Page 62 - Modern physical chemistry

3.10 Corresponding States 51

On each isothenn, the dotted line is drawn so that it cuts off equal areas above and
below. Thus on the labeled curve, area ABC equals area CDE.
The work done on the system when it is taken reversibly from one point on an isothenn
to another equals the negative of the area under the curve -Jp dV. The equal-area rule
ensures that the area under the van der Waals curve between the liquid and the gaseous
states equals the area under the dotted straight line joining those states, Thus, the work
done would be the same. Along the van der Waals curve, the system would go homoge-
neously and reversibly; along the dotted line, it would go heterogeneously and reversibly.
Why the two works should be equal will be considered later.

3. 10 Corresponding States

Describing mathematically the variation in a and b for various substances is beyond
the scope of this book. Instead, we will merely investigate how van der Waals equation
can be generalized to allow for some of the variation.
First, consider that a, b, and R are constant and given by (3.60), (3.61), (3.62). Sub-
stituting into (3.51) yields

[3.63]

which can be reduced to the dimensionless equation

[ p. + ~)[v. _1.) = ~ T. . [3.64]
r 3
3 r
r
2
Vr
Here reduced pressure Pr is the ratio PIPe, reduced volume Vr is the ratio VlVe, and
reduced temperature Tr is the ratio TITc-
In this fonn of van der Waals equation, all references to a specific substance have
dropped out. The reduced volume is a universal function of the reduced pressure and
reduced temperature. Also from van der Waals equation, we have the critical ratio

RTe = ~ = 2.67. [3.65]
PeTe 3
As a generalization, let the reduced volume Vr be an empirical function of the reduced
pressure P r and the reduced temperature T"
Vr = Vr(Pr,Tr), [3.66]

and let the critical ratio be an empirical constant,

RT
__ = const. [3.67]
e
PeVe
Conditions (3.66) and (3.67) constitute a law oj corresponding states. According to it, dif-
ferent substances at the same reduced pressure and reduced temperature correspond; they
have the same reduced volume. We consider the law to apply to both gas and liquid phases.
The constituent molecules must not be too abnormal in their shapes or interactions, however.
For normal substances, it is found that

[3.68]