Concise Physical Chemistry

c02 JWBS043-Rogers September 13, 2010 11:23 Printer Name: Yet to Come

THE LAW OF CORRESPONDING STATES, ANOTHER VIEW 27

2.5 REDUCED VARIABLES

The degree of nonideality for real gases is determined by how near the temperature

is to the critical temperature. We can express the “nearness” of the temperature of

a gas to its critical temperature as the unitless ratioT/Tc. This ratio is called the

reduced temperature

TR=

T

Tc

The other reduced variables are defined in the same way. The (unitless)reduced

pressureandreduced volumeare, for one mole,

pR=

p

pc

and VR=

V

Vc

These new variables are scaling factors by which we take an entire family of isotherms

suggested by the three representative isotherms in Fig. 2.4 and move, stretch, or

compress them until their critical isotherms coincide. Having done that, at the same

TR,thepRand,VRbehavior of all gases fall approximately on the same family

of curves. The gas data have been manipulated intocorresponding states(Section

2.3). Knowing the behavior ofonegas in terms of its reduced variablespR,VR, and

TR, we know the behavior ofanyreal gas, provided only that we know its critical

constantspc,Vc, andTc. Needless to say, industrial chemists and chemical engineers

are delighted by this, and they have devoted considerable effort to construct standard

tables andZ-curves in terms of the reduced variables.

2.6 THE LAW OF CORRESPONDING STATES, ANOTHER VIEW

Replacing the parametersaandband the constantRin the van der Waals equation

for one mole

p=

RT

(V−b)

−

a

V^2

with

b=

Vc

3

, a= 3 pcVc^2 , and R=

8 pcVc

3 Tc

we get, with some algebraic manipulation (see Problems and Exercise 2.1),

[

pR+

3

VR^2

](

VR−

1

3

)

=

8

3

TR