In the Classroom
Phase Separation in Binary Systems: The Possibility of Gas–Gas Equilibrium at Subcritical Conditions Jaime Wisniak Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel
The second law of thermodynamics tells us that at a fixed pressure and temperature a stable state is that which has a minimum Gibbs energy, and a mixture will split into two separate liquid phases if on doing so, it can lower its Gibbs energy. To illustrate these concepts, let us consider a mixture of two liquids at a certain pressure and temperature. The molar Gibbs energy of the mixture, g, is g = x1g1° + x2g2° + ∆G
(1)
where g 1° is the molar Gibbs energy of component i and ∆G the Gibbs energy of mixing. From eq 1 we have ∆G = g – (x1g1° + x2g2°)
(2)
so that ∆G must be negative. Experimental evidence indicates that at a given pressure and temperature the plot of g (x) of liquid mixtures may have the shapes given in Figure 1. Curve I is the more common. It is characterized by being concave for all values of x and having one local minimum. This fact may be expressed mathematically as
∂2 g ∂x21
>0
(3)
Figure 1. Variation of the molar Gibbs function of a mixture with composition.
P,T
for any value of x. Combining eqs 2 and 3 we get the equivalent relation 2
∂ ∆G 2 ∂x 1
>0
(4)
P,T
To understand the meaning of these equations, let us assume a binary mixture with overall composition, x1α and the possibility that it may split into two conjugated phases C and D, with composition x91 and x01. Let us draw a straight line connecting points C and D. From a thermodynamic standpoint the straight line represents the fact that the Gibbs function is an extensive property; the value of the molar Gibbs energy of the total system, g E, will be the weighted value of the Gibbs energy of each phase. Clearly gE > gE9’ , so that the system must be homogeneous. On the other hand, curve II is concave only in parts of the composition range; in the intermediate range (C9D9) it presents a convex portion, which can lead to a substantially different behavior. If points C9 and D9 are joined by a straight line (common tangent) it is seen that for a given overall composition x1α any mixture located on this line has a lower value of Gibbs function than when located in the curved portion. In other words, a lower value of g can be achieved by phase splitting. The difference between mixtures represented by curve I and mixtures represented by curve II may be better understood by plotting the first two derivatives of the latter as a function of composition (Fig. 2). From Figure 2 we learn that a curve of type II has two inflection points (F and G) and that between the corresponding compositions
546
Figure 2. First and second derivatives of g ( x ).
Journal of Chemical Education • Vol. 74 No. 5 May 1997
In the Classroom [(x91 ) inflec and (x01 ) inflec] we have
∂2 g ∂x21
≤0
(5)
P,T
All other compositions satisfy eq 3. The two compositions (x91 ) inflec and (x01 ) inflec define the region of absolute instability; any mixture having an internal composition will always split into two phases of compositions (x91 ) inflec and (x01 ) inflec (partial miscibility). If the function g (x) is plotted at a different temperature level the values of (x91 ) inflec and (x01 ) inflec will change: they will approach or separate according to the manner in which mutual solubility changes with temperature for the system in question. For systems having an upper critical solubility temperature (UCST), an increase in temperature will cause the values of (x1 ) inflec and (x01 ) inflec to approach until they will become identical at the consolute or critical solution temperature. At temperature levels above the UCST, g (x) will behave as curve I and the components will present infinite solubility. Systems having a lower critical solubility temperature (LCST) will behave in exactly the opposite manner. The locus of