CRITICAL BEHAVIOR IX MIXTURES O F FLCIDS
1347
CRITICAL BEHAT’IOR -4KD LIQUID-VAPOR EQUILIBRITJM IS MIXTURES OF FLUIDSL CHARLES A. BOYD
Satial Research Laboratory, Department of Chemistry, University of Wisconsin, Madison, Wisconsin Received March 20, 1950 I. INTRODUCTION
The phase relationships which exist for fluid mixtures in the neighborhood of the critical point have in recent years been given little attention by physical chemists. This paper has, therefore, a twofold purpose: first, to present a survey of the general phase behavior of fluid mixtures at elevated temperatures and pressures; second, to outline a general method of predicting this behavior from thermodynamic principles of equilibria. In the latter part of the nineteenth century, scientists were particularly cognizant of the critical behavior of pure fluids. Andrew (I) reported his classic researches on carbon dioxide in 1869, van der Waals (8) had completed a dissertation on the continuity on the states of matter, and in general a good deal of effort \vas being concentrated on investigations of P-V-T relationships for both pure gases and gaseous mixtures. In the year 1880 Cailletet ( 2 ) made an important discovery in connection with his studies on the liquefaction of carbon dioxide-air mixtures. As might be expected, a mixture of one volume of air and five volumes of carbon dioxide would undergo partial condensation t o the liquid state as t,he pressure was increased to a moderate value, the temperature being held sufficiently low. When the pressure v a s increased further, the temperature being maintained constant, the liquid phase would begin to disappear; when the pressure reached a critical value, the liquid phase \Todd disappear altogether and the system would return to a single vapor phase. Upon lowering the pressure to this critical value, the liquid phase reappeared. This apparently anomalous behavior in mixtures was observed by other investigators, among them van der Waals and Andrem. Contributions t o the interpretat,ion of this behavior were made by Duhem (3) and others. However, Icuenen (€I), reporting on his studies on the system carbon dioxide-methyl chloride, gave the first clear interpretation of the overall phase behavior of binary mixtures in the critical region. He applied the name “retrograde” or backward condensation to the type of anomalous behavior observed by Cailletet and Andrew, and showed how under the proper conditions a similar phenomenon could be produced by holding the pressure constant and varying the temperature. To this second type of behavior he applied the term “retrograde condensation of the second kind.” As has been pointed out by Katz and Kurata (4)in their survey of retrograde Presented a t the Symposium on Critical Phenomena, which was held under the auspices of the Division of Physical and Inorganic Chemistry a t the 116th Meeting of the American Chemical Society at Atlantic City. Kew Jersey, September 22, 1949.
1348
CHARLES A. BOYD
condensation, most physics and physical chemistry texts published since 1910 consider only phase relationships in liquid mixtures a t or near atmospheric pressure and completely ignore the wealth of information existing ifi the earlier literature concerning the influence of elevated pressures upon these relationships. However, in recent years the petroleum industry has been giving much attention to these problems, since retrograde phenomena frequently occur in high-pressure distillation of hydrocarbons and the principle is regularly employed in increasing the yield of petroleum from oil wells by pressurizing. 11. PHASE BEHAVIOR I N BINARY SYSTEMS
The experimental data obtained by Kay ( 5 ) on the ethane-heptane system will be used t o illustrate the phase behavior of binary systems at elevated tem30
250p7-7 , 200
200
w
5100
1 0.2 0.4 0.6 0.8 1.0 MOLE FRACTION ETHANE
0
0 0.2 0.4 0.6
0.8
1.0
MOLE FRACTION ETHANE FIG.2
FIG.1 FIG.1. Distillation curve for ethane-heptane system. Pressure, 100 p.8.i. Experimental data of Kay (Ind. Eng. Chem. SO, 459 (1938)). FIG.2. Distillation curve for ethane-heptane system. Pressure, 800 p.s.i. Experimental data of Kay.
