Page 186 - Advanced Thermodynamics for Engineers, Second Edition

8.5 CONCLUDING REMARKS 173

as the fluid passes from 5 to 7. The regions between 1 and 3, and 5 and 7 can be seen to be stable in
terms of variation of Gibbs energy, whereas that between 3 and 5 is unstable: this supports the
argument introduced earlier based on the variation of pressure and volume. Figure 8.5 also shows
that the values of Gibbs energy and pressure are all equal at the points 2, 4 and 6, which is a line of
constant pressure on Figure 8.4. This indicates that the Gibbs energy of the liquid and vapour
phases can be equal if the two ends of the evaporation process are at equal pressures for an
isotherm. Now considering the region between the saturated liquid and saturated vapour lines; the
Gibbs energy along the ‘isotherm’ shown in Fig. 8.4 will increase between points 2 and 3, and the
substance would attempt to change spontaneously back from 3 to 2, or from 3 to 4, and hence point
3 is obviously unstable. In a similar manner the stability of point 5 can be considered. An isobar at
100bar is shownon fig. 8.5, and it can be seen that it crosses the Gibbs energy line at three points –
the lowest Gibbs energy is at state 1. Hence, state 1 is more stable than the other points. If that
isobar were moved down further until it passed through point 5, then there would be other points of
higher stability than point 5 and the system would tend to move to these points. This argument has
still not defined the equilibrium line between the liquid and vapour lines, but this can be defined by
ensuring that the Gibbs energy remains constant in the two-phase region, and this means the
equilibrium state must be defined by constant temperature and constant Gibbs energy. Three points
on that line are defined by states 2, 4 and 6. The pressure which enables constant Gibbs energy to
be achieved is such that
p
Z 6
vdp ¼ 0 between points 2 and 6 (8.36)
p 2
and this means that
p p
Z 4 Z 6
vdp þ vdp ¼ 0 (8.37)
p 2 p 4
|fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl}
Region I Region II
which means that the area of the region between the equilibrium isobar and line 2-3-4 (Region I) must
be equal to that between the isobar and line 4-5-6 (Region II). This is referred to as Maxwell’s equal
area rule, and the areas are labelled I and II in Fig. 8.4.

8.5 CONCLUDING REMARKS
A number of different equations of state have been introduced which can describe the behaviour of
substances over a broader range than the common perfect gas equation. van der Waals equation has
been analysed quite extensively, and it has been shown to be capable of defining the behaviour of
gases close to the saturated vapour line. The law of corresponding states has been developed, and this
enables a general equation of state to be considered for all substances. The region between the
saturated liquid and vapour lines has been analysed and Gibbs energy has been used to define the
equilibrium state.