Derivation of van der Waals Equation of State

Proceeding classically (except that the molecules are treated as indistinguishable, and the arbitrary parameter is replaced by Planck's constant), we obtain

(8.85) |

(See Sections 7.7 and 8.10.) Thus,

(8.86) |

which yields

(8.87) |

(See Exercise 2.)

The remaining integral extends over the volume, , of the container. To make further progress, we note that there are regions in which because of the strong repulsion between molecules when they approach one another too closely. Thus, the integrand vanishes in these regions, which are assumed to have a total volume . In the remaining volume, , where is assumed to vary relatively slowly with inter-molecular separation, we shall replace by some effective constant average value, . Thus, the previous equation becomes

(8.88) |

It remains to estimate the values of the parameters and . The total mean potential energy of the molecules is . But, because there are pairs of molecules in the gas, it follows that the total mean potential energy is also , where is the potential energy of interaction between a given pair of molecules. Equating the different expressions for the total mean potential energy of the gas, we obtain , or

To estimate the mean potential energy, , between a pair of molecules, let us make the simplistic assumption that the interaction potential takes the form

where is separation between the molecular centers, , and . Thus, the molecules act as weakly-attracting hard spheres of radius . The choice of exponent is usually the most appropriate. (See Exercise 8.) Concentrating attention on a particular molecule, and making the crudest possible approximation, we shall say that another molecule is equally likely to be anywhere in the container that is a distance greater than from the given molecule. Thus, the probability of the distance lying between and is , and

(8.91) |

Let us assume that : in other words, the potential falls off sufficiently rapidly for the previous integral to converge properly. Thus, Equation (8.89) becomes

where

(8.93) |

By Equation (8.90), the distance of closest approach between molecules is . Thus, in each encounter between a pair of molecules, there is a volume excluded to one molecule, by virtue of the presence of the other molecule, that is equal to the volume of a sphere of radius . Because there are pairs of molecules, the total excluded volume is . But, for the sake of consistency, this volume must be equal to , because was understood to be the excluded volume per molecule. Thus, it follows that

where

(8.95) |

is four times the volume of a hard-sphere molecule.

This completes our crude evaluation of the partition function. The equation of state of the gas is given by

(8.96) |

Making use of Equations (8.92) and (8.94), we obtain

(8.97) |

or

(8.98) |

Now,

(8.99) |

where is the molar volume. Thus, our approximate equation of state becomes

where

(8.101) | ||

(8.102) |

Of course, Equation (8.100) is identical to the well-known van der Waals equation of state. (See Section 6.15.) Our analysis allows us to relate the constants, and , appearing in this equation of state, to the inter-molecular potential.