peratures and pressures. Figure 1 shows the ordinary type of distillation curve, where the composition of the liquid and that of the vapor in equilibrium with it are given as functions of the boiling point of the solution. If we consider a mixture of the two components containing 0.2 mole fraction of ethane a t a temperature of -25°C. and under 100 lb./in.* pressure, the situation can be represented by point A in figure 1. As the temperature of the system is increased, the pressure being held constant, the point representing the system follows up the dotted ordinate A D until point B is reached. At this temperature a vapor phase of lower density, represented by B’, appears in equilibrium with the liquid. Further increase in temperature results in an increasing amount of vapor phase and a corresponding decreasing amount of liquid phase until point C is reached, where the last trace of liquid phase, having a composition C’, dis-
CRITICAL BEHAVIOR IN MIXTURES O F FLUIDS
1349
appears. Further increase in temperature has no additional effect on the system, which is not totally in the vapor phase, disregarding, of course, any chemical effects brought about a t elevated temperatures. At higher pressures we obtain similar curves with the liquid and vapor boundaries, corresponding to particular compositions, occurring a t higher temperatures. The variation with pressure of the boiling points of the pure components, given a t 0 and 1.0 mole fraction of ethane in figure 1, follows the regular vapor pressure curves for the two substances. With inciessing pressure the boiling points of the pure components continue to increase until the critical temperature for one of them is reached. In the case of the ethane-heptane system under consideration this occurs for the heptane a t a pressure slightly under 400 lb./in.' The corresponding critical temperature for pure heptane is approximately 250°C. -4t or above this temperature one cannot distinguish between the vapor and liquid phases for pure heptane or mixtures of heptane and ethane up to some limiting composition which is determined by the pressure. The situation a t 800 lb./in.' is represented in figure 2. The heterogeneous area bounded by the liquid and vapor curves does not extend much below 0.3 mole fraction of ethane; consequently, if a mixture containing 0.2 mole fraction of ethane is heated a t constant pressure there will be no phase separation such as occurred a t 100 lb./in.' The liquid boundary in figure 2 goes through a minimum in composition with increasing temperature before it meets the vapor boundary at the point indicated by C.P. Point C.P. then indicates the critical point for a mixture of approximately 0.4 mole fraction of ethane, since only a t this point do the vapor and liquid phases have the same density and composition and thus become indistinguishable. We shall refer to this composition, a t which the critical point occurs for a specified pressure and temperature, as the critical composition. The composition minimum in the liquid boundary gives rise to one type of retrograde condensation. Consider a mixture at this pressure with composition and temperature represented by point A in figure 2. Increasing the temperature of the mixture while holding the composition constant is represented by the motion of a point up the dotted line until point B is reached, where a vapor phase (phase of lower density) appears having a composition given by B'. Upon further increase of the temperature, the amount of vapor phase first increases, reaches a maximum, and then decreases until point C is reached, where the last trace of vapor having a composition represented by C' disappears and the system again returns to the single liquid phase. Thus, by increasing the temperature of the system, the initial vapor formation has been followed by condensation. This is contrary to the behavior which one would expect. Hence, following the nomenclature of Katz and Kurata (4),this disappearance of the vapor phase upon increase in temperature is called retrograde condensation of the second kind. A similar type of behavior occurs in mixtures when the critical temperature of the second component is exceeded. The resulting distillation curves obtained in the ethane-heptane system over a wide range of pressures are shown in figure 3. The experimental data are shown better in a three-dimensional representation, the three axes being pressure, temperature, and composition. A photograph of
250
zoo 150
2
$'a i2 50 w
a
5c 0 0 0.2 09 06 Q8
10
MOLE FRACTION ETHANE
CRITICAL BEHhVIOR I N MIXTURES O F FLUIDS
1351
such a three-dimensional representation for the ethane-heptane system is shown in figure 4, where the heterogeneous region is now represented by a tongueshaped volume bounded on one approach by a liquid surface and on the other by a vapor surface. 1001
I
I
I
FIG.5 . Pressure-composition curve for ethane-heptane system. Temperature, 400°K. Experimental data of Kay.
