## Free Energy Dependence on Pressure

The free energy, G, is defined as

We saw that the enthalpy does not change when the pressure is altered isothermally. Thus, isothermally, pressure can only alter the free energy through its effect on the entropy, S:

When an amount, Q, of heat is added to a system at constant temperature, the entropy increases by Q/T:

From the first law of thermodynamics,

But we saw previously that in an isothermal compression, AU = 0, hence, Q = W. However, the work done per kilomole of gas isothermally compressed (see Section 2.10) is

Thus,

Consequently, the energy of isothermal compression of a gas is entirely free energy. This is an important effect. It is possible to change the efficiency (and the voltage) of a fuel cell by changing the pressure of products and reactants.

### Example 3

A reversible fuel cell, when fed hydrogen and oxygen at RTP, delivers a voltage of 1.185 V. Calculate the voltage delivered by the same cell if air (at RTP) replaces the oxygen.

Air contains roughly 20% of oxygen. Thus, the partial pressure of this gas is 0.2 atmosphere, a 5:1 decompression relative to the pure oxygen case.

The energy of isothermal decompression is

111 6 Wdecompr. = - RT ln - = - x 8314 x 298 x ln0.2 = -2 x 106 J/kmole.

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