Modern Thermodynamics
- Chapter 1
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



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 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.
Previous Page Next Page