Gibbs Free Energy and Ideal Performance
The maximum electrical work (Wel) obtainable in a fuel cell operating at constant temperature and pressure is given by the change in Gibbs free energy (AG)12 of the electrochemical reaction,
where n is the number of electrons participating in the reaction, F is Faraday's constant (96,487 coulombs/gmole electron), and E is the ideal potential of the cell. If we consider the case of reactants and products being in the standard state, then
10. Constraints can limit the degree of part load operation of a fuel cell. For example, a PAFC is limited to operation below approximately 0.85 volts because of entering into a corrosion region.
11. It should be remembered that the actual efficiencies of heat engines and fuel cells are substantially below their theoretical values.
12. Total energy is composed of two types of energy: 1) free energy, G, and unavailable energy, TS. Free energy earns its name because it is the energy that is available or free for conversion into usable work. The unavailable energy is unavailable for work because of the disorder or entropy of the system. Thus, G = H  TS. For changes in free energy at constant T and P, the equation can be written as AG = AH  TAS. This is an important equation for chemical and physical reactions, for these reactions only occur spontaneously with a decrease in free energy, G, of the total system of reactants and products.
where the superscript stands for standard state conditions (25°C or 298K and 1 atm).
The overall reactions given in Table 22 can be used to produce both electrical energy and heat. The maximum work available from a fuel source is related to the free energy of reaction in the case of a fuel cell, whereas the enthalpy (heat) of reaction is the pertinent quantity for a heat engine, i.e.,
where the difference between AGr and AHr is proportional to the change in entropy (ASr). This entropy change is manifested in changes in the degrees of freedom for the chemical system being considered. The maximum amount of electrical energy available is AGr, as mentioned above, and the total thermal energy available is AHr. The amount of heat that is produced by a fuel cell operating reversibly is TASr. Reactions in fuel cells that have negative entropy change generate heat, while those with positive entropy change may extract heat from their surroundings, if the irreversible generation of heat is smaller than the reversible absorption of heat.
Differentiating Equation (225) with respect to temperature or pressure, and substituting into Equation (223), yields dE
which are shown earlier in this section.
The reversible potential of a fuel cell at temperature T is calculated from AG for the cell reaction at that temperature. This potential can be computed from the heat capacities (Cp) of the species involved as a function of T and from values of both AS° and AH° at one particular temperature, usually 298K. Empirically, the heat capacity of a species, as a function of T, can be expressed as
where a, b, and c are empirical constants. The difference in the heat capacities for the products and reactants involved in the stoichiometric reaction is given by
AHt = AH° + J T98 A CpdT (230) and, at constant pressure
then it follows that
AHt = AH° + A a(T  298) + 1/2 A b(T2  2982) + 1/3 A c(T3  2983) (232)
ASt = AS° + A a ln I 2j + A b(T  298) + 1/2 A c(T2  2982) (233)
The coefficients a, b, and c (see Table 103), as well as AS° and AH°, are available from standard reference tables, and may be used to calculate AHT and AST. From these values it is then possible to calculate AGT and E.
Instead of using the coefficients a, b, and c, it is modern practice to rely on tables, such as JANAF Thermochemical Tables (4) to provide Cp, AHT, AST, and AGT for a range of temperatures of various reactants and products.
the free energy change can be expressed by the equation:
When Equations (223) and (224) are substituted in Equation (235),
RT n [reactant activity]
n F n [product activity]
which is the general form of the Nernst equation. For the overall cell reaction, the cell potential increases with an increase in the activity (concentration) of reactants and a decrease in the activity of products. Changes in temperature also influence the reversible cell potential, and the dependence of potential on temperature varies with the cell reaction. Figure 21 illustrates the change in the reversible standard potential for the reaction:
The Nernst equations for this reaction, as well as for CO and CH4 reacting with O2, that can occur in various fuel cells, are listed in Table 22.
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