Page 528 - Modelling in Transport Phenomena A Conceptual Approach

508 APPENDIX A. MATHEMATICAL PRELIMwARTEs

where
(0 is in degrees) (A. 7- 15)

In Eqs. (A.7-12)-(A.7-14) the upper sign applies if N is positive, the lower sign
applies if N is negative.

Example A.4 Cubic equations of state are frequently used in thermodynamics to
describe the PVT behavior of liquids and vapors. These equations are expressed in
the form
RT
p=-- a(T) (A. 7- 16)
P-b Va+pV+y
where the terms a, p, y, and a(T) for diflerent types of equations of state are given
by

vander Waals 2 0 0 lZ

Redlich-Kwong 2 b 0 a/@-
Peng-Robinson 2 2b -b2 a(T)

When Eq. (A.7-16) has three real roots, the largest and the smallest roots cor-
respond to the molar volumes of the vapor and liquid phases, respectively. The
intermediate root has no physical meaning.
Predict the density of saturated methanol vapor at 10.84 atm and 140 "C using
the van der Waals equation of state. The coeficients a and b are given as
a = 9.3424 m6. atm/ kmo12
b = 0.0658 m3/ kmol

The experimental value of the density of saturated methanol vapor is 0.01216 g/ cm3.

Solution

For the van der Waals equation of state, Eq. (A.7-16) takes the form
- a- ab
v2+pv- - =o
P

Substitution of the values of a, b, R, and P into Eq. (1) gives
P3 - 3.1923 V2 + 0.8618 v - 0.0567 = 0