Figure 5 shov-s a constant-temperature section taken through the heterogeneous volume at 400°K. Since this temperature is above the critical temperature of pure ethane, the heterogeneous region bounded by the liquid and vapor lines does not extend over to a composition of 1.0 mole fraction of ethane, and it mill be noted that the vapor curve goes through a maximum in composition before meeting the liquid curve at the critical point. In order t o demonstrate retrograde condensation of the first kind, consider a system represented by point A on figure 5. Only one phase is present until the pressure is lowered (the temperature being held constant) to point B , where a phase of higher density appears having a composition represented by B’. Hence, lowering the pressure at constant temperature causes condensation of a liquid phase; this result is contrary to expected behavior and again we have an example of retrograde or “backward” condensation. In general, it may be stated that retrograde behavior will occur when the path of a point moving in a straight, line through the three-dimensional representation enters and leaves the heterogeneous volume through the same phase boundary whether it be liquid or vapor. This path does not necessarily have to parallel the temperature or pressure axis but can combine changes in both. It should be pointed out that the above discussion has considered only simple solutions and has not considered aaeotropic cases where maxima or minima occur in the distillation curves. However, this situation does not alter the general interpretation.
1352
CHARLES A . BOYD 111. THERMODYKAMIC CONSIDERATIONS
When several phases are in equilibrium in a multicomponent system, one of the thermodynamic criteria for such a state involves the equality of the chemical potentials (partial molal free energies) of the various components in each of the phases, namely:
by
=
p
..
=
(1)
* p;n)]p,T
This equation simply says that the chemical potential, or the partial molal free energy, of the if*component in phase 1 must be equal to the same quantity in phases 2, 3, etc. I n the simpler case of liquid-vapor equilibrium in a binary system, equation 1 reduces to [# = p1 l P , T (2) (U)
hZ(')= pZ(')]P,T (3) where the superscripts 1 and v refer to the liquid and vapor phases and the subscript numbers refer to the components. A general expression for p c a t some specified temperature and pressure is given by : [pi
p?
+ RT In (PXi)+ /'[Pi
- (RT)/Pl d PI
T
(4)
where R, T , and P have their usual significance, X , is the mole fraction of the component being considered, and P, is its partial molal volume. I n order to evaluate the integral in equation 4 an expression for P, as a function of pressure and composition is needed. This requires equations of state for the pure components and of their mixtures, including composition as one of the variables. The equations of state which were used in the present work will be described in the folloming section. Having developed a method of evaluating the integral in equation 4 one can then represent the quantity p - p* or, for convenience, ( p - p*)/RT, for each component as a function of composition at specified temperatures and pressures, thus :
1
[b - r*)/RTI/:: = S[XI, P , TI:::
[
[(P
- p*)/RTI[3 =
m,,P,TI::;
T
(5) (6)
Since we are working with a binary system, the composition can be expressed in terms of the mole fraction of either component. p * , R, and T are constants with respect to composition changes, and consequently equations 5 and 6 can be combined with the equilibrium equations 2 and 3 to give: I,
P , TI:;; = f[Xl, P , TI:;:
1
p , TI::] = f[Xl, P , TIiz";
T
(7) (8)
CRITICAL BEHAVIOR IN MIXTURES OF FLUIDS
1353
For specified values of pressure and temperature which are the same for both liquid and vapor phases, equations 7 and 8 can be solved simultaneously for X i ” and X ! ’ ) . These values then represent the compositions of the liquid and vapor phases in equilibrium at the particular pressure and temperature. At the critical point equations 7 and 8 continue to hold. However, a t this point an additional restraint is placed on the system. For simple binary solutions exhibiting no boiling point maxima or minima, the composition of the vapor and the composition of the liquid become equal. Consequently we h a w a third equation expressing this restraint:
xi” = X‘“’
(9)
Simultaneous solution of equations 7, 8, and 9 allows us in principle to calprovided the fourth culate any three of the four variables P, 2 , X i L ) ,or x{”’, is specified. If one chooses a value of critical temperature for the system, one can then calculate the corresponding critical pressure and composition. In order to apply the equations of the foregoing section to actual cases, it will be necessary to use some equation of state. Although we have not as yet completed actual numerical calculations, it mill be advantageous at this point to introduce the van der Waals equation, which \ye write in the form
P = n R T / ( V - nb) - n % / P
(10)
for n moles. This will enable us to bring out the differences between the treatments required for pure substances and for binary mixtures. This equation was originally proposed t o account for the pressure-volumetemperature relationships of pure substances, including the coexistence of two phases below the critical temperature. van der Waals also applied his equation to mixtures (9), using an expression for b deduced by Lorenta ( 7 ) from theoretical considerations : bmix
=
b1l-X:
+ 2b12XlX2 + bnX:
(11)
where b11 and b22 are the values for the pure components and bl2 is a constant characteristic of the mixture. X1 and X2 refer to the mole fractions of components 1 and 2, respectively. A similar expression was used for the constant amlr,thus: am*= = a u x :
+ 2U12XlX2 + a??x:
(12)
It is of course well known that van der Waals’ equation does not represent the equation of state of real substances with great accuracy. However, it can be considerably improved if the constants are considered to be functions of temperature. It will ultimately be necessary to consider the evaluation of all the constants in equations 11 and 12 as functions of T , and it is hoped by this means (or by use of a better equation of state) to obtain in future work a semiquantitative description of the phenomena. For the present purposes this temperature dependence will not make any difference.
