Modern Thermodynamics

- Chapter 1

23

curve because this curve represents an unstable supersaturated state in which the gas condenses to a

liquid. The actual state of the gas follows the straight line ABC. As T increases, the two extrema move

closer and finally coalesce at T=Tc. For a mole of a gas, the point (p,V) at which the two extrema

coincide are defined as the critical pressure pc and critical molar volume Vmc. For T higher than TC,

there is no phase transition from a gas to a liquid; the distinction between gas and liquid disappears.

(This does not happen for a transition between a solid and a liquid because a solid is more ordered than a

liquid; the two states are always distinct.) Experimentally, the critical constants pc, Vmc and Tc can be

measured and they are tabulated (Table 1.1). We can relate the critical parameters to the van der Waals

parameters a and b by the following means. We note that if we regard p(V,T) as function of V, then for

TTc, the derivative

∂p

∂VT

= 0 at the two extrema. As T increases, at the point where the two extrema

coincide, i.e. at the critical point T=Tc, p=pc and V=Vmc, we have an inflection point . At an inflection

point, the first and the second derivatives of a function vanish. Thus, at the critical point:

∂p

∂VT

= 0

∂2p

∂V2

T

= 0

(1.5.2)

Using these equations one can obtain the following relations between the critical constants and the

constants a and b (exc 1.17):

a =

9

8

RTcV

mc

b =

Vmc

3

(1.5.3)

in which Vmc is the molar critical volume. Conversely we can write the critical constants in terms of the

van der Waals constants a and b (exc 1.17):

Tc =

8a

27Rb

pc =

a

27b2

V

mc

= 3b

(1.5.4)

Table 1 contains the values of a and b and critical constants for some gases.