An electrochemical cell may do work. The idea behind a fuel cell is to extract
maximum work from a chemical reaction on the basis of free energy change rather than
extracting maximum work by the flow of heat. If we were able to make a cell whose
chemistry is the combustion of methane according to the following reactions
we would achieve a free energy of
= - 817.9 kJ/mol and a 
= - 892.4 kJ/mol for a heat loss
of 75 kJ/mol or an ideal efficiency of 90\%. That is to say, we draw 817.9 kJ/mol
of electrical work from the cell. Were we to burn the methane we would extract 892.4
kJ/mol of heat. In the electrochemical cell we are ``burning" the methane but
extracting the 817.9 kJ/mol as useful work. The difference between the enthalpy and
the useful work (free energy) is the heat loss or non-useful work. We might think
of this as the unrecoverable entropy term. When we draw power
from the fuel cell it becomes a non-equilibrium process and the actual efficiency
drops, but so does the efficiency of the heat engine.
We continue to talk about ideal systems. In actuallity, there are no perfect systems.
We see the fundamental advantage of drawing on the free energy, but to do so requires
the development of electrochemical cells. In any electrochemical cell there is the
problem that like charges repell and electron density, in the presence of positive
counter ions, is a complicated one. There is a lot of engineering between the consideration
of chemical principles and the construction of a fuel cell.
Nevertheless, when we burn methane we can use the 892.4 kJ/mol from the combustion.
The best we can do here is to extract work as heat goes from the combustion chamber
to a condenser (either condensing water, dumping the hot water into a river which
is the same thing). The thermodynamic efficiency for this is:
work extracted = (enthalpy change)*(T_2 - T_1)/T_1.
If the combustion temperature was 800K and the condencer temperature was 300K then
the best we could do would be to extract 62.5% of the heat or 558 kJ/mol, considerably
lower than the 817.9 kJ/mole from the fuel cell.