# What other thermodynamic properties besides temperature and pressure may be graphed in phase diagrams?

May 24, 2018

TYPICAL PHASE DIAGRAMS

Consider water's $\text{PT}$ phase diagram as an unusual example of the typical kind of phase diagram:

Here we have:

• Equilibrium (coexistence) curves, where two phases coexist.
• A triple point, where three phases coexist.
• A critical point, where the liquid and vapor phase become a single superfluid phase.

Phase transitions would occur simply by crossing a curve, and while it is going on, you are still on the curve itself.

PV PHASE DIAGRAMS

A common alternative is a $\text{PV}$ phase diagram, generally used to express the variation of molar volume of liquids compared to gases.

[Here, temperature IS a variable, but it extends out of the page, like a $z$ axis. See here for a 3D, $\text{PVT}$ phase diagram.]

Unlike in the usual $\text{PT}$ phase diagrams:

• There are equilibrium regions (instead of equilibrium curves).
• There is a triple line (instead of a triple point).
• The critical point lies on a double-valued curve (instead of at the end of a single-valued curve).

A phase transition from liquid to vapor starts from the righthand intersection above the triple line and ends on the descending curve to the right.

This clearly shows the increase in molar volume at constant pressure (and temperature) while vaporizing, as it should.

• The starting point of the transition marks a solution to the appropriate equation of state for the molar volume of the liquid.
• The ending point of the transition marks a solution to the appropriate equation of state for the molar volume of the gas.

This kind of graph finds more use by graphing the liquid-vapor curve at various temperatures to find the critical natural variables:

The solid horizontal line is the aforementioned vaporization phase transition.

When the cubic curve in the middle coalesces into a flat central slope, we have the critical temperature ${T}_{c}$, pressure ${P}_{c}$, and molar volume ${\overline{V}}_{c}$.

This derivation of the constants $a$ and $b$ used in the van der Waals equation of state makes use of this concept of varying temperature (towards ${T}_{c}$) on a $\text{PV}$ phase diagram.

A $\text{TV}$ phase diagram also exists, and although I won't go too far into it, an example is shown below:

As you may see, in the second $\text{PV}$ phase diagram, the $\text{TV}$ phase diagram is superimposed upside-down.