Van Der Waals Equation - Maxwell Equal Area Rule

Maxwell Equal Area Rule

Below the critical temperature T_R < 1 an isotherm of the Van der Waals equation oscillates as shown.

Along the red portion of the isotherm which is unstable; the Van der Waals equation fails to describe real substances in this region because the equation always assumes that the fluid is uniform while between a and c on the isotherm it becomes more stable to be a coexistence of two different phases, a denser phase which we normally call liquid and a sparser phase which we normally call gas. To fix this problem James Clerk Maxwell (1875) replaced the isotherm between a and c with a horizontal line positioned so that the areas of the two hatched regions are equal. The flat line portion of the isotherm now corresponds to liquid-vapor equilibrium. The portions a–d and c–e are interpreted as metastable states of super-heated liquid and super-cooled vapor respectively. The equal area rule can be expressed as:

where P_V is the vapor pressure (flat portion of the curve), V_L is the volume of the pure liquid phase at point a on the diagram, and V_G is the volume of the pure gas phase at point c on the diagram. The sum of these two volumes will equal the total volume V.

Maxwell justified the rule by saying that work done on the system in going from c to b should equal work released on going from a to b. (The area on a PV diagram corresponds to mechanical work). That’s because the change in the free energy function A(T,V) equals the work done during a reversible process, and the free energy function - being a state variable - should take on a unique value regardless of path. In particular, the value of A at point b should calculate the same regardless of whether the path came from left or right, or went straight across the horizontal isotherm or around the original Van der Waals isotherm. Maxwell’s argument is not totally convincing since it requires a reversible path through a region of thermodynamic instability. Nevertheless, more subtle arguments based on modern theories of phase equilibrium seem to confirm the Maxwell Equal Area construction and it remains a valid modification of the Van der Waals equation of state.

The Maxwell equal area rule can also be derived from an assumption of equal chemical potential μ of coexisting liquid and vapour phases. On the isotherm shown in the above plot, points a and c are the only pair of points which fulfill the equilibrium condition of having equal pressure, temperature and chemical potential. It follows that systems with volumes intermediate between these two points will consist of a mixture of the pure liquid and gas with specific volumes equal to the pure liquid and gas phases at points a and c.

The Van der Waals equation may be solved for V_G and V_L as functions of the temperature and the vapor pressure P_V. Since:

where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as:

Since the gas and liquid volumes are functions of P_V and T only, this equation is then solved numerically to obtain P_V as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes.