Concise Physical Chemistry

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

20 REAL GASES: EMPIRICAL EQUATIONS

Computer speed makes it possible to try very many equations and select the best one,

but there is a point of diminishing returns. An equation may have so many parameters

that no one will ever use it or it may follow random fluctuations in the data set that

tell us nothing about the physics of the actual system. If you are not prudent in your

use of curve-fitting programs, you may be calculating the characteristics of a unicorn

to many significant figures. These caveats apply to any curve-fitting problem, not just

those of real gases.

A very nice balance that avoids daunting complexity but achieves good accuracy

is the series equation

y=a+bx+cx^2 +dx^3 + ···

which has an infinite number of terms but which is cut off ortruncatedat some

reasonable number of terms, usually 3 or 4. As applied to real gases, this series is the

virial equation of state:

pVm=RT+B 2 [T]

(

RT

Vm

)

+B 3 [T]

(

RT

Vm

) 2

+B 4 [T]

(

RT

Vm

) 3

+ ···

The parametersB 2 [T],B 3 [T], andB 4 [T]are called the second, third, fourth,...

virial coefficientsand the notationVm=V/nis used to remind us that the volume

taken is amolarquantity. The notationB 2 [T],B 3 [T],B 4 [T],...is used to indicate

temperature dependence of the virial coefficients. The square brackets do not indicate

multiplication. By a simple algebraic manipulation, it is possible to express the virial

coefficients in terms of the van der Waals constants and findB 2 [T]=b−a/RT.By

another simple manipulation, one obtains thecompressibility factor.

2.3 THE COMPRESSIBILITY FACTOR

The difference between ideal and real gaseous behavior can be made clearer if we

define acompressibility factor Z, a way of indicating the degree of nonideality of

agas

Z=

pVm

RT

=

pVm

(pVm)ideal

IfZis less than one, nonideality is largely due to attractive forces between molecules.

IfZis greater than one, the nonideal behavior can be ascribed to the volume taken

up by individual molecules treated as hard spheres or to repulsive forces, or both.

An ideal gas would show a compressibility factor of 1.00 at all pressures. At high

temperatures, the total volume is large for any selected pressure. Molecular crowding

becomes less significant, and attractive or repulsive forces are weaker because they

act over longer distances. The gas approaches ideal behavior andZapproaches the

constant value of 1.00 aspapproaches zero.