The overall chemical reaction for hydrogen fuel cells is:

.

If the water is produced as steam, then the chemical reaction is of low heating value (LHV); if it is produced in liquid form, then the reaction is of high heating value (HHV).

The enthalpy or heat of reaction (J/mol) of the chemical reaction

is the difference between the enthalpies of the product

and reactants

and

:

.

The enthalpy of component

(

,

or

) is the sum of standard enthalpy of formation

and sensible enthalpy

, namely:

.

can be found in thermodynamic tables and

is obtained using the following formula:

,

where

is the heat capacity at a constant pressure.

(J/mol K) changes with temperature:

,

where

,

,

,

and

are experimentally determined coefficients. Likewise, the entropy (J/mol K) of the chemical reaction

is given by:

,

,

.

The Gibbs free energy

, which represents the electric work of the fuel cell, is given as:

.

The reference thermodynamic voltage

could be then obtained from:

,

where

is the number of electrons transferred in the reaction for one mole of

(here equal to 2) and

is the Faraday constant (96485

). The thermodynamic (Nernst) voltage, which incorporates the effects of pressure and composition of the reacting mixture, is obtained using the Nernst equation:

,

where

is the universal gas constant (8.314 J/mol K) and

is the activity of the component involved in the reaction. The activity of an ideal gas is the ratio between the partial pressure of the gas and the reference pressure. For example, the activity of

is given as:

,

where

is the mole fraction of hydrogen,

is the pressure at the anode and

is the reference pressure (1 atm for ideal gases). The activity of water (

) is normally approximated by 1. Based on the preceding, the Nernst equation becomes:

.

Finally, the thermodynamic efficiency is given using:

.

[1] M. M. Mench,

*Fuel Cell Engines*, Hoboken, NJ: John Wiley and Sons, 2008 Chapter 3.

[2] F. Barbir,

*PEM Fuel Cells: Theory and Practice*, 2nd ed., Boston: Elsevier/Academic Press, 2013 Chapter 2.

[3] R. H. Perry and D. W. Green (eds.),

*Perry's Chemical Engineers' Handbook*, 7th ed., New York: McGraw-Hill, 1997 pp. 2–174.