# 31.18: Efficiency and Reaction Conditions

The proton exchange membrane is a solid-state electrolyte that functions at around 80 °C. Compared to the 1000 °C at which the solid oxide electrolytes become conductive, this is a low temperature.

## What temperature is preferable?

One of the key banners under which fuel cells are marketed is their efficiency. We must consider how this efficiency comes about and what factors influence it. To do this we must look at the thermodynamics governing the fuel cell.

• Considering the energy of the system - the Gibbs free energy change.

Let's consider the energy of a hydrogen fuel cell system as follows:

INPUTS:

PROCESS:

OUTPUT:

Hydrogen

FUEL CELL

Electrical Energy = VIt

Heat

Oxygen

Water

It’s easy to calculate the electrical power and energy output of the system:

Power = VI ; Energy = VIt

The “chemical energies” of inputs and outputs are a little more difficult to define. It is the change in Gibbs free energy that we must consider in this case (or more precisely Gf – the energy of formation - because we use the convention of comparison to pure elements in their standard states), which is the energy available to do external work. In the case of a fuel cell system this external work is pushing electrons through the external circuit, and past whatever impedances we put in their way. Work done by changes in volume and temperature between inputs and outputs is not harnessed by the fuel cell (although this is possible in turbine hybrid systems).

The Gf of both O2 and H2 is zero, a useful result when dealing with a hydrogen oxygen fuel cell. ΔGf refers to the difference in Gibbs free energy of formation between the inputs and the outputs, and is therefore a specific measure of the energy released by the reaction.

ΔGf = Gf of products – Gf of reactants

We usually consider this quantity to be per a mole of chemical. Let us find the chemical energy released during a nominal fuel cell reaction:

H2 + ½O2H2O

The Gibbs free energy of a system is defined as:
G = HTS

Which leads to the change in free energy being expressed as:

$\Delta \bar g_f = \Delta \bar h_f - T \Delta \bar s$

Note that we’ve gone lower-case and added little lines above the letters. This is to signify that we are dealing with molar quantities, so that the units will be Joules per mole or something similar.

The value Δhf of is the difference between hf of the products and hf of the reactants. So for the reaction H2 + ½O2 → H2O, we have:

$\Delta \bar h_f = (\bar h_f)_{H_2O} - (\bar h_f)_{H_2} - \frac{1}{2}(\bar h_f)_{O_2} \Delta \bar h_f = (\bar h_f)_{H_2O} - (\bar h_f)_{H_2} - \frac{1}{2}(\bar h_f)_{O_2}$

And Δs is the difference between entropy of the products and reactants so that:

$\Delta \bar s = (\bar s_{H_2O} - (\bar s)_{H_2} - \frac{1}{2}(\bar s)_{O_2} \Delta \bar s = (\bar s)_{H_2O} - (\bar s)_{H_2} - \frac{1}{2}(\bar s)_{O_2}$

These values of s and hfvary with temperature and pressure according to the equations given below. A full derivation of these equations is beyond the scope of this TLP but can be found in thermodynamics textbooks. It should be noted that we use 298 K as the standard temperature, which is necessary as an integration limit. The “T” subscript to the enthalpy, h, means the enthalpy at temperature T.