Physical Chemistry Third Edition

32 1 The Behavior of Gases and Liquids

For a van der Waals gas, the compression factor at the critical point is

Zc

PcVmc

RTc



3

8

 0. 375 (1.4-7)

Exercise 1.11

Verify Eqs. (1.4-6) and (1.4-7).

Equations (1.4-5) and (1.4-6) can be solved foraandb:

a 3 Vmc^2 Pc

9 RVmcTc

8



27 R^2 Tc^2

64 Pc

(1.4-8)

b

Vmc

3



RTc

8 Pc

(1.4-9)

There are two or three formulas for each parameter. Since no substance exactly fits

the equation different values can result from the different formulas. The best values of

aandbare obtained by usingPcandTcas independent variables. The values of the

parameters for any two-parameter or three-parameter equation of state can be obtained

from critical constants.

Exercise 1.12

a.Show that for the Dieterici equation of state,

Vmc 2 b, Tc

a

4 bR

, Pc

a

4 b^2

e−^2 (1.4-10)

b.Show that for the Dieterici equation of state,

Zc 2 e−^2  0. 27067

c.Obtain the formulas giving the Dieterici parametersaandbas functions ofPcandTc. Find

the values ofaandbfor nitrogen and compare with the values in Table A.3.

The parametersaandbin the Redlich–Kwong equation of state can be obtained

from the relations

a

R^2 Tc^5 /^2

9

(

21 /^3 − 1

)

Pc

, b

(

21 /^3 − 1

)

RTc

3 Pc

(1.4-11)

The value of the compression factor at the critical point according to the Redlich–

Kwong equation of state is 1/3.

Exercise 1.13

Find the values ofaandbin the Redlich–Kwong equation of state for nitrogen.

Figure 1.8 shows schematically a more complete view of the three-dimensional

graph of Figure 1.7, including the solid–liquid and solid–gas phase transitions. There