Fugacity & fugacity coefficient

Introduction
Concepts of Fugacity
Effect of Temperature & pressure on Fugacity
Important relation of Fugacity Coefficient
Vapour Liquid Equilibrium for pure species
Fugacity & Fugacity coefficient: Species in solution
Reference

1. Chemical Engineering Thermodynamics
-II(2150503)
GUJARAT
TECHNOLOGICAL
UNIVERSITY
Fugacity & Fugacity Coefficient

2. Fugacity & Fugacity
Coefficient

3. Contents
1. Introduction
2. Concepts of Fugacity
3. Effect of Temperature & pressure on Fugacity
4. Important relation of Fugacity Coefficient
5. Vapour Liquid Equilibrium for pure species
6. Fugacity & Fugacity coefficient: Species in solution
7. Reference

4. INTRODUCTION
• The concept of Fugacity was introduced by Gilbert
Newton Lewis.
• Fugacity is widely used in solution thermodynamics to
represent the behaviour of real gases.

5.  Fugacity is derived from Latin word ‘fleetness’ or the
‘Escaping Tendency’.

Fugacity has been used extensively in the study of phase
and chemical reaction equlibria involving gases at high
pressures.

6. Concepts of Fugacity
 For an infinitesimal reversible change occurring
in the system under isothermal condition
dG = -SdT + VdP reduces to,
dG = VdP
 For one mole of an ideal gas V in the above
equation may be replaced by RT/P,
dG = RT (dP/P) = RT d(ln P)
 Above equation is applicable only to ideal gas.

7.  For representing the influence of present on
Gibbs free energy of real gases by a similar
relationship, then the true pressure in the equation
should be replaced by an ‘effective’ pressure,
which we call fugacity f of the gas.
 Hence, fugacity has the same dimensions as
pressure.

8. The following equation, thus provides the partial
definition of fugacity. It is satisfied by gases whether ideal
or real.

dG = RT d(ln f)

Integration of above equation gives,
G = RT ln f + C
where C is the constant that depends on temperature and
nature of the gas.

9. Concepts of Chemical potential:-
 The chemical potential μ
i
provides the
fundamental criteria for phase equilibria. This is
true as well for chemical reaction equilibria.
The Gibbs energy, and hence μ
i
, is defined in
relation to the internal energy and entropy.
Because absolute values of internal energy are
unknown and same it is true for μ
i
.

10.  Moreover, -----(1)
shows that approaches negative infinity
when either P or y
i
approches zero.

This is true not just for an ideal gas but for any
gas.
 Although these characteristics do not preclude
the use of chemical potentials, the application of
equilibrium criteria is facilitated by introduction of
the fugacity, a property that takes the place of μ
i

but which does not exhibit its less desirable
characteristics.

11.  The origin of the fugacity concept resides in
equation-1, valid only for pure species i in the
ideal-gas state.
 For a real fluid, an analogous equation that
defines fi, the fugacity of pure species i:
--------- (2)
The fugacity of pure species i as an ideal gas is
necessarily equal to its pressure. Subtraction of
eq. (1) from eq. (2), both written for the same T
and P,

12. • we know that, is the Residual Gibb’s
energy G
i
R
,
where the dimensionless ratio fi/P has been defined as
another new property, the fugacity coefficient, given by
the symbol
ɸ
i :

•Fugacity Coefficient:- The ratio of fugacity to pressure
is referred to as fugacity coefficient and is denoted by
ɸ
i
. It is dimensionless and depends on nature of the gas,
the pressure and the temperature.

13. Effect of Temperature &
Pressure on Fugacity
 By integrating eq. dG = RT d(ln f) between
pressure P and P
0
.
 G
0
and f
0
refer to the molar free energy and
fugacity respectively at a very low pressure where
the gas behaves ideally. This equation can be
rearranged as,

14. • Differentiate this with respect to temperature at constant
pressure.
• Substituting the Gibbs-Helmholtz equation,
into the above result and observing that f
0

is equal to the pressure.
•H is the molar enthalpy of the gas at the given pressure
and H
0
is the enthalpy at a very low pressure. H
0
– H can
be treated as the increase of enthalpy accompanying the
expansion of the gas from pressure P to zero pressure at
constant temperature.
•Above equation indicates the effect of temperature on the
fugacity.

15.  The effect of pressure on fugacity is evident from
the defining equation for fugacity.
dG = V dP = RT d (ln f)
which on rearrangement gives,

16. Important relations of Fugacity
coefficient
• The identification of ln i with G
ɸ
i
R
/ RT by eq.
permits its evaluation by the eq.

 Compressibility factor is given by,

(Const T)
(Const T)

17. Fugacity & Fugacity coefficient:
Species in solution
• The definition of the fugacity of a species in
solution is parallel to the definition of the pure
species fugacity. For a species i in a mixture of
real gases or in a solution of liquids,

-----------(A)
where f
i
^
is the fugacity of species i in
solution, replacing the partial pressure y
i
P.
•This does not make it a partial molar property,
therefore identified by circumflex rather than by
an over bar.

18.  As we know for the Chemical potential, here also
all phases are in equilibrium at the same T,
(i = 1,2,...,N)
 Multiple phases at the same T and P are in
equilibrium when the fugacity of each species is
the same in all phases.
 for the specific case of vapour/liquid equilibrium
above equation becomes:
(i = 1,2,...,N)

19.  The residual property is,
where M is the molar value of a property and M
ig
is
the value that the property would have for an ideal gas of
the same composition at the same T and P.
(For n mole)
 Differentiation with respect to n
i
at constant T, P and n
j
,
In the terms of Partial molar property,
(Partial Residual Gibbs
Energy)

20.  Eq. (A) subtracting from,
 The identity gives,

where by definition:-
 The dimensionless ratio i^ is called the fugacity
ɸ
coefficeint of species i in the solution.