Chapter 4 Phase Diagrams
4.4 Graphical Representation, Ideal and Regular Solutions
By repeating the procedure for each of the defined thermodynamic variables, we have implies that the partial molar Gibbs free energy and the chemical potential, one of the most important properties in thermodynamics and chemistry, are the same in quantity.
In real systems, it is usually difficult to hold the entropy fixed, since this involves good thermal insulation. It is therefore more convenient to use μ as the partial derivative of F or G with respect to ni with constants T, V , and T, P , respectively. Under isobaric and isothermal conditions, knowledge of μi yields every property of mixture as it completely determines Gi.
μi as a fundamental parameter is conjugate to the composition, being used extensively in the treatment of the thermodynamics of solutions and of chemical reactions. In modern statistical physics, μi is the Lagrange multi-plier (see Eq. (2.18)) for the average particle constraint, when maximizing S. This is a precise and abstract scientific definition, just like T is defined as the Lagrange multiplier for the average energy constraint. μ at T = 0 K of a system of electrons is also called the Fermi level.
4.4 Graphical Representation, Ideal and Regular Solu-tions [1]
If we know a property of a two-component solution as a function of ni, we can determine the partial molar values of that property. Taking Vm as an
130 Chapter 4 Phase Diagrams
example, we have Vm= VAxA+ VBxB. From this equation, the result would look something like the solid line in Fig. 4.4. Because xA+ xB = 1, dxA+ dxB
= 0, and dxA/dxB=−1, Eq. (4.20) can also be written as (∂Vm/∂xB)T,P = VB−VA. Multiplying this equation by xAand xB, respectively, two equations are present. Subtracting them from Vm, it reads
VA= Vm− xB
∂Vm
∂xB
T,P
, (4.22a)
VB= Vm+ xA
∂Vm
∂xB
T,P
. (4.22b)
This is a kind of abstract at this point and it may be difficult to realize what this is telling us. Equation (4.22a) says that VA can be thought of as the intersection of the tangent of Vm as a function of xB with Vm axis at xB= 0. VBon the other hand is the intersection of that tangent with Vmaxis at xB = 1. Figure 4.4 shows this graphically.
Fig. 4.4 Plot of Vmversus xBto show geometrical relationships to an A-B solution.
Substituting G for V in Eqs. (4.22a) and (4.22b) and combining the rela-tionship of GA= μA, the chemical potentials of each species in solution as a function of composition are got,
μA= Gm− xB
∂Gm
∂xB
T,P
, (4.23a)
μB = Gm+ xA
∂Gm
∂xB
T,P
. (4.23b)
The functions that are of special importance for expressing μ in detail and to construct a phase diagram for a multicomponent material are G, H, S. Thus, we will concentrate on them. There are various standard models for solutions that provide approximations to the above functions. Let us consider firstly the thermodynamics of ideal solution.
4.4 Graphical Representation, Ideal and Regular Solutions 131
Ideality of solutions is analogous to that of gases, with the important difference that intermolecular interactions in condensed phases are strong and can not simply be neglected as they can in an ideal gas. What is meant by
“ideal” in these cases is that the interactions between the constituents of the solution are the same, regardless of their nature. More formally, for a mix of molecules of A and B, the interaction energy between unlike neighbors (εAB) is the same as that between neighbors εAAand εBB, i.e. εAB = εAA= εBB. It follows that there is no enthalpy change (enthalpy of mixing ΔmixH = 0) when the substances are mixed, then the solution is automatically ideal.
The more dissimilar the nature of A and B, the more strongly the solution is expected to deviate from ideality.
Why do the components of the ideal solution mix with each other? The reason is the existence of the entropy of mixing of an ideal solid solution, which consists of atoms labeled A and B. Using Boltzmann equation written for each of the components and the mixture, all the configurations are at the same energy level, all are equally probable, and then the positioning of atoms on a lattice is random where ΔmixH = 0. For one mole of atoms (the sum of NA and NB is Avogadro’s number of atoms, Nm), the way number of microstates is Nm! to introduce the atoms onto the lattice. Not all these different configurations are distinguishable. To account for this, we must di-vide by the number of different ways that the NA atoms can be distributed on their sites. Using an argument similar to the one above, rearranging the atoms labeled A on the lattice can be done in NA! ways. A similar relation-ship applies for NB. The entropy of this configuration, the entropy of mixing NAand NB, is ΔmixSm= klnΩ = kln[Nm!/(NA!NB!)] = kln[Nm!-NA!-NB!].
Using Stirling approximation,
ΔmixSm= k[Nmln Nm− Nm− NAln NA+ NA− NBln NB+ NB]
= k[(NA+ NB) ln Nm− NAln NA− NBln NB]
=−k[NAln(NA/Nm) + NBln(NB/Nm)]
=−kNm[(NA/Nm) ln(NA/Nm) + (NB/Nm) ln(NB/Nm)]
=−R[xAln xA+ xBln xB].
The above expression can be generalized to a mixture of i components, starting withΩ = Nm!/(
iNi!), namely, ΔmixSm=−R
i
xiln xi. (4.24)
Since ΔmixHm= 0, there is ΔmixGm= ΔmixHm−T ΔmixSm= RT
i
xiln xi, or for a two-component solution, ΔmixGm= RT (xAln xA+ xBln xB). Since the mole fraction is always smaller than unity, the ln terms are negative, and ΔmixGm< 0. This becomes more negative as T increases. As a result, ideal solutions are always completely miscible.
