24.9: Gibbs Energy of Mixing of Binary Solutions in Terms of the Activity Coefficient
Activity and activity coefficients
In the ideal case we have seen that the thermodynamic potential can be written as:
\[μ_2^{sln}=μ^*_2 + RT \ln \,x_2=μ^*_2 + RT \ln \left(\dfrac{P_2}{P^*_2}\right)\]
One approach to nonideality is to simply redefine the problem away and say:
\[μ_2^{sln} \equiv μ^*_2 + RT \ln\, a_2\]
≡ indicates this is actually a definition. The newly defined variable a_{2} is known as the activity. Alternatively we can define it as:
\[[a_2] \equiv \dfrac{P_2}{P^*_2}\]
As at high enough values of the mole fraction we know that we can still apply Raoult law. So \(a_2\) must approach \(x_2\) in this limit, but for other concentrations this will no longer hold.
Often this is expressed in terms of an activity coefficient γ:
\[[a_2] = \gamma_2 x_2\]
For high values of x_{2}, γ_{2} will approach unity.
If we model the nonideality with a Margules function we see that:
 P_{2}=x_{2}P*_{2}.f_{Mar}

 [a_{2}] = [P_{2}/P*_{2}]=[x_{2}P*_{2}.f_{Mar}/P*_{2}]=[x_{2}.f_{Mar}]
Obviously the activity coefficient and the Margules function are the same thing in this descripion.
Regular solutions
A special and simplest case of a Margules function is the case where all but one Margules parameters (α) can be negelected. Such a system is called a regular solution
In this case we can write
 a_{1}= x_{1}. exp[αx_{2}^{2}]
We can use GibbsDuhem to show that this implies (See example 248):
 a_{2}= x_{2}. exp[αx_{1}^{2}]
Gibbs free energy of regular solutions
Consider the change in Gibbs free energy when we mix two components to form a regular solution:
Δ_{mix}G = n_{1}μ_{1}^{sln}+n_{2}μ_{2}^{sln}[n_{1}μ_{1}*+n_{2}μ_{2}*]
Using:
 μ_{j}^{sln} ≡ μ*_{j} + RT ln [a_{j}]
and:
 [a_{j}] = γ_{j}.x_{j}
we get:
 Δ_{mix}G/RT = n_{1}lnx_{1}+n_{2}lnx_{2}+n_{1}lnγ_{1}+n_{2}lnγ_{2}
If we divide by the total number of moles we get
 Δ_{mix, molar}G/RT = x_{1}lnx_{1}+x_{2}lnx_{2}+x_{1}lnγ_{1}+x_{2}lnγ_{2}
For a regular solution:
 lnγ_{1}= lnf_{Mar} = αx_{2}^{2}
 lnγ_{2}= lnf_{Mar} = αx_{1}^{2}
This gives:
 Δ_{mix, molar}G/RT = x_{1}lnx_{1}+x_{2}lnx_{2}+x_{1}αx_{2}^{2}+x_{2}αx_{1}^{2}
 Δ_{mix, molar}G/RT = x_{1}lnx_{1}+x_{2}lnx_{2}+α[x_{1}+x_{2}]x_{1}.x_{2}
 Δ_{mix, molar}G/RT = x_{1}lnx_{1}+x_{2}lnx_{2}+α.x_{1}.x_{2}
 ([x_{1}+x_{2}] = 1 by definition)
In this expression we see that we have an additional term to the entropy of mixing term we had seen before. Its coefficient α is dimensionless but represents the fact that the (strong!) interactions between the molecules are different depending on who is the neighbor. In general α can be written as W/RT where W represents an energy (actually enthalpy) that brings the difference in interaction energies into account. W does not depend strongly on temperature. We could look at W as the difference in average interaction energies:
 W= 2U_{12}U_{11}U_{22}
Rearranging we get:
 Δ_{mix, molar}G/W = RT/W*[x_{1}lnx_{1}+x_{2}lnx_{2}]+ x_{1}.x_{2}
As you see the two terms will compete as a function of temperature. The mixing entropy will be more important at high temperatures, the interaction enthalpy at low temperatures. The entropy term has a minimum at x_{1}=0.5, the enthalpy term a maximum if W is positive. So, one tends to favor mixing, the other segregation and we will get a compromise between the two.
Depending on the value of RT/W (read: temperature) we can either get one or two minima. This means that at low temperatures there will be a solubility limit of 1 into 2 and vice versa. At higher temperatures the two components can mix completely. At the transition between these two regimes we will have critical or consolute point
Notice that even though we used the vapor pressures of the gas to develop our theory, they are conspicuously absent from the final result. The same thing we said about melting points hold true here. Because we are dealing with the miscibility behavior of two condensed phases, the outcome should not depend very strongly on the total pressure of our experiment.
Although in regular solutions the consolute point is predicted to be a maximum in temperature, we can find them as minima as well in practice. The nicotinewater system even has two consolute points, an upper and a lower one. When heating up a mixture of these we first observe mixing, then segregation and then mixing again. Obviously this behavior is far more complicated than we can describe with just one Margules parameter.
Partial molar volumes
What we said above about volumes simply being additive in the ideal case is no longer true here.

 ∂ΔG_{mix}^{regular}/∂P _{T} = ΔV_{regular}^{ideal}

 ∂ΔH_{mix}+RT(n_{1}ln[x_{1}]+n_{2}ln[x_{2}])/∂P _{T} = ΔV_{mix}^{ideal}

 ∂ΔH_{mix}/∂P _{T} = ΔV_{mix}^{ideal}
In general the enthalpy of mixing does depend on pressure as it is related to the interactions between the molecules in solution. (W depends on the distance between them). This means that partial molar volumes now become a function of composition and volume is no longer simply aditive
Real solutions
Notice that the curves are symmetrical around x=0.5. This implies that it is as easy (or not) dissolving A into B as vice versa. In many cases this is not realistic. Many systems diverge more seriously from ideal behavior that the regular one. Up to a point we can model that by adding more terms to the Margules function. For example, adding a βterm undoes the symmetry (see example 247). However, many systems are so nonideal that the Margules expression become unwieldy with too many parameters.
Boiling nonideal solutions
Azeotropes
For ideal solutions we have seen that there is a lense shaped twophase region between the gas and the liquid phase. For nonideal systems the twophase region can attain different shapes. In many cases there is either a minimum or a maximum. As such a point the phase gap closes to a point that is known as an azeotrope. It represents a composition of the liquid that boils congruently. That means that the vapor and the liquid have the same composition for a change. Azeotropes impose an important limitation unto distillation: they represent the end point of a distillation beyond which we can not purify by this method.
Eutectics
Another point to be made is that in the diagram with the consolute point we are assuming the pressure to be constant. If we lower the pressure this would affect the boiling points strongly: the whole gasliquid gap would come down in temperature (see animation). The mixing behavior is only weakly affected. (The reason is that one involves the volume term of the gas, the other only of the liquid(s)). At lower pressures it is possible therefore that the consolute point is above the gasliquid gap. In other words: the mixtures will boil before they get a chance to mix. The boiling points will be lower there than for the pure compounds. There will be a composition for which the boiling point is at a minimum and where the mixture boils congruently (i.e. to a vapor with the same (overall) composition).
Notice that the mutual solubility limits increase as temperature increases, just as happens in the critical mixing case, but that due to the competition from the vapor phase this process comes to an end at the eutectic temperature.
At this temperature one liquid boils aways completely, the other one in part. At the eutectic composition they both boil away simultaneously.