Page 180 - Advanced Thermodynamics for Engineers, Second Edition

8.3 LAW OF CORRESPONDING STATES 167

PROBLEM
Evaluate the isothermal compressibility, k, and coefficient of expansion, b, of a van der Waals gas.

8.3 LAW OF CORRESPONDING STATES

If it is assumed that all substances obey an equation of the form defined by the van der Waals equation
of state then all state diagrams will be geometrically similar. This means that if the diagrams were
normalised by dividing the properties defining a particular state point (p, v, T) by the properties at the
critical point (p c ,v c ,T c ) then all state diagrams based on these reduced properties will be identical.
Thus there would be a general relationship

v R ¼ f p R ; T R ; (8.16)
where v R ¼ v=v c ¼ reduced volume;
p R ¼ p=p c ¼ reduced pressure;
T R ¼ T=T c ¼ reduced temperature:
To be able to define the general equation that might represent all substances, it is necessary to
evaluate the values at the critical point in relation to the other parameters. It has been seen from Fig. 8.1
that the critical point is the boundary between two different forms of behaviour of the van der Waals
equation. At temperatures above the critical point the curves are monotonic, whereas below the critical
point they exhibit maxima and minima. Hence, the critical isotherm has the following characteristics:
2 2
ðvp=vvÞ and ðv p=vv Þ ¼ 0.
T c T c
Consider van der Waals equation
<T a
p ¼ : (8.15)
v b v 2
This can be differentiated to give the two differentials, giving

vp RT 2a
¼ þ ; (8.17)
vv 2 v 3
T ðv bÞ
2
v p 2RT 6a
¼ 3 4 : (8.18)
2
v v T ðv bÞ v
and at the critical point
2a
RT c
þ ¼ 0; (8.19)
2 3
ðv c bÞ v c
6a
2RT c
¼ 0: (8.20)
3 4
ðv c bÞ v c
giving
8a a
v c ¼ 3b; T c ¼ ; p c ¼ : (8.21)
27bR 27b 2