4.3.1 A Simple Phenomenological Model for Liquid-liquid Coexistence
The van der Waals approach is applicable to gas-liquid phase separation in a one- component system. Another type of phase separation is observed in binary mixtures.
Depending on thermodynamic conditions the components may be miscible or not. A simple model describing this is based on the following molar free enthalpy approx- imation
g=x(Al)gA+x(Bl)gB+x(Al)lnx(Al)+x(Bl)lnx(Bl)+χx(Al)x(Bl). (4.77) HeregA =μ∗A(l)/RT andgB =μ∗B(l)/RT are the reduced molar free enthalpies of two pure liquid components AandB. Mixing AandB gives rise to the mixing free enthalpy described by the ln-terms. Note that the mole fractions are x(Al) and x(Bl)=1−x(Al). We had obtained this contribution earlier, cf. (3.30), for mixtures of ideal gases. Here we consider liquids. Nevertheless we still assume ideal behavior.
The last term is new. It introduces an additional interaction free enthalpy proportional to the two mole fractions. The quantityχis a parameter in this theory.
Figure4.20showsgfor different values of theχ-parameter. Ifχ is less than a critical value thengis a convex function ofx(Al)(orx(Bl)). This situation is analogous to the free energy in the van der Waals theory for temperatures above the critical temperature. Ifχ=χcthen the curvature ofgatx(Al)=1/2 becomes zero. For still larger values ofχa “bump” develops - again analogous to the free energy in the van der Waals theory for temperatures less than the critical temperature. Driven by the second law the system now lowers its free enthalpy by separating into two types of regions, which over time will coagulate into two large domains, one depleted of A and one enriched with A. The resulting phase diagram is shown in the lower right panel in Fig.4.20. The binodal line is obtained via a common tangent construction applied to the free enthalpy (cf. the lower left panel) akin to the common tangent construction used in the van der Waals case. The common tangent is the lowest possible free enthalpy in betweenxA,poor andxA,r i ch. For a givenxAin this range the quantity(xA−xA,poor)/(xA,r i ch−xA,poor)is the fraction of Ain theA,r i ch- phase relative to the total amount of Ain the system. The second special line is the spinodal line. It marks the stability limit
∂2g
∂x2A T,P
=0 (4.78)
(cf. the third stability condition in Eq. (3.16)). Both lines meet at the critical point where
0.8 1.2
0 0.2 0.4 0.6 0.8 1
gB
g gA
< c
0.8 1.2
0 0.2 0.4 0.6 0.8 1
gB
xA
g gA
xA,poor xA,rich
> c
0.8 1.2
0 0.2 0.4 0.6 0.8 1
gB
xA
g gA
critical point = c
0.4 0.5
0 0.2 0.4 0.6 0.8 1
T
xA spinodal
line
binodal line critical point
"tie line"
Tc
Fig. 4.20 Schematic of thexA-dependence ofgfor differentχ-values. Thecurvesshown here are forχ =1 (upper left),χ =2 (upper right), andχ =3 (lower left) usinggA=1.5 andgB=1.0.
Lower right:T-xA-phase diagram of our model of a binary mixture, where we assume thatT=1/χ
∂3g
∂x3A T,P
=0, (4.79)
because at the critical point the curvature obviously changes sign.
Note that temperature here enters via the assumed proportionality χ ∝ 1/T. This assumption accounts for the observation that phase separation usually occurs upon lowering temperature. Nevertheless this is purely empirical and more complex descriptions ofχcan be found.
290 300 310 320 330 340
0.0 0.2 0.4 0.6 0.8 1.0
x1 T [K]
Fig. 4.21 Liquid-liquid equilibria data for the binary mixtures water/phenol (solid squares) and methanol/hexane (solid circles)
Figuer4.21shows experimental liquid-liquid equilibria data for the binary mix- tures water/phenol (solid squares) and methanol/hexane (solid circles).21Herex1is the mole fraction of water and methanol, respectively. Notice that while both systems show the basic behavior predicted by our theory, only the second system also exhibits the symmetry aroundx=0.5. Nevertheless, the solid lines are “theoretical” results, which where obtained using
χ= c0+c1x
T +c2. (4.80)
Herec0,c1, andc2are constants, which are adjusted so that the theory matches the data points. In particular thec1-term breaks the symmetry aroundx =0.5. While it is quite common to introduce such expressions for χ, it is not easy to provide reasonable physical explanations of the individual terms. In addition, the “best fit”
usually does not correspond to a unique set of values forc0,c1, andc2. We return to this in the context of polymer mixtures.
4.3.2 Gas-liquid Coexistence in a Binary System
Can this type of phase separation be used to physically separate components Aand B? In principle yes—but not entirely and usually not as a practical means. Let us look at the binary mixture from another angle. The above model did not include possible distribution of components Aand Bin phases corresponding to different states of matter, e.g. gas, liquid or solid. Here we want to study a situation when gas and liquid coexist containing bothAandB. This is depicted in Fig.4.22—which we encountered before (cf. Fig.3.7).
21data from HTTD.
