Equilibrium with constraints that are more practical

Consider that a system with internal degrees of freedom placed inside a giant reservoir at constant $P_R$ and $T_R$ (i.e. the atmosphere or the conditions inside a furnace, etc.) ( $R$ for reference or reservoir)

Figure 19-1: System with internal degrees of freedom in contact with a pressure and temperature reservoir.

We have already shown that for equilibrium:

$T_{sys} = T_R$ and $P_{sys} = P_R$ for a system that can exchange energy and volume with its environment.

The condition that a spontaneous change can take place is (in other words, the system is not in equilibrium):

$$\input{equations/13-3A}$$ (19-3)

Apply the first law to the entire system:

$$\input{equations/13-3B}$$ (19-4)

so that:

$$\input{equations/13-4A}$$ (19-5)

or

$$\input{equations/13-4B}$$ (19-6)

or, comparing with Equation 19-1,

$$\input{equations/13-4C}$$ (19-7)

Note that the sum in Equation 19-7 is really a placeholder for the ``internal degrees of freedom'' for a system in equilibrium with a reservoir of constant $P$ and $T$. Equation 19-7 as it is written, with chemical potentials and changes in the compositions, is the ``normal case'' and the one that is the best to remember.