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 *μ*_{i} 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*.

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Tags: Chemical Potential, Entropy, Friday Physics, Gibbs Free Energy, Ideal Gas, physics, Specific Heat Capacity

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