all pairs [(x91 ) inflec and (x01 ) inflec] describe the spinodal curve of the system. The spinodal curve has a maximum (or minimum) value that coincides with the critical solution temperature (equal sign in eq 5). Points F and G, which define the limits of the range of absolute instability, do not define the two-phase equilibrium compositions x91 and x01 . In Figure 1, segments CF and GD represent states of metastable equilibrium where singlephase behavior can in principle occur. However, these states are unstable to large perturbations. It is important to realize that stability analysis can tell us only whether a system can or cannot exhibit phase splitting at the given temperature. That is, if we have an expression for ∆GE , stability analysis can determine whether there is some range of composition where two phases exist. It does not tell us what the composition range is. To find the range of compositions where two phases can exist at equilibrium requires the following more elaborate calculation. Equation 5 may also be written
∂ 2∆G ∂x21
≤0
(6)
P,T
= g – RT [x1 ln x1 + x2 ln x 2]
P,T
(11)
Tc = A / RT
(12)
and The critical temperature given by eq 12 must be an upper critical solution, since for T > Tc eq 10 is always satisfied. We see that the regular solution model is capable of predicting partial miscibility but cannot describe systems having an LCST. We now find the boundary separating the metastable states from the unstable states. From eq 10 the equation for this boundary is T = (2A / R) x (1 – x)
(13)
T = 4Tc x (1 – x)
(14)
or Note that eq 14 provides the composition (x) of only one of the two conjugated phases. From symmetry conditions the composition of the second phase must be (1 – x). Finally, we need to know the curve of coexistence of the two layers in true equilibrium. On this curve (binodal curve) we must have µ 19 = µ10 (15) µ29 = µ20
(16)
where the chemical potential µ is given by µ i = µ i° + RT ln (x i γi )
(17)
For a regular solution the activity coefficients are RT ln γ1 = A x22
(18)
RT lnγ 2 = A x12
(19)
RT ln (1 – x91 ) + A( x91 )2 = RT ln x91 + A (1 – x91 )2
+ xRT ≤0 1x 2
T = 2Tc
(8)
1 – x′1 x′1
(21)
Again, the composition of the second phase is obtained by symmetry considerations. Gas–Gas Equilibrium
(9)
Parameter A may be independent of temperature (strictly regular solution) or dependent on it (regular solution). Applying now the condition for instability (eq 8) we get
1 2A ≤ RT x (1 – x)
1 – 2x′1 ln
(7)
We will now illustrate the use of the above equations by considering the simple case of a regular solution for which gE = Ax1x2
(20)
so that
to yield
∂2 g E ∂x21
(x2)c = 0.5
From eqs 15 and 16 we get
Equation 6 may be expressed in terms of the excess Gibbs function gE
temperature; if its value is sufficiently large then inequality 10 cannot be satisfied at all concentrations. The minimum value of the left-hand side is 4; hence for values of A / RT smaller than 2 the condition given by eq 10 will not be satisfied and the system will split into two liquid phases of different composition. The conditions at the critical point may be found by applying eq 10 to yield
(10)
Note that parameter A / RT will always be a function of the
The mutual solubility of liquids is decreased by chemical dissimilarity of the components as reflected by an increase in their separate activity coefficients. Where the activity coefficients become sufficiently great, the solution separates into two liquid phases. Liquid–liquid equilibrium is a phenomenon that is well documented in the literature (1, 2); but not so the thermodynamic conditions under which a gaseous system may split into two phases. The possibility of gases exhibiting limited miscibility was predicted on theoretical grounds by Van der Waals; it has been observed experimentally only at high pressures, normally supercritical, where gases are at liquidlike densities (3, 4).
