Modern Thermodynamics

- Chapter 1

28

Molecular Forces and the Law of Corresponding States

From a molecular point of view, van der Waals equation has two parameters, a and b, that describe

molecular forces, often called van der Waals forces. These forces are attractive when the molecules are

far apart but are repulsive when they come into contact, thus making the condensed state (liquid or solid)

hard to compress. It is the repulsive core that gives the molecule a non-zero volume. The typical

potential energy between two molecules is expressed by the so-called Lennards-Jones energy:

ULJ(r) = 4ε

σ

r

12

−

σ

r

6

(1.5.8)

Fig. 1.7 shows a plot of this potential energy as a function of the distance r between the centers of the

molecules. As the distance between the molecules decreases ULJ decreases, reaches a minimum, and

sharply increases. The decreasing part of ULJ is due to the term –(σ/r)6, which represents an attractive

force, while the sharply increasing part is due to the term, (σ/r)12, which represents a repulsive core. The

Lennard-Jones energy reaches a minimum value of -ε when r=21/6σ (exc 1.20). The two van der Waals

parameters, a and b, are related to ε and σ respectively, the former being a measure of the molecular

attractive force and the later a measure of the molecular size. In fact, using the principles of statistical

thermodynamics, for a given ε and σ, the values of a and b can be calculated. Such a relationship

between the molecular interaction potential and the parameters in van der Waals equation of state gives us

an insight into the limitations of the law of corresponding states, which depends on just two parameters, a

and b. If more than two parameters are needed to adequately describe the forces between two molecules,

then we can expect the equation of state to also depend on more than two parameters. Lennard-Jones type

potentials that use two parameters are good approximations for small molecules; for larger molecules the

interaction energy depends not only on the distance between the molecules but also on their relative

orientation and other factors that require more parameters. Thus, significant deviation from the law of

corresponding states can be observed for larger molecules.