Physical Chemistry , 1st ed.

GG° nRTln 

p

p

°

 (4.55)

° RTln 

p

p

°

 (4.56)

The second equation shows that the chemical potential varies with the natural

logarithm of the pressure. A plot ofversus pwould have a general logarith-

mic form, as shown in Figure 4.4.

However, measurements on real gases show that the relationship between 

and pisn’t so exact. At very, very low gas pressures, all gases approach ideal be-

havior. At moderate pressures, for a given chemical potential, the pressure is

lower than expected. This is because real gas molecules do attract each other

slightly, and the measured pressure is lower than ideal. At very high pressures,

for a given chemical potential the pressure is higher than expected, because the

gas molecules become so densely packed that they begin to repel each other.

The actual behavior of the chemical potential versus the real pressure of a gas

is shown in Figure 4.5.

For real gases, thermodynamics defines a scaled pressure called fugacity, f,as

fp (4.57)

where pis the pressure of the gas and is called the fugacity coefficient.The

fugacity coefficient is dimensionless, so fugacity has units of pressure. For real

gases, the fugacity is the proper description of how the gas behaves, and so the

equation in terms of the chemical potential is better written as

° RTln 

p

f

°

 (4.58)

As the pressure gets lower and lower, any real gas behaves more and more ide-

ally. In the limit of zero pressure, all gases act as ideal gases and their fugacity

coefficient equals 1. We write this as

limp→ 0 (f) p;limp→ 0  1

How do we determine the fugacity experimentally? We can start with the

fundamental thermodynamic equation given in equation 4.48:

dGS dT V dp (^)
0
i
idni
For a single component (so that the summation is just one term) undergoing
an isothermal process, this becomes
dGV dp dn
Since dGis an exact differential (see section 4.5), we get the relation
/ p
V/ n. The second expression is the partial molar volume of the substance,V.
That is,




p

V

which leads to

dVdp

For an ideal gas, this would be

didealVidealdp

4.9 Fugacity 111



p

Figure 4.4 An idea of what a plot of the chem-
ical potential versus pressure pshould look like
for an ideal gas.



p

Ideal

Actual

Figure 4.5 For real gases, at high pressures the
chemical potential is higher than expected due to
intramolecular repulsions. At intermediate pres-
sures, the chemical potential is lower than ex-
pected due to intramolecular attractions. At very
low pressures, gases tend toward ideal behavior.