Physical Chemistry , 1st ed.

(We will see in a minute why ideal gases are brought up again.) Subtracting

these two:

ddideal(VVideal) dp

where we have factored dpout of both terms on the right. Integrating:

ideal

p

0

(VVideal) dp

ideal

p

0

Vdp

p

0

Videaldp (4.59)

If we understand that equation 4.56 gives the chemical potential of an ideal gas

idealin terms of pressure and equation 4.58 gives the chemical potential of

our real gas in terms of fugacity, we can use them to evaluate ideal:

ideal° RTln 

p

f

°

° RTln 

p

p

°



RTln 

p

f

°

ln 

p

p

°



RTln 

p

f/

/

p

p

°

°

RTln 

p

f



Therefore, substituting into the left side of equation 4.59:

RTln 

p

f

p

0

Vdp

p

0

Videaldp

Rearranging:

ln 

p

f

ln 

R

1

T



p

0

Vdp

p

0

Videaldp (4.60)

This might seem to be a complicated expression, but consider what it is. An in-

tegral is an area under a curve. The first integral is the area under a plot of the

partial molar volume versus pressure. The second integral is the area under a

plot of the ideal molar volume versus pressure. The subtraction of the two in-

tegrals, then, is simply the difference in areas of the two plots between p 0 and

some nonzero value of p.Divide this value by RTand you have the logarithm

of the fugacity coefficient . Fugacities are therefore determined by simply

measuring the volumes of known quantities of gases under isothermal con-

ditions and comparing them to the expected ideal volume. Figure 4.6 is an

example of what a graphical representation of such an investigation might

look like.

Equation 4.60 can also be evaluated in terms of the compressibility Zfor a

real gas. We won’t derive it here but simply present the result. (For a deriva-

tion see P. W. Atkins and J. de Paulo,Physical Chemistry, 7th ed., Freeman, New

York, 2002, p. 129.)

ln 

p

0



Z

p

1

dp (4.61)

If you know the equation of state for a gas and its compressibility in terms of

the equation of state, you can substitute for Zin equation 4.61 and evaluate

the integral. Or, the compressibility can be plotted and the integral determined

by numerically measuring the area under the plot of (Z1)/pversus p.

112 CHAPTER 4 Free Energy and Chemical Potential

V

p p

0

Ideal

Actual

Figure 4.6 A simple way of determining the
fugacity coefficient of a real gas is to plot the real
volume of the gas at various pressures and com-
pare it to the expected ideal volume of the gas.
The fugacity coefficient is related to the difference
in the area under the curves (indicated by the
shaded portion of the diagram). See equation 4.60.