Page 60 - Modern physical chemistry

3.9 Gas-Liquid Isotherms 49

3.9 Gas-iliquid Isotherms

Multiplying van der Waals equation by V2IP and rearranging leads to the form

v 3 -[b+ RT)V 2 +~V _ ab =0. [3.52]
P P P

At a given temperature T and pressure P, this is a cubic equation in volume V. At high
temperatures, only one root is real. But at a critical temperature and a critical pressure,
the complex roots become real and equal. Going below this critical point on a pressure-
volume plot, the three real roots gradually spread out. Equation (3.52) can be rewritten
in the form
(v - VI Xv -v 2Xv - V3 )=0, [3.53]
in which VI> V 2 , V3 are the three roots. At the critical point, where

[3.54]
Equation (3.53) reduces to
[3.55]

whence
3
322
V -3VcV +3Vc V - Vc = O. [3.56]
Comparing (3.52) with T = Tc and P = Pc to (3.56) yields

3V =b+ RTc [3.57]
c
P. '
c
a [3.58]
p'
c
[3.59]

Solving for a, b and R gives us
[3.60]

b= Vc [3.61 ]
3'

R= 8PcVc. [3.62]
3T c
With these formulas, one can calculate a and b from the critical constants Pc and Vc
for a given substance. Alternatively, one can substitute experimental values of P, V, and
T for a particular region into (3.52) and solve for a and b. Because they vary, these para-
meters should be determined for the temperature-pressure region to be represented.
Experimental data for a particular substance are plotted in figure 3.2. Each curve
shows how the pressure varies with volume at a given temperature. Because the tem-
perature is fixed along it, each curve is called an isotherm.