Pressure and chemical potential

The argument developed above can be repeated pretty much in the same way for the other two constraints. For example, we could allow the partitioning wall to move, leaving it still impermeable to transfer of particles, and realise a condition in which the energies of the two subsystems are not changed. This will give us the equilibrium condition for the change of entropies w.r.t to changes of volume $V_1$ and $V_2$. However, it may be difficult to realise a situation in which moving the wall keeps the energies $E_1$ and $E_2$ constant, because changing the volumes involves work in the system and one would have to realise a situation in which this work is exactly balanced by a heat flow. However, we could realise a situation in which we allow both energy and volume to change, and we maximise the entropy w.r.t both of these variations. We obtain:

$$\begin{aligned} dS = dS_1 + dS_2 & = \left ( \frac{\partial S_1... ...tial S_2}{\partial V_2}\right )_{E_2,N_2} dV_2 = 0. \end{aligned}$$

Since this has to be valid for any variation of energy and volume, we obtain again , but with the additional condition:

$$\left ( \frac{\partial S_2}{\partial V_2}\right )_{E_2,N_2} = \left ( \frac{\partial S_1}{\partial V_1}\right )_{E_1,N_1}.$$ (3.17)

The wall will not move if the pressures are the same in the two sub-volumes, $P_1 = P_2$, and therefore pressure must be related to $\left ( \frac{\partial S}{\partial V}\right )_{E,N}$. We shall see that by setting:

$$P = T \left ( \frac{\partial S}{\partial V}\right )_{E,N}$$ (3.18)

we obtain a definition of pressure which is identical to the conventional one (i.e. force divided by an area).

The final condition of equilibrium w.r.t. exchange of particles is obtained by adding variations w.r.t. to $N_1$ and $N_2$ to , which results in

$$\left ( \frac{\partial S_2}{\partial N_2}\right )_{E_2,V_2} = \left ( \frac{\partial S_1}{\partial N_1}\right )_{E_1,V_1},$$ (3.19)

from which we define the chemical potential:

$$\mu = -T \left ( \frac{\partial S}{\partial N}\right )_{E,V}.$$ (3.20)

Just as temperature and pressure control the flow of heat and the change of volume, the chemical potential controls the flow of particles, which move in the direction of reducing their chemical potential. This means that if we introduce a constraint that makes the chemical potential different in the two subsystems, removal of that constraint will result in a flow of particles towards the system that had the lower chemical potential when the constraint was on, and the system will adjust to a configuration compatible with having the chemical potential equal (no constraint) in the two subsystems.