Chapter 7
Real Gases
7.1
Equations of State
1. van der Waals:
 =

(−)
−

2
From this analytic function for  we can (integrate

at constant  ) to infer () [Recall  = − 
) ]
The main assumption to recall is that there is
ONE  (molar volume) or its inverse  (the density).
This is often called the “uniform density approximation”. This is why the vdw’s logic fails in the
two-phase region. (van der Waals completely understood this! He knew when his eq. worked, when
it failed and why !)
This equation played an important role in the development of theory of real fluids. It captures two
physical concepts and makes one major assumption.
(a)  represents the intermolecular attraction
(b)  represents the hard-core repulsion and thus
not all the actual volume is available for motion. If the hard core of the potential is denoted by  = 2  the excluded volume per
mole is =  43 (2 )3 2 = 4 where  is the
volume of a mole of units. The vdw EoS does
NOT deal with the shape of the potential, only
the net influence of the a) attraction and b)
hard core repulsion.
vdw EoS. Note the inset and the mechanical
equilibrium condition connecting phases with a
common tangent and thus equal pressures.
There is a temperature at which these two effects cancel out, (see plot in virial EoS section.)
This temperature is called the Boyle temperature T and has the analytic value of
 

T =
∴ T % with %  and & 
(c) The assumption is that the density must be
uniform! This is clear as there is a (singular)
molar volume. This is not the case when two
phases coexist!
(d) The “vdw” loop is composed of two parts.
50
One of the very nice features of the virial expansion
is that the coeficients are derivable from the intermolecular potential V(r). For example, the coefficient B2 (as
can be shown in a proof suitable for a graduate class) is,
R∞
2 ( ) = −2 0 2 (− ()  − 1)
i. The metastable branches, connecting the
stable branches to the extremum. These
branches can be studied for a finite period
of time, as any metastable condition can, if
the system is not jolted (over the barrier.)
ii. The unstable branch with − isothermal compressibility ( ≡ − 1 ( 
 ) =
1 
(
)

where

=
1).
This
branch
  
does not exist in the real world. It is only a
result of an analytic form of the vdw EOS.
MOTHER NATURE KNOWS BETTER !
Exercise 7.1.1 Show that NATURE’s equal area
construction connects the phases with equal Gibbs
free energy per mole,   =   . (In the thought
process, it is very helpful to draw a plot of ( ),
draw vertical lines [horizontal in P(v)], and think
about the areas under the curve.)
2. Other analytic EoS’s

 2 )( − ) = 
 2
9  
or   =  [1 + 128
  (1 − 6(  ) )]

)( − )
Redlich-Kwong: ( +  05 (+)
(a) Berthelot: ( +
(b)
Virial coefficient 2 ( ) Lim →∞ 2 = 4 (4 times
the hard-core molar volume.)
7.1.1
All phase diagrams are characterized by ONE critical
point and one or more triple points. Triple points (for
one component, C = 1 systems) are fixed points in nature. (You will see why when we discuss the Gibbs phase
rule, for now appreciate that with C = 1 and three phases
in equilibrium, nature picks all the thermodynamics variables, you have no control !)
The critical point is the terminus of the l-v equilibrium line. (There is no terminus to the s-l line.) This
critical point is characterized by
2
( 
 ) = 0 and (  2 ) = 0
The first relations implies that that the isothermal compressibility is infinite (i.e. the system is infinitely compressible at T .) Approaching this point from below, on
the l-v equilibrium line, the distinction between l and v
disappears ( ⇔  .)
7.1.2
= 
3. Expansion EoS
Virial:  =  + 2 ( )2 + 3 ( )3 
The first correction term varies from − to +
as the temperature increases. At low  the attractive
part of the intermolecular potential ( in vdw) is more
important while at high  the excluded volume or hard
core ( in vdw) is more important. The temperature at
which the two effects cancel (in ) is called the Boyle
temperature. At  both the attractive and hardcore features are important, they just happen to cancel
leaving an approximately ideal EoS.
Critical points
Corresponding states
The fact that simple analytic EoS, like the vdw equation,
can explain the behavior of so many fluids, suggests that
there is a greater truth. This truth is even more strongly
suggested by creating a reduced  ( = 1) diagram for
many fluids. (By fluids I mean both the gas and liquid phases.) By dividing the actual temperature and
pressure by the values at the critical point, the “corresponding states” figure shown is obtained. The reason
that all these substances look the same, in this reduced
units plot, is that all the interactions look the same when
plotted in reduced units, i.e. the energy scaled by T and
the distance scaled by some critical molecular dimension.
All this points out, is that the EOS (for a C = 1 system)
is a function of two variables.
51
division by 1 bar. ) At low pressure (where all gases
become ideal) the fugacity becomes the pressure,
lim →0


