Implications

Consider an isolated system with two separate regions at uniform (but potentially different) pressures and temperatures.

Figure 17-4: Region $A$ with pressure and temperature $P_A$ and $T_A$ enclosing a subregion $B$ at $P_B$ and $T_B$.

Suppose that there is a ``virtual'' change (i.e. any one of an infinite number of possible changes) in the system such that $\ensuremath{\delta}U = 0$ and $\ensuremath{\delta}V = 0$.

Any change in $B$ ( $\ensuremath{\delta}U_B$, $\ensuremath{\delta}V_B$) can be chosen as a virtual changes as long as we also pick for system $A$, ( $\ensuremath{\delta}U_A = - \ensuremath{\delta}U_B$ and $\ensuremath{\delta}V_A = - \ensuremath{\delta}V_B$).

Then

$$\input{equations/11-6A}$$ (17-4)

Because $\ensuremath{\delta}U_A$ is independent of $\ensuremath{\delta}V_B$ we can find a $\ensuremath{\delta} S_{total} > 0$ unless

$$\input{equations/Tconst}$$ (17-5)

$$\input{equations/Pconst}$$ (17-6)

These are the necessary conditions for equilibrium in a heterogeneous isolated system: no spatial variation pressure can exist if the volume can move from region to region ( $\ensuremath{\delta}V$) and no spatial variation in temperature can exist if energy can flow from region to region ( $\ensuremath{\delta}U$).¹