My textbook Thermal Physics by Kittel and Kroemer [1, p. 124] says:
When external potential steps are present, we can express the total chemical potential of a system as the sum of two parts:
$$\mu = \mu_\mathrm{tot} = \mu_\mathrm{ext} + \mu_\mathrm{int}\tag{16}$$
Here $\mu_\mathrm{ext}$ is the potential energy per particle in the external potential, and $\mu_\mathrm{int}$ is the internal chemical potential* defined as the chemical potential that would be present if the external potential were zero.
I am very confused since
$$\mu = \frac{\partial F}{\partial N},$$
where $F$ is Helmholtz free energy.
As there is only internal energy in Helmholtz free energy, why can we define something related to the external potential energy (such as gravitational potential energy) from the internal energy? Would anyone explain this to me?
Reference
- Kittel, C.; Kroemer, H. Thermal Physics, 2nd ed.; W. H. Freeman: San Francisco, 1980. ISBN 978-0-7167-1088-2.