Page 179 - Advanced thermodynamics for engineers

8.2 VAN DER WAALS EQUATION OF STATE 165

The negative sign is introduced because an increase in pressure produces a decrease in volume and

vv
hence < 0 (see Section 7.4); it is more convenient to have a coefficient with a positive value
vp
T
and the negative sign achieves this. The isothermal compressibility of a fluid is analogous to the
Young’s modulus of a solid.
The other coefficient that can be defined is the coefficient of expansion, b. This is defined as the
‘volumetric strain’ produced by a change in temperature, giving

1 vv 1 vV
b ¼ ¼ : (8.12)
v vT V vT
p p
The coefficient of expansion is analogous to the coefficient of thermal expansion of a solid
material.
Hence, Eqn (8.4) may be written

dv dV
¼ ¼ kdp þ bdT (8.13)
v V
The ideal gas law applies to gases in the superheat phase. This is because when gases are super-
heated they obey the kinetic theory of gases in which the following assumptions are made:
• molecules are solid spheres;
• molecules occupy a negligible proportion of the total volume of the gas;
• there are no forces of attraction between the molecules, but there are infinite forces of repulsion
on contact.

If a gas is not superheated the molecules become closer together and the assumptions are less valid.
This has led to the development of other models which take into account the interactions between the
molecules.

8.2 VAN DER WAALS EQUATION OF STATE
In an attempt to overcome the limitations of the perfect gas equation a number of modifications have
been made to it. The two most obvious modifications are to assume
• the diameters of the molecules are an appreciable fraction of the mean distances between them,
and the mean free path between collisions. This basically means that the volume occupied by the
molecules is not negligible;
• the molecules exert forces of attraction which vary with the distance between them, while still
exerting infinite forces of repulsion on contact.
First, if it is assumed that the molecules occupy a significant part of the volume occupied by the gas
then Eqn (8.10) can be modified to
<T
p ¼ ; (8.14)
v b
where b is the volume occupied by the molecules. This is called the Clausius equation of state.