## Physics Friday 81

For a thermodynamic system consisting of n different species of particles, we can define for each species a chemical potential, which is the increase in internal energy of the system with the addition of a single particle of that species, with volume, entropy, and the number of particles of other species held constant. Thus, for the i-th species, the chemical potential μi is defined as
.
To change from constant volume and entropy to constant pressure and temperature, we use the Gibbs free energy , so that

(we have exact differentials, so one can use the chain rule to prove the above).

For an ideal gas of a single species, the internal energy U at constant volume is , where  is the dimensionless specific heat capacity at constant volume, and f is the number of available degrees of freedom for a molecule of the gas (see here). We previously found that (for high enough temperatures) the entropy of an ideal gas can be expressed as  for some positive undetermined constant φ. We can rewrite this as
,
where , and  is the dimensionless specific heat capacity at constant pressure.
Then
.
To rewrite in terms of pressure instead of volume, we use , so

and so
.
Thus, for a single-species ideal gas, G=μN.