1354
CHARLES A , BOYD
At this point it might be of use to consider the meaning of the critical relationships of van der Waals’ equation which express a and b in terms of the critical temperature, pressure, and volume. As is well known, when we apply van der Waals’ equation to any one-component system a t a temperature below the critical value, the pressure of the liquid phase must equal the pressure of the vapor phase in order for equilibrium to be established. Consequently, the tie-lines connecting the liquid and vapor states in equilibrium are parallel to the volume axis, as shown in figure 6 for the temperatures
P
e FIG.6. van der Waals isotherms for a. one-component system at various pressures
TI and TO. As the temperature is raised from TI through T2to T,, the equilibrium pressure rises but the molar volumes of the liquid and vapor phases approach each other in value until the critical temperature is reached, a t which point the two molar volumes become identical, indicating that the two phases can no longer be distinguished from one another. This behavior leads to the folloming mathematical consequences at the critical point :
(aP/av), = 0 (a*P/av2)T= 0
(14)
Substitution of van der Waals’ equation into these two expressions and solving for a and b leads to equations relating these two quantities to the critical temperature, pressure, and volume, such as:
b =
;v,
(15)
1355
CRITICAL BEHAVIOR I S MIXTURES O F FLUIDS
The situation is not so simple in the case of a two-component system, since in addition to the variables P , V, and T we have another variable: namely, composition. For simple solutions with no azeotrope formation the composition of the liquid phase differs from that of the vapor phase in equilibrium with it, and consequently the isotherm for the liquid is not a simple continuation of that for the vapor. This is illustrated in figure 7 . The isotherm represented by A‘ABB‘ corresponds to some definite composition, F . However, the vapor phase represented by B is in equilibrium with a liquid phase represented by D on a different iso-
E
F COMPOSITION
G
FIG.i . van der Waals’ equation, showing vapor-liquid equilibria for a two-component system at. a constant temperature above the critical temperature of the second component.
