Fugacity

Fugacity of pure substances
It has been shown in section 6.3 that the chemical potential provides a fundamental description of phase equilibria. As we shall further see in chapter 8, it also proves an effective tool for depicting chemical reaction equilibria. Nevertheless, its direct usage is restricted, as it is not easy to directly relate the chemical potential to thermodynamic properties amenable to easy experimental determination, such as the volumetric properties. The definition of a new function called fugacity, itself related to the chemical potential, helps bridge the gap.

The concept of fugacity is advanced based on the following thermodynamic relation for an ideal gas. For a single component closed system containing an ideal gas we have (from eqn. 5.7):                                                                                                                   
At constant temperature, for a pure ideal gas ‘i’ the above equation reduces to: 
(6.85)
             [Where, Gi(T) is the constant of integration]

Utilizing the essential simplicity of eqn. 6.85 we apply it a real fluid but by replacing pressure with fugacity, since it is not valid for a real fluid:
 (At const T)
Thus, ;  
Hence      
Since fi has the units of pressure, it is often described as a “fictitious pressure”. It may be noted that the definition of fugacity as provided by eqn. 6.86 is completely general in nature, and so can be extended to liquids and solids as well. However, the calculation of fugacity for the latter will differ from that for gases. This equation provides a partial definition of fi, the fugacity of pure species ‘i’. Subtracting eqn. 6.85 form 6.86 gives: 
(6.87)
The dimensionless ratio  is termed fugacity coefficient  
Thus:(6.88) 
Where,  (6.89)
Clearly for an ideal gas the following relations hold: 
However, by eqn. 5.38: 
Thus, using the last relation in eqn. 6.78:
          (At const. T)
Fugacity-based phase equilibrium criterion for pure component system
The general criterion of thermodynamic equilibrium has been defined by eqn. 6.38. Applying it to, for example, a vapour (V) and liquid (L) system of a pure component ‘i’ we have:
(6.91)
However, for a pure component system:    
Thus:  and   (6.92)
Thus, using eqn. 6.91 and 6.92 we have:
(6.93)
The above equation may be generalized for any other types of phases. However, the eqn. 6.93 is rendered more easily applicable if the chemical potential is replaced by fugacity. Thus integrating eqn. 6.86 between vapour and liquid states of a pure component: 
(6.94)
     
Now applying eqn. 6.93 to 6.95 it follows  
Or:
(6.96)
In eqn. 6.86 indicates the value for either saturated liquid or saturated vapor, this is because the coexisting phases of saturated liquid and saturated vapor are in equilibrium. Since under such condition the pressure is  we can write:


Thus, employing eqn. 6.86 again, it follows:
(6.97)
Both eqns. 6.93, 6.96 and 6.97 represent equivalent criterion of vapor/liquid equilibrium for pure species.
Equation 6.103 is a generalized expression for obtaining pure component fugacity from a pressure explicit EOS. We show below the expressions for fugacity coefficients that derive on application of the above equation to various cubic EOSs. (The reader may refer to section 2.3.3 for various forms of cubic EOS).

VdW EOS:
(6.104)
RK-EOS:
(6.105)
SRK EOS:
(6.106)
PR-EOS:

Fugacity expressions for pure gases
Fugacity coefficient (and hence fugacity) of pure gases may be conveniently evaluated by applying eqn. 6.80 to a volume-explicit equation of state. The truncated virial EOS is an example of the latter type, for which the compressibility factor of pure species (i) is given by:

Or 
Thus, on using eqn. 6.80:  (at const T)
Hence,  
Eqn. 6.80 is, however, not amenable to use for obtaining expressions using cubic EOSs. The general equation for such purposes is relatively more invloved and we derive it below.
Example 6.4
Estimate the fugacity of ethane at 122.2 K and 5 bar using the truncated virial EOS. For ethane Tc = 305.4K, Pc = 48.84 bar, ω = 0.099
(Click for solution)
Derivation of fugacity coefficient expression for cubic EOS:
Starting from eqn. 5.38
(6.99)
(6.100)
Using Eqn. 6.100 in 6.99:
On simplifying:
(6.101)
(6.102)
Using eqn. 6.102 in 6.101 one arrives at:

Equation 6.103 is a generalized expression for obtaining pure component fugacity from a pressure explicit EOS. We show below the expressions for fugacity coefficients that derive on application of the above equation to various cubic EOSs. (The reader may refer to section 2.3.3 for various forms of cubic EOS).

VdW EOS:
(6.104)
RK-EOS:
(6.105)
SRK EOS:
(6.106)
PR-EOS:

The above set of equations may well be used for estimating the overall fugacity coefficient of a gaseous mixture. As in earlier instance (sections, 2.3 and 5.7) one makes use of pseudo-critical properties that are estimated from the following set of linear relations: 
;(2.32)
The subscript ‘i’ runs over all the species present in the mixture ‘m’. The above relations allow the mixture to be treated as single substance; the methodology outlined above for pure species may then be used to compute the fugacity coefficient of the mixture.

(6.107)
(6.103)
(6.98)
(6.107)
(6.95)
(6.90)
(6.86)