THE PHASE RULE
9.4 PARTIAL MOLAR VOLUME
Prior to Gibbs, thermodynamics was largely about the transfer of heat in the process of driving an engine. It was an age justly called the age of steam. Gibbs’s departure was to focus on the transfer of matter. That is why Gibbs’s work is so important to chemists. Our science is largely devoted to the transfer of matter from a reactant state to a product state. In classical physical chemistry, we ask whether the transfer occurs (thermodynamics) and, if it does, we want to know how long it takes (kinetics).
To answer the first of these questions, Gibbs used partial molar thermodynamic state functions. To start out, we shall consider the volume of a system and we shall restrict mixtures to two components. The volume of a system is easy to visualize, and restricting the system to two components simplifies the arithmetic. Later we shall release these restrictions.
The volume of an ideal two-component liquid solution at constant p and T depends on how much of each component is present:
V = f (n1, n2)
If the molar volume of pure component 1 is Vm1◦ and the molar volume of pure component 2 is Vm2◦ , then an ideal solution of equal molar amounts of 1 and 2 will have a molar volume that is the average of the two
Vm1◦ +Vm2◦ /2 (Fig. 9.4). Solutions of other ratios of components 1 and 2 would be arrayed along a straight line connecting Vm1◦ to Vm2◦ , provided that no shrinkage or swelling takes place when 1 and 2 are mixed.
In general, for different ratios of the components, we obtain V =Vm1◦ n1+Vm2◦ n2
0 1 X2
Vm1
Vm2
V
FIGURE 9.4 Total volume of an ideal binary solution. X2is the mole fraction of compo- nent 2.
In reality, life is not that simple. The total volume of the solution will not usually be the sum of the molar volumes weighted by their relative amounts. The volume actually occupied by the solute in the solution is called the partial molar volume, which can be greater than or smaller than its molar volume in the pure state. The volume of the solution will then be greater or smaller than the sum of its parts.
Addition of some potassium salts to one mole of pure water results in a volume of solution that is even smaller than the initial volume of water (shrinkage occurs).
These three possibilities are shown in Fig. 9.5.
9.4.1 Generalization
We have been completely arbitrary in designating component 2 with amount n2as the solute and n1as the number of moles of solvent, so we can switch designations just as arbitrarily. Thus everything we have said about component 2 in a binary solution also applies to component 1 treated as though it is the solute in solution with component 2.
In a solution of completely miscible liquids, it is conventional to take the lesser
V
n2
Von2
FIGURE 9.5 Volume increase (or decrease) upon adding small amounts of solute n2to pure solvent. Three cases are shown for Vn2>Vn2◦, Vn2<Vn2◦, and Vn2<0.
PARTIAL MOLAR VOLUME 131
V
n2
FIGURE 9.6 Partial molar volume as the slope of V vs. n2. The lower line gives Vm2 as n2→0, and the upper tangent line gives Vm2at a specific concentration n2=0.
component as the solute and the greater component as the solvent. In many cases, the choice is obvious, for example, in KCl solutions in water, KCl is clearly the solute.
Evidently, from Fig. 9.6, the partial molar volume of one component, which we have chosen to call the solute, is the slope of one of the solid lines found by measuring the volume increment upon adding small amounts of component 2 to large amounts of component 1. In real solutions, these lines need not be straight; the partial molar volume of the solute is found in the limit as dn2→0.
Recognizing Vm2as the slope of the function of V vs. n2means that we have the definition
Vm2= ∂V
∂n2
T,p,n1
This slope can be found anywhere on the entire curve of experimentally measured total volume vs. X2, where Xn2 =n2/(n1+n2). Now Vm2 is defined as the volume change found upon adding an infinitesimal amount of component 2 to a solution of composition X2 specified by a horizontal distance along the X2 axis; for example, X2=0.20 in Fig. 9.7.
0 1
X2 Vm1
Vm2
V
0.20
FIGURE 9.7 Volume behavior of a nonideal binary solution. X2 is the mole fraction of component 2.
The definition of Vm2should come as no surprise. It comes from the condition on perfect differentials. In the case of the volume, we have
d V = ∂V
∂p
T,ni,nj
d p+ ∂V
∂T
p,ni,nj
d T +
ni
∂V
∂ni
T,p,nj
dni
This generalization involving the sum
ni
∂V
∂ni
T,p,nj
dni
gets us away from the restriction that the solution be binary, a condition we imposed at the beginning for simplicity. Now the concepts developed can be applied to any number of components. Recalling that volume is a thermodynamic property, we have
V = f ( p,T,n1,n2, . . . ,nj)
In view of the first two terms in the sum, the functions are sensitive to variations in both pressure and temperature hence one or both may be held constant in the phase diagrams discussed below. The variation of total volume with composition in curves like Fig. 9.7 gives rise to the term excess volume as the volume above the straight line expected from a simple sum of molar volumes in the pure state Vm◦. The excess volume can be negative leading to a nonideal curve below the straight line in Fig. 9.7.
The volume in a real binary system corresponds to the sum in which V =Vm1◦ n1+Vm2◦ n2
is replaced by
V =V¯m1n1+V¯m2n2
where the molar volumes ¯Vm1 and ¯Vm2are no longer volumes in the pure phase but are partial molar volumes, unique to the ratio of n1and n2in the solution. In general,
V¯mi = ∂V
∂ni
T,p,nj
Each partial molar volume must be determined experimentally. There are, of course, simplified equations containing empirical constants that work more or less well for real (nonideal) solutions just as there are for nonideal gases.
The greatest generalization in this field, however, is to recognize that nothing in the arguments made is specific to volume alone. The general rule is that partial molar
PARTIAL MOLAR VOLUME 133 quantities analogous to the partial molar volume exist for all thermodynamic state variables. This rule gives us the partial molar energy,
U¯mi = ∂U
∂ni
T,p,nj
the partial molar enthalpy,
H¯mi = ∂H
∂ni
T,p,nj
the partial molar entropy,
¯Smi = ∂S
∂ni
T,p,nj
and the partial molar Gibbs free energy µmi =
∂µ
∂ni
T,p,nj
This last partial molar quantity is so important that it is given a unique name and symbol. It is called the Gibbs chemical potentialµmi. It should be clear that in real (nonideal) systems, each of these functions has a corresponding excess function; the excess energy, the excess enthalpy, the excess entropy, and so on.
In generalizing thermodynamics to many-component systems, Gibbs brought about an immense expansion of the scope of the subject. All of the classical ther- modynamic equations apply to partial molar quantities as well. For example, by analogy to
d G
T
d 1
T =H
for the Gibbs free energy of ideal components (Section 6.6), we have d
µ T d
1 T
=H¯m
relating the partial molar Gibbs free energy and the partial molar enthalpy. ¯Hm.