Entropy maximization implies equality of pressure

For two weakly coupled subsystems allowed to exchange volume at fixed totals, the entropy maximum implies equality of (generalized) mechanical intensities; with thermal equilibrium this becomes equality of pressures.

Entropy maximization implies equality of pressure

Statement (pressure equality from entropy maximization)

Let subsystems 1 and 2 be separated by a movable boundary so they can exchange volume. Let $S_i(U_i,V_i,N_i)$ be the entropy of subsystem $i$ .

Assume:

the composite is isolated with fixed totals $U_{\mathrm{tot}}$ and $V_{\mathrm{tot}}$ ,
particle numbers $N_1,N_2$ are fixed,
the subsystems are weakly interacting so entropy is additive:
$S_{\mathrm{tot}}(U_1,V_1)=S_1(U_1,V_1,N_1)+S_2(U_{\mathrm{tot}}-U_1,V_{\mathrm{tot}}-V_1,N_2),$
$S_1,S_2$ are differentiable, and equilibrium corresponds to an interior maximizer of $S_{\mathrm{tot}}$ .

Then the entropy maximization conditions imply

\left(\frac{\partial S_1}{\partial V_1}\right)_{U_1,N_1} ={} \left(\frac{\partial S_2}{\partial V_2}\right)_{U_2,N_2}.

Using the thermodynamic identity (definition of pressure )

\left(\frac{\partial S}{\partial V}\right)_{U,N}=\frac{P}{T},

one obtains the equilibrium condition

\frac{P_1}{T_1}=\frac{P_2}{T_2}.

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

P_1=P_2.

Key hypotheses

Two weakly coupled subsystems, entropy additive.
Exchange of volume allowed; totals $U_{\mathrm{tot}}$ and $V_{\mathrm{tot}}$ fixed.
Differentiability of $S_i$ in $V_i$ and an interior entropy maximizer.

Key conclusions

The entropy maximum enforces equality of the mechanical intensity $P/T$ :
$P_1/T_1 = P_2/T_2.$
With simultaneous thermal equilibrium ( $T_1=T_2$ ), pressures equalize:
$P_1=P_2.$

Cross-links to relevant definitions

Entropy and equilibrium: thermodynamic entropy , thermodynamic equilibrium .
Pressure and temperature: pressure , temperature .
Internal energy: internal energy .
Stability conditions ensuring a maximum: thermodynamic stability .

Proof idea / significance (sketch)

Treat $S_{\mathrm{tot}}(U_1,V_1)$ as a function to be maximized under $U_2=U_{\mathrm{tot}}-U_1$ and $V_2=V_{\mathrm{tot}}-V_1$ . Stationarity in $V_1$ forces equality of the partial derivatives $(\partial S/\partial V)_{U,N}$ across the subsystems. The identity $(\partial S/\partial V)_{U,N}=P/T$ converts this into $P_1/T_1=P_2/T_2$ , and combined with temperature equalization yields $P_1=P_2$ . This characterizes pressure as the intensive variable that equalizes under mechanical contact.