= 1
2. Thus for a pure material,
∆ = (2 ) − (1 ) =  ln 21
and if the lower pressure in the reference
∆ =  −   =  ln 
3. The trends in  , or the fugacity coefficient  ≡  
can be displayed on a universal figure and simply
explained.
 ≡   = 1
 ≡    1
 ≡    1
The EoS’s of most fluids is idential if represented in
reduced units. To get appropriate reduced units, the 
 and  (or ) are scaled by the critical point values.
4. Exanding the CP
 ≡  +  ln  +  ln  
From another point of view, as there are only two
parameters in the vdw’s equation, it is not surprising
that the values of two values that define the critical
point, T and  (or its inverse  ) are sufficient to define a fairly universal EoS. The critical paramenters for
the vdw EOS are
 

 8
 = 27
 = 3
 = 27

2
Fig. potential scaling to universal form
7.2
ideal gas
decreased escaping tendency
increased escaping tendency
= ref + ideal scaling + real corrections.
See Fig. 7.9 pg 167.
5. What might reduce the “escaping tendency”? Consider a case where the intermolecular attraction is
large (vdW “” term more important than the “”
term.) The generalized force to leave all a molecule’s “friends” behind and go into the gas phase is
reduced. On the other hand, the “hard-core” part
of the interaction enhances the tendency to escape
from the condensed phase.
Fugacity of a RG
G.N. Lewis wanted to preserve the functional forms
appropriate for an IG when dealing with a RG. To do
so, he defined a variable, a “pressure-like”-quantity, such
that if we use this new quantity in the functional form
appropriate for 0  we get the correct generalized chemical force.
1. Following G.N., we DEFINE the fugacity1 by,
 ≡


+  ln  =  +  ln  

Fugacity coefficient   as a function of reduced
pressure  and temperature  
(7.1)
(Remember, arguments of ln and exp are unitless,
when I write a form like the last, I am implying
1 Fugacity

6. The rate of change of  with pressure, ( 
) = 
For a vdw gas,

( − ) =
is from the latin fugere meaning to escape.
52

( + 2 ) 
(a) Let  = 0 and consider the error we make in
calculating how  changes with  if we were
to neglect a finite “” term. The ln  expression (as we have thrown out everything) would
be incrementing too much as we would be neglecting a finite term in the denominator. The
real growth of  with P is slower than the IG,
i.e. we need a correction factor multiplying 
inside the ln that is  1
3. Taking the difference and integrating from some low
pressure 1 (= 0) to some target 2 
{ } − [ ] = ( −  )
R
R 2 =
{ } − [ ] = 12 ()
1 =0
{ (2 ) −  (1 )} − [ (2 ) −  (1 )] =
4. As the real gas is ideal at low 
R2
{ (2 )} − [ (2 )] = 1 ()
(b) Let  = 0 and considered the error made by
neglecting a finite We would not provide for
a fast enough increase in  with  Now the
correction factor, multiplying  inside the ln,
must be  1, i.e. the growth of (P) is faster
than the IG ln(P) dependence.
7.3
7. As  =  
ln  = ln   =
Z
0

 −1


(7.3)
All you need to do to get the fugacity coefficient
( ≡  ) is integrate the difference of the compressibility from 1 (ideal) weighted by 1 If the attractive interactions dominate, the escaping tendency
will be reduced, making  =    1. If the hard
core is more important (always the case if the target pressure is high enough),   1 and the gas will
have an enhanced escaping tendency. The fugacity coefficient and the activity coefficient (used for
solutions) have the same physical content.
Gross Compressibilities
A family of universal curves, like the plot of (   )
can be drawn from the readily measurable gross compressibility (   ),
  

