Subsections

• General Theory
• Law of Corresponding States
• Maxwell Construction

Phase Transitions in Van der Waals Gas

General Theory

We know from previous lectures :

$$P = \frac{NkT}{V-Nb} - \frac{N^2a}{V^2}$$

Stability condition:

$$\left(\frac{\partial P}{\partial V}\right)_{T,N}<0$$

Is this always true for VdW gas?  

$$\left(\frac{\partial P}{\partial V}\right)_{T,N} = -\frac{N^2kT}{(V-Nb)^2} + \frac{2N^2a}{V^3}$$ (4)

1.

If T is large enough, the first term in (4) dominates--system is always stable

2.

If T is small, first term wins only for $V\approx Nb$ and $V\to\infty$--system is unstable at intermediate V!

3.

Borderline case: $\partial P/\partial V=0$ just in one point (critical point K)

Phases:

1.

At T>T_c one phase (fluid)--no phase transitions

2.

At T<T_c two phases (liquid & gas)--first order phase transition

3.

At T=T_c--second order phase transition

Equation for T_c:

$$\left(\frac{\partial P}{\partial V}\right)_{T,N}=0,\quad \left(\frac{\partial^2 P}{\partial V^2}\right)_{T,N}=0$$

or

$$\begin{aligned} -\frac{N^2kT}{(V-Nb)^2} + \frac{2N^2a}{V^3}... ... \frac{2N^3kT}{(V-Nb)^3} - \frac{6N^2a}{V^4} &=0 \end{aligned}$$

Result:

$$T_c = \frac{8a}{27bk},\quad P_c = \frac{a}{27b^2},\quad V_c = 3bN$$

Law of Corresponding States

Let us measure everything in critical units:

$$T' = T/T_c,\quad V' = V/V_c,\quad P' = P/P_c$$

Van der Waals law:

$$\left(P'+\frac{3}{{V'}^2}\right)(3V'-1)=8T'$$

This is same for all van der Waals gases! All PVT data lie on same master curve. This is called law of corresponding states.

Maxwell Construction

Let us determine binodal for van der Waals gas. In coexisting phases $\mu_1-\mu_2=0$ or

$$\int_1^2 d\mu = 0$$

If $N=\mathit{const}$, $T=\mathit{const}$ we have $d\mu=N\,dG = V\,dP$, or

$$\int_1^2 V\,dP = 0$$

It means that dashed areas below are equal!