Vol. 74 No. 5 May 1997 • Journal of Chemical Education
547
In the Classroom When fluid–fluid phase separation occurs at a temperature higher than the gas–liquid critical point of either pure component, the phenomenon is known as gas–gas immiscibility (5). We will now consider the thermodynamic conditions for phase splitting in a gas under subcritical conditions. From the Maxwell relationships we have dG = { SdT + VdP
(22)
Applying eq 22 to the excess Gibbs function ∆GE we get
Table 1. Virial Coefficients and Parameters for Typical Gas Mixture, mL·mol{1 (6 ) System
T (K)
B 11
Argon + carbon tetrachloride
323.15
{11.02
Chloroform + methyl formate
323.2
P δ12 / RT 0.49
{990
{770
{1521
{1282
{0.24
Methyl bromide + ethyl bromide 313.2
{402
{669
{881
{691
{0.13
Carbon dioxide + helium
298.2
{117.6
Fluorobenzene + benzene
453.2
{627
Hydrogen + helium
323.2
Acetonitrile + cyclohexane
326
Methane + sulfur hexafluoride
313.16
Chloroform + acetone
333.2
Chloroform + diethylamine
323.2
(23)
d∆G E = ∆G E(P) =
P 0
∆V dP
(24)
In order to calculate the value of the volume of mixing, ∆V, let us assume that each of the pure gases as well as their mixture behaves according to the virial equation of state truncated to the second term (pressures below, say, 5 atm)
Pv = 1 + B v RT
(25)
11.8 {584
15.2
∆V = v – Σ yivi
(26)
vi = RT + B ii P
(27)
v = RT + B P
(28)
{3500
(29)
The second virial coefficient of a binary gas mixture B is related to the second virial coefficients of the pure components by B = y1B11 + y2B22 + y1y2δ12
(30)
δ12 = 2 B12 – B11 – B22
(31)
where
{598
15
15.8
4.8
0.007 0.002 0.0009
{1300
{600
{253
{85
{910
{1330
{2005
1770
{0.32
{1020
{1200
{2063
1906
{0.36
{37.9
3600 120.9
0.67 0.025
ing eq 32 in eq 24 and integrating yields ∆GE = ∆V = ∆B = y1y2δ 12P
(33)
Inspection of eq 33 yields the interesting result that a binary mixture of gases at pressures up to, say, 5 atm behaves as a regular solution, since ∆GE = Ay1y2
(34)
where A = δ 12 P. Since A is a function of temperature, a binary mixture of gases is not strictly regular. Let us now consider the stability properties of such a mixture. Replacing eq 34 in eq. 8 yields
2A ≥ 1 RT y1 y2
(35)
Analysis of the right-hand side of eq 35 indicates again that the equation will have a real root whenever
2A > 1 RT 4
(36)
A ≥2 RT
(37)
Pδ 12 ≥2 RT
(38)
is fulfilled the binary gas mixture may split in two phases. The minimal value Pδ12 / RT = 2 corresponds to incipient stability and the temperature corresponding to that condition is the consolute temperature. If we assume that P = 5 atm and T = 400 K, we obtain that δ12 > 13,000 mL mol {1 in order for the phenomenon to occur at subcritical pressures. The value of δ12 is extremely large, so that although phase splitting in a gas binary mixture is theoretically possible, it will occur rarely if at all. In Table 1 are reported the pertinent parameters for 10 binary systems of different polarities at 10 atm. It is seen that all of them are far from the required condition for phase separation given by eq 38. Literature Cited
so that from eq 29 ∆V = B – y1B11 – y2B22 = ∆B
33.6
In other words, if the condition
so that
∆V = RT + B – Σ yi RT + B ii P P
{36.1
11.7
where B represents the second virial coefficient of the pure gas or of the mixture. We have
(32)
Since the second virial coefficients of the pure components are a function of temperature alone, eq 32 indicates that for a mixture of constant composition the volume of mixing, ∆V, is a function of the temperature alone. Replac-
548
δ12 2616
Where ∆S E and ∆VE represent the excess entropy and excess volume of mixing, respectively. Let us calculate the value of ∆GE for a binary gas mixture that performs an isothermal process from pressure 0 to any pressure P. Remembering that ∆VE = ∆V we obtain
0
B 12 {63
d∆GE = {∆SEdT + ∆VEdP
P
B 22 {2731
Journal of Chemical Education • Vol. 74 No. 5 May 1997
1. Novak, J. P.; Matous, J.; Pick, J. Liquid–Liquid Equilibria; Elsevier: Amsterdam, 1987. 2. Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures; Butterworth Scientific: London, 1982; pp 115–119. 3. Streett, W. Can. J. Chem. Eng. 1974, 52, 92–97. 4. Schneider, G. M. Adv. Chem. Phys. 1970, 17, 1. 5. Van Konynenburg, P. H.; Scott, R. L. Phil. Trans. Roy. Soc. London 1980, 298A, 495. 6. Dymond, J. H.; Smith, E. G. The Virial Coefficient of Gases and Mixtures; Clarendon: Oxford, 1980.
