Partial Molar Property

In all the preceding chapters we have focused primarily on thermodynamic systems comprising pure substances. However, in all of nature, mixtures are ubiquitous. In chemical process plants – the ultimate domain of application of the principles of chemical engineering thermodynamics – matter is dominantly processed in the form of mixtures. Process streams are typically comprised of multiple components, very often distributed over multiple phases. Separation or mixing processes necessitate the use of multiple phases in order to preferentially concentrate the desired materials in one of the phases. Reactors very often bring together various reactants that exist in different phases. It follows that during mixing, separation, inter-phase transfer, and reaction processes occurring in chemical plants multi-component gases or liquids undergo composition changes. Thus, in the thermodynamic description of such systems, in addition to pressure and temperature, composition plays a key role. 

Further, whenever multiple phases are present in a system, material and energy transfer occurs between till the phases are in equilibrium with each other, i.e., the system tends to a state wherein the all thermal, mechanical and chemical potential (introduced earlier in section 1.4) gradients within and acrossall phases cease to exist. 

The present chapter constitutes a systematic development of the concept of a new class of properties essential to description of real mixtures, as well of the idea of the chemical potential necessary for deriving the criterion of phase and chemical reaction equilibrium. Such properties facilitate the application of the first and second law principles to quantitatively describe changes of internal, energy, enthalpy and entropy of multi-component and multiphase systems.

Of the separate class of properties relevant to multi-component and multi-phase systems, the partial molar property and the chemical potential are particularly important. The former is used for describing behaviour of homogeneous multi-component systems, while the latter forms the fundament to description equilibrium in multi-phase, as well as reactive systems. 

As in the case of pure gases, the ideal gas mixture acts as a datum for estimating the properties of real gas mixtures. The comparison of the properties of the real and ideal gas mixtures leads to the introduction of the concept of fugacity, a property that is further related to the chemical potential. Fugacity may also be expressed as a function of volumetric properties of fluids. As we will see, the functional equivalence of fugacity and the chemical potential provides a convenient pathway for relating the temperature, pressure and phase composition of a system under equilibrium.   

In the last chapter it was demonstrated that residual properties provide very suitable means of estimating real gas properties. But as pointed out its usage for description of liquid states is not convenient. This difficulty is overcome by the formulation of a concept of ideal solution behaviour, which serves as a datum for estimating properties of real liquid solutions. The departure of the property of a real solution from that of an ideal one is termed as excess property. In other words, the excess property plays a role similar to that of residual property. In the description of solution behaviour at low to moderate pressures, we employ yet another property, the activity coefficient; which originates from the concept of fugacity. The activity coefficient may also be related to the excess Gibbs energy. It is useful not only as a measure of the extent of non-ideality of a real solution but also, more significantly in describing phase equilibria at low to moderate pressures.   

6.1 Partial Molar Property
We consider first the case of a homogenous (single-phase), open system that can interchange matter with its surroundings and hence undergo a change of composition. Therefore, the total value of any extensive property   is not only a function of T and P, but also of the actual number of moles of each species present in the system. Thus, we may write the following general property relation: 

Taking the total derivative for both sides of eqn. 6.1: 
Where, subscript n indicates that all mole numbers are held constant and subscript  that all mole numbers except ni are held constant. This equation has the simpler form:
(the subscript x denotes differential at constant composition
Eqn. 6.4 defines the partial molar property  of species i in solution. It represents the change of total property ‘nM’ of a mixture resulting from addition at constant T and P of a differential amount of species ‘i’ to a finite amount of solution. In other words it also signifies the value of the property per mole of the specific species when it exists in solution. In general, the partial molar property of a substance differs from the molar property of the same substance in a pure state at the same temperature and pressure as the mixture or solution.  This owing to the fact that while in a pure state the molecules interact with its own species, in a solution it may be subjected to different interaction potential with dissimilar molecules. This may render the value of a molar property different in mixed and pure states.  
Also:    (6.6)
Substituting eqns. 6.5 and 6.6 in eqn. 6.3 leads to: 
On re-arranging:

While deriving eqn. 6.8 no specific constraints on the values of either  have been applied. This suggests that the equation is valid for any arbitrary values of these two variables. Thus n and dn are independent of each other. Therefore, eqn. 6.8 can only be valid if the coefficients of these two variables are identically zero. On putting the coefficients to zero the following equations obtain:
From eqn. 6.10, it follows, that     (6.11)
And also:      (6.12)
Equating (6.19) and (6.21) yields the well-known Gibbs/Duhem equation (GDE):

The GDE must be satisfied for all changes in P, T, and in  caused by changes of state in a homogeneous phase. For the important special case of changes at constant T and P, it simplifies to:

Or, taking any arbitrary species ‘j’:       (at const T, P)    (6.15)
Equations (6.14 and 6.15) implies that the partial molar properties of the various species () are not independent. Some key properties of partial molar properties are defined as follows: 

Select additional relations among partial properties are demonstrated in the Appendix 6.1.
Based on the foregoing considerations one may define an isothermal molar property change of mixing as follows:
Or:     (6.17)
(M can be = V, U, H, S, A, G)
Typical examples of molar volume change of mixing for a number of binary solutions are shown in figs. 6.1 and 6.2. Clearly then there can be substantial variation of this property depending upon the nature of the constituent molecules.