132 Chapter 4 Phase Diagrams
Gibbs free energy of the mixture equals the sum of Gibbs free energy of individual components and Gibbs free energy of mixing. This yields a fundamental expression for μ as a function of xi. Now by mixing A and B to form an ideal solution, it reads Gm = Ginitial+ ΔmixGm = xAGm,A+ meaning. The absolute value of μi depends on the location of zero of the potential energy scale, which is also called the standard state. For an ideal solution system, it is convenient to choose the pure component as its standard state. Note that the standard state is defined for each T .
Figure 4.5 shows the graphs of Gm as a function of xB, with the two points representing Gm,Aand Gm,Bof pure A and B taken at different levels.
Using the partial molar operator, we can convert general thermodynamic relationships to the coefficient one,
and the change in the partial molar volume is ΔmixVi= Vi− Vm,i=
Now, if the mixture is ideal, we have ΔmixVi =
For an ideal solution, the change of the partial molar volume is absent as we form the mixture. In other words, the partial molar volume is independent of composition. If that is the case, the volume of the mixture is just the weighted average of the molar volumes of the pure substances. Thus, for an
4.4 Graphical Representation, Ideal and Regular Solutions 133
Fig. 4.5 Plot of Gm versus xBto show their geometrical relationships.
ideal mixture, there is no change in V , U , or H from the total values of the pure states of all the components.
Better model of the Gibbs free energy of solution comes from consideration of the interactions between atoms in the system. The simplest version is called the regular solution model–it averages the interactions between like and unlike atoms to calculate ΔmixH and uses the ideal solution to calculate ΔmixS. The regular solution model can predict “unmixing” and this results in a spinodal phase diagram.
For a solid solution consisting of randomly distributed A and B on the lattice sites, ΔmixS of A and B is equal to the entropy of mixing of an ideal solution. Each atom will be assumed to have an energy interaction only with its nearest neighbors numbered by CN or z. We will mix NA
atoms of A with NB atoms of B. Before mixing, the number of NAAbonds among the A atoms (in pure A) is NAz/2. We arrive at this conclusion by counting the number of bonds emanating from each A atom (z), multiplying by NA, then dividing by 2 because the bonds are doubly counted. After mixing, the number of AA bonds will be (NA)2z/2N with N = NA+ NB. To count the number of AA bonds we multiply NA by two factors: z and the probability that its nearest neighbor is an A atom, NA/N . The same procedure is followed for calculating the number of BB bonds and AB bonds.
Thus, ΔmixU = z[(NA)2zεAA/(2NT)+(NB)2zεBB/(2NT)+NANBzεAB/NT− NAεAA/2− NBεBB/2]. The resulting energy after mixing is listed in Table 4.2.
Table 4.2 Bond energy after mixing atoms A and B in a solid solution Number of bonds Energy per bond Energy
NAB NANBz/N εAB NANBzεAB/N
NAA (NA)2z/(2N ) εAA (NA)2zεAA/(2N ) NBB (NB)2z/(2N ) εBB (NB)2zεBB/(2N ) Since xA = NA/N , the above equation can be rewritten as ΔmixU =
134 Chapter 4 Phase Diagrams
zN [(x2A− xA)εAA+ (x2B− xB)εBB]/2 + xAxBεAB= zN xAxB[εAB− (εAA+ εBB)/2], or
ΔmixU = ωxAxB (4.26)
where ω = zN [εAB− (εAA+ εBB)/2] is called the interaction parameter.
All ε values are negative, decreasing in absolute value with reducing r or decreasing N [14 – 16]. If decreasing rates of different ε values are distinct, ω is size-dependent. According to general quantum chemistry consideration, all thermodynamic quantities are roughly a linear function of 1/r that cor-responds to the surface/volume ratio of particles. ω is assumed to have the same relationship. Because ΔHm(2r0) = 0 in terms of Eq. (3.88), ω(r) should have the same limit and thus is assumed to have the following form:
ω(r)/ω(∞) = 1 − 2r0/r. (4.27) Equation (4.26) for the ΔmixU is equal to ΔmixH for solids because the volume change influencing the PV term in the definition of enthalpy is very small. Therefore,
ΔmixGm= ΔmixHm−T ΔmixSm= RT (xAln xA+xBln xB)+ωxAxB. (4.28) From Eq. (4.26) it can be seen that the solution will behave as ideal (ΔmixH
= 0) when |εAB| = (|εAA| + |εBB|)/2. If |εAB| > (|εAA| + |εBB|)/2, ω < 0 and ΔmixH < 0. (Remember that ε < 0). A negative ΔmixH favors stable homogeneous solutions and a positive ΔmixH makes a system separate and has a miscibility gap at lower T .
To make adjustment for the non-ideality of the solution, a new property called the activity is used instead of x in the fundamental equation for μ. If materials A and B form a continuous set of solutions, but are non-ideal, the activity of material B is usually expressed as aB= fBxB. Consequently,
μB= μ∗B+ RT ln aB= μ∗B+ RT ln xB+ RT ln fB (4.29) where f is called the activity coefficient. If the solution itself is ideal, aB= xB
and fB =1 in light of Eqs. (4.25) and (4.29). Thus, departure of f from unity indicates a non-ideal behavior of the regular solution from the ideal solution, and f should account for the changes in the enthalpy during mixing.
Substituting Eq. (4.28) into Eqs. (4.23a) and (4.23b), μ of the individual components in a regular solution is shown to be
μA= μ∗A+ RT ln xA+ ωx2B, (4.30a) μB= μ∗B+ RT ln xB+ ωx2A. (4.30b) Comparing Eq. (4.29) and Eq. (4.30), it reads
fA= expωx2B
RT , fB = expωx2A
RT . (4.31)