up through ion channels by the electrical component of Δp, which equals –120 mV. Using the Nernst equation (see textbook p. 147), we can calculate that an external K⁺ concentration of 1 mM and a ΔE value of –120 mV will equilibrate with an internal K⁺ concentration of 100 mM. Thus, if the electrical component of Δp is used, the cell can generate a 100-fold K⁺ gradient across the membrane.

The proton motive force can also be used to take up sucrose against a concentration gradient, usually via a proton–sucrose symporter. Because sucrose is cotransported with a proton, both components of the gradient of electrochemical potential (–238 mV in our example) can be used for sucrose uptake (Harold 1986). At equilibrium, (see textbook Equations 6.4 and 6.5) Δμ_H⁺ will be equal to FΔp; thus we can calculate that an external sucrose concentration of 1 mM and a Δp of –238 mV will equilibrate with an internal sucrose concentration of 10 M. In real life, however, such concentration gradients would not exist: Sucrose would diffuse back out of the cell, and regulatory mechanisms at the membrane would repress the function of the symporter after certain critical concentrations were attained.