Electro Motive Force (EMF) → Work (Joule) / Charge (Coulomb)
E = Wel / e-
As a fuel cell is supplied with fuel and oxidizer on the anode and cathode side respectively, electrons start to accumulate on the anode side resunting in
the establishment of a gradient (difference in the concentration of electrons) of electrons across the electrodes. If the anode and cathode are not connected through an external circuit, the gradient continues
to build upto a certain threshold magnitude. Once the threshold magnitude is attained with no current being drawn (when the circuit is open), no further accumulation
of electrons takes place. This gradient of electrons establishes a driving potential OR voltage across the electrodes known as the Electro Motive Force (EMF) having units of
volts. Thus, essentially, the EMF is the potential difference established across the anode and cathode when they are not electrically connected and hence no current is drawn.
It is important to note that, as fuel and oxidizer are supplied continuously to the anode and cathode side, electron and hence negative charge build-up on the
anode side takes place. If these electrons are not evacuated (as in open circuit mode) the negative charge continues to build resulting in increased resistance to
further oxidation of fuel. At a sufficiently high negative charge, practically no further nett oxidation of fuel (and hence release of electrons) takes place as equilibrium sets in.
At equilibrium, every forward reaction is accompanied by an equivalent reverse reaction, resulting in no nett change. Thus, even in the face of continuous and infinite
supply of fuel and oxidizer, equilibrium restricts the electron build-up and hence the EMF to a finite value.
When the anode and cathode are electrically connected through and external circuit, it is the EMF (measured in volts) that drives the electrons from the anode
to the cathode across the load. The EMF is thus, basically the work done (Joules) in moving a unit charge (Coulombs) across the applied load.
If a single fuel molecule is oxidized releasing one electron, the EMF generated can mathematically be represented as;
If, on oxidation of a single fuel molecule, n electrons are released then the total charge correspond to ne-. Similarly, oxidation of one mole of fuel molecules
(Avogadro constant → NA = 6.02214076 x 1023 number of molecules) will result in a total charge of NAne-. The product of Avogadro number
and charge on an electron is basically the Faraday constant F (= NAe-). Thus, the total charge will be nF. The expression for EMF developed when a mole of fuel is oxidized
with each molecule releasing 2 electrons will be;
E = Wel / nF
The maximum electrical work developed by a fuel cell under constant temperature and pressure conditions basically corresponds to the change in the Gibbs free energy dG; essentially dG = - Wel [Click HERE for the derivation].
The expression for the EMF thus takes the form;
E = -dG / nF
Under standard state conditions the expression takes the form;
Eo = -dGo / nF
- Faraday constant F → 96,485 Coulombs / mole
- Avogadro constant NA → 6.02214076 x 1023 per mole
- Charge on an electron e- → 1.60217662 x 10-19 coulombs