Consider the phase diagram in the P-v plane depicted in Fig.3.7. [cf. Fig3.4-6].
Along an isothermal
the straight line segment AB inside
the vapor-liquid coexistence region
According to eq(2.179), mechanical stability requires
Consider again the isotherm in Fig.3.7. If we extend the gaseous state beyond point 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 and , where the subscript 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
is the number of moles of phase i substance present and
is the molar energy of the substance when it
is in the pure phase i with molar
volume v and
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 , we have
3. Finally, to calcualte , we begin with
Putting everything into (3.30) gives
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
is infinite for a mixture in the coexistence
region. Thus, adding heat at constant
pressure to a mixture in the coexistence region only converts some liquid into