PA+PB A(g) + B(g)
A(l) + B(l)
xA(g), xA(l)
P PA*
PB*
liquid
gas
0 1 Fig. 4.22 Gas-liquid coexistence in a binary system
In equilibrium we haveμ(Ag)=μ(Al)andμ(Bg) =μ(Bl), cf. (3.38). Thus we may also writedμ(Ag) =dμ(Al)anddμ(Bg)=dμ(Bl). Concentrating on componentAand using Eqs. (3.27) and (3.45) we have
d(μ(Ag) T,PA∗
+RTlnx(Ag))=d(μ(Al) T,PA∗
+RTlnx(Al)), (4.81) i.e. from the start we assume that both the gas as well as the liquid are ideal. This may be reshuffled to yield
dμ(Ag) T,PA∗
−dμ(Al) T,PA∗
=RT dln x(Al)
x(Ag). (4.82) Note that we work at constant temperature.
Combination of the Gibbs-Duhem equation (2.168) at constant temperature with Eq. (2.170) yields
∂μi
∂P
T =vi = ∂V
∂ni
T,P,nj(=i). (4.83) Heredμ(Ag)
T,PA∗
anddμ(Al) T,PA∗
refer to the pure componentA, i.e.v∗Ais the molar volume of Ain the gaseous and liquid states, respectively. In particular we may neglectv∗(Al)in comparison tov∗(Ag). Therefore Eq. (4.82) becomes
v∗(Ag)d P ≈RT dln x(Al)
x(Ag). (4.84)
Integration, after insertion of the ideal gas law, i.e.v∗(Ag)=RT/P, yields
P PA∗ ≈ x(Al)
x(Ag), (4.85)
where the reference state is pure Aandx(Al)/x(Ag) =1. Of course AandB may be interchanged and thus
P PB∗ ≈ x(Bl)
x(Bg). (4.86)
Because for ideal gasesP x(Ag)=PAandP x(Bg) =PB, wherePAandPBare partial pressures, Eqs. (4.85) and (4.86) become
PA
PA∗ ≈x(Al) and PB
PB∗ ≈x(Bl). (4.87) In addition we may use Dalton’s law, P = PA+PB, which allows to express P entirely throughx(Al)/Borx(Ag/)B. The first relation is
P≈x(Al)(PA∗−PB∗)+PB∗, (4.88) where we usex(Al)+x(Bl)=1. Next we replacex(Al)withx(Ag)via Eq. (4.85) obtaining
P≈ PB∗
1−(1−PB∗/PA∗)x(Ag). (4.89) Equations (4.88) and (4.89) both are shown in the right panel of Fig.4.22. The straight line is Eq. (4.88), whereas the curved line is Eq. (4.89). Their meaning is as fol- lows. The dashed vertical line corresponds to a fixed xA. Above its intersection with Eq. (4.88) we are in the liquid state of the mixture. Below its intersection with Eq. (4.89) the mixture is a homogeneous gas. In between however liquid and gas do coexist, and the mole fractionsx(Al)andx(Ag)are given by the intersections of the horizontal dashed lines (depending on P) with Eqs. (4.88) and (4.89).
There is a simple law connecting the total amount of liquid, n(l), and the total amount of gas, n(g), with xA, x(Al), and x(Ag)—the lever rule. Note that (i)nxA=n(l)x(Al)+n(g)x(Ag), wherex(Al)=nA(l)/n(l)andx(Ag)=nA(g)/n(g), and (ii)nxA=n(l)xA+n(g)xA. Combining (i) and (ii) yields the lever rule:
x(Ag)−xA
xA−x(Al) = n(l)
n(g). (4.90)
Notice thatxAis indeed bracketed byx(Ag)andx(Ag)as we had assumed.
Fig. 4.23 Left1-chlorbutane/toluane;rightwater/ethanol
Experimental isothermal gas-liquid equilibria are shown in Fig.4.23. The left panel shows the system 1-chlorbutane/toluene atT =298.16K. This system is well described by the above Eqs. (4.88) and (4.89) shown as solid lines. But there are other systems, like water/ethanol atT =323.15Kshown on the right, which are not as ideal.
One may wonder about the difference between our two treatments of binary mix- tures. The first one basically is a model composed of the ideal free enthalpy of mixing supplemented by a temperature dependent phenomenological “interaction”
free enthalpy. Here the mixture may phase separate into regions of different com- ponent concentration depending onT. The second approach assumes a (first order) transition between different states of matter and describes the distribution of com- ponents AandBbetween phases corresponding to those different states (gas/liquid etc.) in terms of pressure. Aside from the assumed coexistence of phases ideality is used throughout. In reality a combination of both approaches may be necessary.
However, it is worth noting in this context that a complete theory for the full phase diagram of a real system (or material) does not exist.
4.3.3 Solid-liquid Coexistence in a Binary System Solubility
Based on the simple model expressed in Eq. (4.77) we want to study the situation depicted in Fig.4.24. The figure shows a solution of B in Aand a pile of solid B on the bottom. This can happen for instance if we try to dissolve too much sugar or salt in water. The following is a rough calculation of the maximum mole fraction, xB(T), which we can dissolve at a given temperature,T.
