Entropy maximization implies equality of chemical potential

Statement (chemical potential equality from entropy maximization)

Let subsystems 1 and 2 be able to exchange particles of a given species through a permeable interface. Let $S_i(U_i,V_i,N_i)$ be the entropy of subsystem $i$, where $N_i$ is the particle number of that species.

Assume:

• the composite is isolated with fixed totals $U_{\mathrm{tot}}$, $V_{\mathrm{tot}}$, and $N_{\mathrm{tot}}$,

• the subsystems are weakly interacting so entropy is additive:

  $$S_{\mathrm{tot}}(U_1,V_1,N_1)=S_1(U_1,V_1,N_1)+S_2(U_{\mathrm{tot}}-U_1,V_{\mathrm{tot}}-V_1,N_{\mathrm{tot}}-N_1),$$

• $S_1,S_2$ are differentiable in $N$, and equilibrium corresponds to an interior maximizer of $S_{\mathrm{tot}}$.

Then entropy maximization implies

Using the thermodynamic identity (definition of chemical potential )

one obtains the equilibrium condition

In particular, if the subsystems are also in thermal equilibrium (so that temperatures are equal ), then

Key hypotheses

• Particle exchange is allowed; total particle number is fixed.
• Weak coupling so entropy is additive.
• Differentiability of $S_i$ in $N_i$ and an interior entropy maximizer.
• (For the conclusion $\mu_1=\mu_2$) thermal equilibrium, i.e., $T_1=T_2$.

Key conclusions

• The entropy maximum enforces equality of $\mu/T$:

  $$\mu_1/T_1=\mu_2/T_2.$$

• With temperature equalization ($T_1=T_2$), chemical potentials equalize:

  $$\mu_1=\mu_2.$$

Cross-links to relevant definitions

• Chemical potential and temperature: chemical potential , temperature .
• Entropy and equilibrium: thermodynamic entropy , thermodynamic equilibrium .
• Internal energy: internal energy .
• Stability conditions: thermodynamic stability .

Proof idea / significance (sketch)

Maximize $S_{\mathrm{tot}}$ with respect to $N_1$ subject to $N_2=N_{\mathrm{tot}}-N_1$. Stationarity gives equality of the derivatives $(\partial S/\partial N)_{U,V}$ across subsystems, which equals $-\mu/T$. Hence $\mu_1/T_1=\mu_2/T_2$, and with thermal equilibrium this reduces to $\mu_1=\mu_2$. This is the thermodynamic basis for chemical potential as the intensive variable that equalizes under diffusive contact.