Editing Electromotive force (section)

==In (electrochemical) thermodynamics==
When multiplied by an amount of charge <math>dQ</math> the emf <math>\mathcal{E}</math> yields a thermodynamic work term <math>\mathcal{E}\,dQ</math> that is used in the formalism for the change in [[Gibbs free energy|Gibbs energy]] when charge is passed in a battery:

<math display="block">dG = -S\,dT + V\,dP + \mathcal{E}\,dQ\ , </math>

where <math>G</math> is the Gibbs free energy, <math>S</math> is the [[entropy]], <math>V</math> is the system volume, <math>P</math> is its pressure and <math>T</math> is its [[absolute temperature]].

The combination <math>(\mathcal{E}, Q)</math> is an example of a [[conjugate variables (thermodynamics)|conjugate pair of variables]]. At constant pressure the above relationship produces a [[Maxwell relation]] that links the change in open cell voltage with temperature ''<math>T</math>'' (a measurable quantity) to the change in entropy ''<math>S</math>'' when charge is passed [[isothermally]] and [[isobarically]]. The latter is closely related to the reaction [[entropy]] of the electrochemical reaction that lends the battery its power. This Maxwell relation is:<ref name=Finn/>

<math display="block">
\left(\frac{\partial \mathcal{E}}{\partial T}\right)_Q =
-\left(\frac{\partial S}{\partial Q}\right)_T
</math>

If a mole of ions goes into solution (for example, in a Daniell cell, as discussed below) the charge through the external circuit is:

<math display="block"> \Delta Q = -n_0 F_0 \ , </math>

where <math> n_0 </math> is the number of electrons/ion, and <math> F_0 </math> is the [[Faraday constant]] and the minus sign indicates discharge of the cell. Assuming constant pressure and volume, the thermodynamic properties of the cell are related strictly to the behavior of its emf by:<ref name=Finn/>

<math display="block">\Delta H = -n_0 F_0 \left( \mathcal{E} - T \frac {d\mathcal{E}}{dT}\right) \, , </math>

where <math> \Delta H </math> is the [[standard enthalpy of reaction|enthalpy of reaction]]. The quantities on the right are all directly measurable. Assuming constant temperature and pressure:

<math display="block">\Delta G = -n_0 F_0\mathcal{E}</math>

which is used in the derivation of the [[Nernst equation]].