Concise Physical Chemistry

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

THE VIRIAL EQUATION: A PARAMETRIC CURVE FIT 19

increases directly with the volume occupied by the gas; thus, for the attractive force

between particles, he wrote

F=

a

V^2

The conversion factor from distance to volume is not needed because it is included in

the parametera, which is determined experimentally. Pressure is force per unit area

on the walls of a container, so van der Waals added the force term to the pressure and

rewrote the ideal gas equation for one mole of a real gas as the corrected pressure

times the volume remaining after subtracting the volume of the particles:

(

p+

a

V^2

)

(V−b)=RT

The van der Waals equation is asemiempiricalequation because the ideal gas law on

which it is based can be derived from pure theory (see below), butaandbare empirical

parameters found by trial and error. One can start with a pair of plausible estimates

foraandb, vary them, compare the results with measuredp, V, Tbehavior, and select

the values ofaandbthat give the best agreement with experimental measurements.

Computer routines are available that make many thousands of estimates and give the

best curve fit in a matter of seconds.

A knowledge ofbpermits one to calculate order-of-magnitude radii of molecules.

For example, the van der Waals radius of methane is 190 pm (picometers, 10−^9 m) as

compared to the spectroscopic value (obtained many years later) of 109 pm for the

C–H bond length of methane. Numerous similar calculations give comparable results.

This rough agreement supports van der Waals’s qualitative picture of the excluded

volume of real gases.

2.2 THE VIRIAL EQUATION: A PARAMETRIC CURVE FIT

Thevirialequation is an example of a more general and frequently more accurate

curve fitting routine than the van der Waals equation, but it that gives less insight into

possible causes of nonideal behavior than the van der Waals equation does. Any data

set can be graphed and fit by an analytical equation (an equation that can be written

out in terms of a limited number of variables and some accompanying parameters).

A parameter is a number entering into an equation that takes on a fixed value for one

system but may change to some other fixed value for another system. For example,

what is usually called the Boyle’s law “constant”pV=kis, in fact, a parameter

because it is valid only for a specified, fixed temperature. Change the temperature

and the parameter takes on a different value, but it acts like a constant as long as you

maintain the temperature fixed. The gas law constantRis a true constant; for one

mole of an ideal gas it is always the same.

Of any two equations, one will be a better fit to a given data set than the other.

Of three equations, one will be a better fit than either of the other two, and so on.