Page 181 - Advanced Thermodynamics for Engineers, Second Edition

168 CHAPTER 8 EQUATIONS OF STATE

Hence,
v c 8 p c v c
a ¼ 3p c v c ; b ¼ ; R ¼ : (8.22)
3 3 T c
The equation of state for a gas obeying the law of corresponding states can be written
8T R 3
p R ¼ ; (8.23)
ð3v R 1Þ v 2
R
or
!
3
p R þ 2 ð3v R 1Þ¼ 8T R : (8.24)
v
R
Equation (8.24) is the equation of state for a substance obeying van der Waals equation: it should be
noted that it does not explicitly contain values of a and b. It is possible to obtain a similar solution
which omits a and b for any two-parameter state equation, but such a solution has not been found for
state equations with more than two parameters. The law of corresponding states based on van der
Waals equation does not give a very accurate prediction of the properties of substances over their
whole range, but it does demonstrate some of the important differences between real substances and
perfect gases. Figure 8.1 shows the state diagram for water evaluated using parameters in van der
Waals equation based on the law of corresponding states. The parameters used for the calculation of
Fig. 8.1 will now be evaluated.
The values of the relevant parameters defining the critical point of water are
3

v c ¼ 0:00317 m =kg; p c ¼ 221:2 bar; T c ¼ 374:15 C
which gives
v c 0:00317 3 2 2 6 2
b ¼ ¼ ¼ 0:0010567 m =kg; a ¼ 3p c v ¼ 3 221:2 0:00317 ¼ 0:0066684 bar m =kg
c
3 3
8 221:2 0:00317 3
R ¼ ¼ 0:002889 bar m =kg K
3 ð374:15 þ 273Þ
Thus, the van der Waals equation for water is
0:002889T 0:0066684
p ¼ 2 : (8.25)
ðv 0:0010567Þ v
Note that the value for ‘R’ in this equation is not the same as the specific gas constant for the
substance behaving as a perfect gas. This is because it has been evaluated using values of parameters at
the critical point, which is not in the region where the substance performs as a perfect gas.
It is possible to manipulate the coefficients of van der Waals equation to use the correct value of R
for the substance involved. This does not give a very good approximation for the critical isotherm, but
is reasonable elsewhere. Considering Eqn (8.22), the term for v c can be eliminated to give
2 2
27R T c RT c
a ¼ ; and b ¼ : (8.26)
64p c 8p c