Editing Electrochemical potential (section)

== Thermodynamic definition ==

Formally, the electrochemical potential <math>\bar \mu_i</math> for species <math>i</math> is defined the same way as the [[chemical potential]], as the partial molar [[Gibbs free energy]]:<ref name="guggeinheim-book">{{cite book|last=Guggenheim|title=Thermodynamics|edition=8|year=1985}}</ref>
:<math>\bar\mu_i = \left(\frac{\partial G}{\partial N_i} \right)_{T,P, N_{j \ne i}},</math>
where <math>G</math> is the total Gibbs free energy that does not neglect electrostatic energy.

The above definition is somewhat delicate: when charged species are added to a thermodynamic body, they alter its total charge. The excess charge does not spread homogeneously throughout the body, but instead interacts with itself at long distances and tends to accumulate at surfaces while the interiors of bodies remain charge-neutral. The total <math>G</math> that permits charge imbalance is therefore necessarily inhomogeneous, containing electrostatic surface effects that do not scale proportional to the volume of the system, and are sensitive to the electric environment around the body.<ref name="guggeinheim-book"/>

The bulk interior of an electrochemical fluid meanwhile remains charge neutral and homogeneous. Since the interior is charge neutral, it is not possible to vary the interior numbers of ions independently of each other. The constraint on the interior <math>N_i</math>s is:
:<math>\sum N_i z_i = 0,</math>
where <math>z_i</math> is the [[ionic charge]] (−2, +1, etc.) for the species <math>i</math>. A charge-neutral Gibbs free energy can be defined for the interior, and it will be insensitive to the electrostatic offset. But, due to the charge-neutrality constraint on the <math>N_i</math>, it is also insufficient for defining any one ion's <math>\bar \mu_i</math> using the formal definition above. The charge-neutral <math>G</math> however does define all charge-neutral combinations of chemical potentials, such as <math>\bar\mu_{\mathrm H^+} + \bar\mu_{\mathrm{Cl}^-}</math>. In this manner, the electrochemical potentials of all the ions are determined in relation to each other, though they all have an undetermined electrostatic offset. Once any ion's <math>\bar \mu_i</math> is fixed, the rest are fixed as well (for a given composition, temperature, and pressure).<ref name="guggeinheim-book"/>