=
=1


()
6. Multiplying the RHS by 
and employing Z,

R

1
( − 1) 
ln   = 
0
Compressibility⇒Fugacity
≡
1
5. Plugging in for the  0  and appreciating that the
references are the same,
R2
 ln 2 −  ln 2 = 1 ( −  )
The simpliest way to generate an approximate fugacity
is from a plot like the one above  (   ) To do this
all one needs to know are the coordinates of the triple
point.
7.3.1
R2
(7.2)
I will make you show that
Such a plot again provides a verification of corresponding
ln 
ln 
(  
) =  − 1
and
(  
) = 1 − 
states and the trends are similarly explained. (That is,
the vdw “a” dominates at low  and  thus   1 Therefore, using the later, one can relate the isothermal
and   1 at both high  and  due to an increased compressibility to the fugacity (with the gross compressimportance of the vdw “b” term. See Fig. 7.5, pg 161.) ibility as an intermediate result.)
7.3.2
f(P) or (Z)
7.3.3
ln 
f(T) or ( 
)
Both the gross and the differential compressibilities can What about the temperature dependence of the fugacbe used to generate the fugacity coefficient . Lets run ity? First the logic. The attractive features of the potenthrough the former.
tial become less and less important at high  (You have
to slow down to enjoy the attraction. Go too fast and
all you do is bump into walls.) OK nice words - but WE
1. We know both
WANT NUMBERS, numBERS, numBERS! The crowd



(  ) = 
and
 =  +  ln(   ).
is screaming for numbers. OK, to get numbers, we need
equations.
2. Consider both an real and an ideal gas:
To get  ( ) we have to first get a different form of


the
Gibbs-Helmholtz
equation. Consider an isothermal
 =  
(a)  =  +  ln(   )
process
converting
a
mole
of gas from low pressure 

+  ln(  )
 =  
(b)  = 
(where  =  ) to some high target  (where  6= )
53
3. Integrating and using  =   
1. The change in   is,

∆ =  −  =  ln


⇒


−


=  ln

 
2. The temperature dependence is captured by a form
of the Gibbs-Helmholtz equation: the dependence
of  with  at fixed 
[

( 

)
] − [
− 2 +

( 
)


2

 ln 
] = {(  ln
 ) − (  ) }

2

(  ln
 )

= {(  ln
 ) − 0}
=
− 1 2 (
or
−  )
4. A result that can be integrated from one temperature (1 ) to another (2 ) to get,
R 2
R
1
2)
 ln  = ln  (
(1 ) = 1 −  2 ( −  )
Recall that the quantity ( −  ) is the difference
between the enthalpy at the target pressure to that
at low pressure.
5. Fortunately, we can get the ()[= (how  changes
with  ) ] inside the integral above from one of our
standard experiments. Considering H(T,P):


−1 = ( 
 ) (  ) (  ) .
 ≡

( 
)
=
Indentifying

−( 
) ( 
 )
Therefore, measurement of

)
a)  ≡ ( 
and
= − 1 ( 
 ) 
b) 
( 
 ) 
allows for the extraction of
This partial can
be integrated to get the difference needed to get the
temperature dependence of 
R  
( )  =  −  
  
7.4
Mixtures of RGs
You need  to get  for a pure gas, while one needs a

quantity  ≡ ( 
)  to get  in a mixture. The dif
ferential is an example of a “partial-molar quantity”,
in this case a “partial molar volume”
1. While for a pure material

( 
) = 
2. for a mixture

( 
 ) =  and
 =  +  ln  
As the first term is a constant
ln 
(  
) =



54
1

) 
4. This leads to a fugacity coefficient at a total pressure
of P of
ln  =

 ln 
= {(  ln
 ) − (  ) }
3. Now at very low pressure the fugacity equals the
pressure and therefore the fugacity or ln  does not
change with anything at fixed pressure. Therefore,
− 2 +

  )
(  ln(
) = 
− 1 

R

−
ln(  − 0) = 0  ( 
R 
0

( 
−
1

) 