Neglecting theχ-parameter in Eq. (4.77) we may write for the chemical potential of BinA:
μB(l)=μ∗B(l)+RTlnxB. (4.91)
Fig. 4.24 A solution of B in A including solid B at the bottom
solution A + B
solid B
Ifμ∗B(s)is the chemical potential of the pure solid B, then we have at coexistence μ∗B(s)=μB(l), i.e.
μ∗B(s)=μ∗B(l)+RTlnxB. (4.92) This can be rewritten into
lnxB = −μ∗(T)
RT +μ∗(Tm)
RTm , (4.93)
whereμ∗(T)=μ∗B(l)−μ∗B(s). Notice thatTmis the equilibrium melting temper- ature of pureBand thusμ∗(Tm)=0. Why have we added this term? To see this we writeμ=h−Ts, wherehandsare molar enthalpy and entropy changes.
Assuming (!) thath∗(T)≈h∗(Tm)≡melt,Bhands∗(T)≈s∗(Tm), i.e.
these quantities depend only weakly onT, we immediately find lnxB≈ −melt,Bh
R 1
T − 1 Tm
. (4.94)
The added term has essentially eliminated the entropy and all we need to know is the transition enthalpy (here the melting enthalpy) of pure Bas well as its melting temperature.
Notice that the indexBin Eq. (4.94) can be replaced by an indexi, wherei = A,B.
This means that the roles ofAandBmay be interchanged. Figure4.25shows what we get for a mixture of tin (Sn) and lead (Pb). For tin we havemelt,Snh=7.17 kJ/mol andTm =231.9◦C (HCP). In the case of leadmelt,Pbh =4.79 kJ/mol andTm = 327.5◦C. UsingxPb =1−xSn we can combine both graphs ofT versusxSnand T versusxPbaccording to Eq. (4.94) into one plot. They intersect atxSn =0.485 andTe =81.6◦C. Below the intersection both lines are continued as dashed lines.
AboveTeand between the solid lines the mixture is a homogeneous liquid. The solid lines are the solubility limit of SninPbor, abovexSn =0.485,PbinSn.Te, the eutectic temperature , is the lowest temperature at which a mixture of Snand Pb can exist as a homogeneous liquid. The intersection of the lines marks the so called eutectic point.
Predictions of this simple approach, even though they are helpful for our under- standing, are neither quantitative nor complete. The true eutectic point of theSn/Pb- system is atxSn =0.73 andTe = 183◦C. The real phase diagram of this system
0.0 0.2 0.4 0.6 0.8 1.0 0
50 100 150 200 250 300 350
xSn
T[°C]
homogeneous liquid mixture
Fig. 4.25 Approximate solubility limits of tin and lead in a binary system
0.0 0.2 0.4 0.6 0.8 1.0
50 100 150 200 250 300
T[°C]
liquid
homogeneous liquid mixture
liquid
xSn
Fig. 4.26 The real phase diagram of the tin/lead-system
(at P =1bar) is shown in Fig.4.26(based on data in HCP). As in Fig.4.25there is a homogeneous liquid mixture with an eutectic point shown as black dot. The regions markedαandβcorrespond to homogeneous solid phases rich inPbandSn, respectively. The remaining regions are phase coexistence regions.
4.3.4 Ternary Systems
Figure4.27explains how to read triangular composition phase diagrams of ternary systems. A ternary system contains the there componentsA,B, andC. By definition the side lengths of the equilateral triangleA BCare equal to one. At the point labeled Q the system has the composition xA, xB, and xC. The position of Q within the triangle is described via the dashed lines possessing the respective lengthsxA,xB,
Q
A B
C
xA
xB
xC
Fig. 4.27 Triangular composition phase diagram of ternary systems
10 20 30 40 50
80 90 60
50 40 30 20
Fe 10 70 Ni
Mass Percent Nickel
Mass Percent Chromium Mass Percent
Iron
90 80
70 60
50 40
30 20
10 90
80 70
60 Cr
(Cr)+( Fe,Ni) (Cr)+
(Cr)
+(Fe,Ni)
( Fe,Ni) 18-8 Stainless steel (Cr)+
(Fe,Ni)
Fig. 4.28 Experimental example of a ternary phase diagram
andxC. The dashed linexAis parallel toA B,xBis parallel toBC, andxCis parallel to AC. This definition satisfies the necessary conditionxA+xB +xC = 1. The validity ofxA+xB +xC = 1 is verified easily using the division of the original triangle into a lattice of smaller equilateral triangles. There is no loss of generality due to this meshing, because the coarse mesh in our example can be replaced by one which is arbitrarily fine.
An experimental example is depicted in Fig.4.28showing the iron-chromium- nickel ternary phase diagram at 900◦C (adapted from HCP (Fig. 23 of Sect. 12 p.
199, 89th edition)).
Exercise: Determine from Fig.4.28the composition of 18-8 stainless steel (open circle).
Fig. 4.29 Binary mixture of linear polymers represented by paths on a lattice