Physical Chemistry Third Edition

34 1 The Behavior of Gases and Liquids

Thereduced temperatureis the ratio of the temperature to the critical temperature:

Tr

T

Tc

(1.4-14)

Using the definitions in Eqs. (1.4-12), (1.4-13), and (1.4-14) and the relations in Eqs.

(1.4-5) and (1.4-6) we obtain for a fluid obeying the van der Waals equation of state:

P

aPr

27 b^2

, Vm 3 bVr, T

8 aTr

27 Rb

When these relations are substituted into Eq. (1.3-1), the result is

(

Pr+

3

Vr^2

)(

Vr−

1

3

)



8 Tr

3

(1.4-15)

Exercise 1.14

Carry out the algebraic steps to obtain Eq. (1.4-15).

In Eq. (1.4-15), the parametersaandbhave canceled out. The van der Waals equation

of state thus conforms to the law of corresponding states. The same equation of state

without adjustable parameters applies to every substance that obeys the van der Waals

equation of state if the reduced variables are used instead ofP,Vm, andT. The other

two-parameter equations of state also conform to the law of corresponding states.

Figure 1.9 is a graph of the experimentally measured compression factor of a number

of polar and nonpolar fluids as a function of reduced pressure at a number of reduced

Methane

Isopentane

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.^0

Reduced pressure, Pr

Tr
2.00

1.50

1.30

1.20

1.10

1.00

Ethylene

n-Heptane

Ethane n-Butane

Nitrogen

Propane

Carbon dioxide Water

Z

PV

m

RT

Figure 1.9 The Compression Factor as a Function of Reduced Pressure and

Reduced Temperature for a Number of Gases.From G.-J. Su,Ind. Eng. Chem., 38 , 803

(1946). Used by permission of the copyright holder.