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```					5. Non-Ideality in 1-component Systems
Pure, Non-ideal Gases
The ideal gas assumption:
PV = RT
where V = molar volume holds
only for low pressures, where
molecular interactions are
negligible and molecular volume
need not be considered.

At higher pressures, we have used
the compressibility factor, Z, to
characterize gas behaviour.

Z = PV / RT
= 1 for ideal gases

CHEE 311                       Lecture 13   1
Gibbs Energy of Pure Gases
For any pure gas, ideal or non-ideal, the fundamental equation
applies:
dG = VdP - SdT

At constant T, changes in the Gibbs energy of a pure gas arise only
from changes in pressure, and:
dG = VdP                          (constant T)

We can integrate between two pressures, Pref and P to obtain:
P
G( T,P)  G(T,Pref )   VdP
Pref

For an ideal gas, we can substitute for the molar volume, V=RT/P
P RT
G (T,P)  G (T,Pref )  
ig        ig
dP
Pref P

 P 
P 
 RT ln     
CHEE 311                      Lecture 13
 ref    2
Gibbs Energy of Pure, Ideal Gases
For the ideal gas case, we have

Gig (T,P)  Gig (T,Pref )  RT ln P
 P 

 ref 
If we consistently select unit pressure (1 bar, 1 psi, etc) as our
reference state, we can simplify the expression:
11.28
G (T,P)  i (T )  RT ln P
ig

where i(T) is only a function of temperature.

This expression provides the Gibbs energy per mole of a pure, ideal
gas at a given P and T
 We would like to develop an analogous expression for non-
ideal systems, for which V RT/P
 Like all non-ideal systems, we can’t predict how V,T and P
relate, but we can perform experiments and correlate our data
CHEE 311                        Lecture 13                     3
Gibbs Energy for Pure, Non-ideal Gases
The utility of Equation 11.28 leads us to define a direct analogue

11.31
where              G(T,P)  i (T)  RT ln fi
i(T) the same function of temperature
fi is a defined intensive variable called the fugacity (units of
pressure)
Fugacity is used to describe the Gibbs energy of non-ideal gases.
In these cases, Gibbs energy does not vary with ln(P), so we define
a new “chemical pressure” such that the Gibbs energy varies
directly with ln fi.
Equation 11.31 is the first part of the definition of fugacity. The
second part specifies that as the pressure approaches zero (and
the pure gas becomes more ideal) the fugacity approaches the
pressure.
As P  0 : fi  P
CHEE 311                        Lecture 13                      4
Pure Gases: Fugacity and Fugacity Coefficient
In summary, the fugacity of a pure, non-ideal gas is defined as:

G(T,P)  i (T)  RT ln fi
with the specification that:
As P  0 : fi  P
Together, these definitions allow us to quantify the Gibbs energy of
non-ideal gases.

A closely related parameter is the fugacity coefficient, defined by:
fi
fi 
P
such that
G(T,P)  i (T)  RT ln fi P
Note that a gas behaving ideally is defined as having fi = 1, in
which case the expression reduces to equation 11.28.

CHEE 311                           Lecture 13                 5
Calculating the Fugacity of a Pure Gas
The simplest means of calculating the fugacity of a pure gas is to
compare its behaviour to an ideal system. We will do this
frequently in our treatment of non-ideality.

For the non-ideal gas:
P
G(T,P)  G(T,Pref )   VdP  RT ln( fi / fi,ref )
Pref

For the ideal gas:
P
G (T,P)  G (T,Pref )   V igdP  RT ln(P / Pref )
ig            ig

Pref

Taking the difference of these equations:
P                       fi fi,ref 
 ( V  V )dP  RT ln 
ig

Pref                   P Pref    

CHEE 311                          Lecture 13                    6
Calculating the Fugacity of a Pure Gas
We can simplify this relation by an appropriate choice of Pref. As
pressure goes to zero, a real gas approaches ideality. Therefore,
As       Pref  0 :        fi,ref / Pref  1
With Pref = 0, we have:
P
 ( V  V )dP  RT ln ( fi / P)
ig

0
or
1 P
ln ( fi / P)      ( V  V )dP
ig
RT 0
Substituting V = ZRT/P and Vig = RT/P, we arrive at:

P ( Z  1)
ln ( fi / P)             dP              11.35
0    P

CHEE 311                            Lecture 13                      7
Calculating the Fugacity of a Pure Gas
Equation 11.35 is commonly written in terms of the fugacity
coefficient:
P ( Z  1)
at a given T.
ln fi             dP
0   P

To calculate the fugacity of a pure, non-ideal gas, all we need is
information on the relationship of Z as a function of P at T.
 Experimental data
 Equations of State
 Generalized correlations

CHEE 311                         Lecture 13                   8

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