Page 184 - Advanced Thermodynamics for Engineers, Second Edition

8.4 ISOTHERMS OR ISOBARS IN THE TWO-PHASE REGION 171

• the Dieterici equation.
RT a
p ¼ e = RTv (8.31)
v b

Returning to the van der Waals gas, examination of Eqn (8.22) shows that

p c v c 3
z c ¼ ¼ ¼ 0:375 (8.32)
RT c 8
This value, z c (sometimes denoted m c ) is the compression coefficient, or compressibility factor, of a
substance at the critical point. In general, the compression coefficient, z, is defined as
pv
z ¼ (8.33)
RT
It is obvious that, for a real gas, z is not constant, because z ¼ 1 for a perfect gas (i.e. in the
superheat region), but it is 0.375 (ideally) at the critical point. There are tables and graphs which show
the variation of z with state point, and these can be used to calculate the properties of real gases. These
were originally derived by Obert (1960), and are reproduced in, for example, Bejan (1988) and Moran
and Shapiro (1988).

8.4 ISOTHERMS OR ISOBARS IN THE TWO-PHASE REGION

Equation (8.15) predicts that along an isotherm the pressure is related to the specific volume by a cubic
equation, and this means that in some regions there are three values of volume which satisfy a
particular pressure. Since the critical point has been defined as an inversion point then the multi-valued
region lies below the critical point, and experience indicates this is the two-phase (liquid þ vapour)
region. It is known that pressure and temperature are not independent variables in this region, and that
fluids evaporate at constant temperature in a constant pressure chamber; hence these isotherms should
be at constant pressure. This anomaly can be resolved by considering the Gibbs energy of the fluid in
the two-phase region. Since the evaporation must be an equilibrium process then it must obey the
conditions of equilibrium. Considering the isotherm shown in Fig. 8.4, which has been calculated
using Eqn (8.27) for water at 300 C, it can be seen that in the regions from 3 to 1, and from 7 to 5 a

decrease in pressure results in an increase in the specific volume. However, in the region between 5 and
3 an increase in pressure results in an increase in specific volume: this situation is obviously unstable.
It was shown in Chapter 2 that equilibrium was defined by vGj 0: Now
p;T
dg ¼ vdp sdT (8.34)

and hence, along an isotherm the variation of Gibbs energy from an initial point, say, 1 to another
point is
p
Z
Dg ¼ g g 1 ¼ vdp (8.35)
p 1