Consider the phase diagram in the *P-v* plane depicted in Fig.3.7. [cf.
Fig3.4-6].

Along an isothermal *AB* inside
the vapor-liquid coexistence region *D* of molar volume

where *x*

Þ

According to eq(2.179), mechanical stability requires

Consider again the isotherm *A* into the coexistence region (see dashed line), mechanical
stability can still be maintained as long as condition (a) is satisfied. However, since the free energy is not a
minimum, the system is only metastable. These states are called supercooled
(vapor) states. Similarly, metastable
states resulting from the continuation of the liquid state past point *B* into the coexistence region are called
superheated (liquid) state.

It is possible to extend the superheated liquid states into the negative pressure region. In which case, no wall is required to contain the system.

Note that at the critical point *C*, the molar volumes of both phases are
equal. Since the relevant symmetries are
also the same, these phases become indistinguishable. Also, there are no metastable states.

The liquid-vapor coexistence curve in the *T-ρ* plane can be plotted in terms of
reduced quantities *C* denotes a
critical value. Guggenheim found that
such curves more or less coincide for a large number of pure substances (see
Fig.3.8). This is an example of the **law of corresponding states**, which
postulates that all pure classical fluids obey the same equation of state
involving reduced quantities.

The curve in Fig.3.8 is given by Guggenheim as the solutions to the following equations

In the
coexistence region, the total internal energy of a mixture (point *D* in Fig.3.7 ) is

where *i* substance present and *i* with molar
volume *v* and *T*. Dividing by the total number of moles, we
obtain the molar total internal energy

so that the molar heat capacity at constant volume is

where we've used

The following steps are required to related (3.30) with directly measurable quantities.

1.
From

Similarly,

2.

3.
Finally, to calcualte

so that

i.e.,

Putting everything into (3.30) gives

Now, from

we have

Also, using the Maxwell relation

we can write

Putting (3.38-9) into the terms included in one of the square brackets in (3.37a), we get, in the coexistence region,

Hence, (3.37) simplifies to

with all quantities on the right side measurable.

Since the isotherm is also an isobar in the
coexistence region,