therm corresponding to a different) composition, E . Likewise, the liquid phase of composition F represented at A is in equilibrium with a vapor phase, P, having a composition between F and G. I n figure 7 composition G is the critical composition because at the critical point, C, the liquid phase, bounded by the bubble-point line, and the vapor phase, bounded by the dev-point line, have the same composition, G, and the same molar volume, Trc. The dew-point line, BPST’C, bends back to meet point C in such a fashion that a tie-line, represented by TT’, connects to a vapor phase having a composition to the right of G. This behavior, of course, is necessary in order to have retrograde condensation of the first kind. Although the isotherm at composition G can be called the critical isotherm
1356
CHARLES A . BOYD
since it serves to define the critical point, it also specifies a vapor phase S of composition G in equilibrium with a liquid phase L of different composition, Projections of the bubble-point and dew-point lines upon the pressure-composition plane will result in a figure similar to that shown in figure 5 . The expressions relating a and b to critical temperature, pressure, and volume, which were derived for a one-component system, do not necessarily hold in a two-component system. Consequently one cannot use these expressions to calculate the critical constants for all compositions directly from the values of a and b obtained from combining expressions 12 and 13. The critical constants for mixtures must be calculated in the manner outlined in the previous section. Using equations 11 and 12, which give the constants a and b as a function of composition, and the van der Waals law given in equation 10 we can obtain an expression forvi = aV/ani, where ni is the number of moles of component i. To do this we differentiate both sides of equation 10 with respect to ni with T,P , and all other nk constant, so that aP/ani = 0 and an/ani = 1, and we recall that
aXi/ani
=
(1 - Xi)/ni
for a binary system. ?We can then substitute V , in the integral in equation 4, use equation 10 to get d p in terms of dV, and obtain finally:
[lp[?< -
( R T / P ) ]d P
=
-RT(ln [P(V - b ) / R T ] )
+ [ ( R T ) / ( V- b)lBi - A i / V ] ,
(17)
where V is the molar volume of the mixture obtained from van der Waals' equation and:
+ (1 - xi)(ab/axi) = 2a + (1 - xi)(aa/axi)
B~ = b
(18) (19)
The quantities a and b and their derivatives refer to the values for the mixture and can be obtained directly from equations 11 and 12. Equation 17 can then be combined directly with equation 4 to give: [(pi -
p:)/RT = iln [ ( R T X i ) / ( V- b ) l ]
+ [Bi/(V - b)l - [ A i / ( v R T ) l l ~ (20)
Equation 20 can then be used in the manner described in Section I11 to obtain the equilibrium compositions and the critical compositions as functions of temperature and pressure. If the pressure is sufficiently low, and the temperature is below the critical temperature, van der Waals' equation when solved for volume may give three real roots. The smallest of these corresponds to the liquid phase, and the largest corresponds to the vapor phase. The intermediate root is only a mathematical consequence of the form of the equation and does not correspond to any real situation. If the pressure is sufficiently high there will be only one real solution
CRITICAL BEHAVIOR I N MIXTURES O F FLUIDS
1357
for volume which, of course, would correspond to the liquid phase, assuming the temperature to be lower than the critical value. Correspondingly, depending upon the exact value of the temperature, there may be only one real solution for volume representing the vapor phase if the pressure is sufficiently low. Therefore, from van der Waals’ equation for a temperature below the critical point, we ran obtain two sets of molar volumes as a function of pressure, one set describing the liquid phase and the other describing the vapor phase. These sets will overlap in one pressure region. The ratio ( p i - p:)/RT will represent either the liquid or the vapor, depending upon which value of V is taken for substitution in equation 17. In this manner it is possible to set up the simultaneous equations represented in equations 7 and 8 of Section 111. V. SUMMARY
A general discussion is given of the liquid-vapor equilibrium in a binary system, with particular attention to the phenomenon of retrograde condensation. By way of illustration, the behavior of the ethane-heptane system is elucidated by a series of phase diagrams. The thermodynamics of the liquid-vapor equilibrium is developed, and the van der Waals equation of state is introduced. In this way, we have shown the difference in the treatment required for pure systems and for binary systems, and have laid the ground for future calculations of a semiquantitative nature. The author would like to thank Professor J. 0. Hirschfelder and Professor C.
F. Curtiss of this laboratory, and Professor 0. A. Hougen of the Department of Chemical Engineering, for their many helpful suggestions brought out in discussions during the course of this work. REFEREKCES ANDHEWS, T . : Phil. Trans. Roy. Sco. 169, 575 (1869). CAILLETET, L : Compt. rend. Bo, 210 (1880). DUHEM, P . : Trav. mem. facult6 Lille 3, K O 13 (1893). KATZ,D. L., AND KURATA, F . : Ind. Eng. Chem. 32,817 (1946). KAY,W, B . : Ind. Eng. Chem. SO, 459 (1938). (6) KUENEN,J . P.: Arch. neerland sci. 26, 354 (1893). (7) LORENTZ: Wied. Ann. 12, 127, 860 (1881). (8) VAN DER WAALS,J . D.: Thesis, Leyden, 1873. (9) VAN DER WAALS,J. D . : Die Konlinuitdi des yasfbrmigeri und flussigen Zuslandes, Part 11. Barth, Leipzig (1900).
(1) (2) (3) (4) (5)
