# Equivalence of different definitions for chemical potential

In some books, the chemical potential is defined in term of the Gibbs energy as $$\mu_i \equiv \left( \frac{\partial G}{\partial n_i} \right)_{P,T,n_j}$$ where $n_j$ indicates all mole numbers except the ith species mole number.

In other books, it is defined in term of the Helmholtz energy as

$$\mu_i \equiv \left( \frac{\partial A}{\partial n_i} \right)_{V,T,n_j}$$

Looking up at the Physics Stack Exchange, I could also find:

$$\mu_i \equiv \left( \frac{\partial U}{\partial n_i} \right)_{S,V,n_j}$$

And I suppose (by analogy) that one may also define it as

$$\mu_i \equiv \left( \frac{\partial H}{\partial n_i} \right)_{S,P,n_j}$$

Are all these definitions equivalent? Why?

Or they give rise to different quantities and one should be consistent when using one definition or the other?

I have tried to work out one definition from the other using thermodynamics identities but I could not make it out.

• They are just simplifications depending on what is taken to be constant. Commented Jan 2, 2015 at 18:06

Let's start from the second TD law: $$\tag 1 dU(S,V) = TdS - pdV,$$ In case of variable number of particles it is modified by the adding $\mu dn$ term to the right side: $$dU(S, V, n) = TdS - pdV + \mu d n.$$ The other TD potentials are related with $U$ by Legendre transformation with variables $T, S, p, V$, but not $\mu$ or $n$. So it's obviously that the relation $$\mu = \left( \frac{\partial Y (n, X)}{\partial n}\right)_{X = const}$$ is true for all of four TD potentials $Y$.