J. Phys. Chem. 1995, 99, 4779-4786
4779
Condensed-Phase Transitions in Binary Systems during Dynamic Vaporization Experiments: Effusion and Transpiration Jimmie G. Edwards* Department of Chemistry, University of Toledo, Toledo, Ohio 43606
Hugo F. Franzen Department of Chemistry and Ames Laboratory-DOE, Iowa State University, Ames, Iowa 5001I Received: September 22, 1994; In Final Form: January 13, 1995@
During a condensed-phase transition at equilibrium in a vaporization experiment, three phases are present. In this paper, equations relating vapor pressure, temperature, and compositions of the vapor and condensed phases are derived for systems undergoing such transitions. Previously observed unusual phenomena, such as vapor pressures that increase at constant temperature and vapor pressures that increase with decreasing temperature, are explained. It is shown that equilibrium condensed-phase transitions in the presence of the vapor are always hysteretic in the temperature; the transition occurs at a higher temperature in the increasingtemperature direction than in the decreasing-temperature direction. The particular cases of effusion and transpiration experiments are treated in detail.
Introduction Effusion and transpiration experiments are the most often used methods for studying vapor arising from condensed phases at high temperatures. Static methods for studying vaporizing substances are difficult to apply at temperatures above room temperature, and at increasingly higher temperatures they become increasingly more difficult. Effusion and transpiration methods at high temperatures have the advantage of removing the vapor from an enclosure where it is practically at equilibrium with the condensed phase in a manner that allows determination of its pressure and examination of its properties, e.g., optical or mass spectrum, with apparatus at room temperature. These methods are called dynamic, as opposed to static, methods of vapor study. We treat here cases of both congruent and incongruent vaporization. In the former the overall elemental compositions of the vapor and condensed phases are the same, and in the latter they are different. In dynamic methods, when vaporization is incongruent, as is the case with many chemically binary systems, the composition of the condensed phase changes during the progress of a study of the vapor. If the vaporization is congruent, but a firstorder phase transition occurs in the condensed phase within the temperature range of the vaporization study, then during the course of the transition three phases are at equilibrium. It has been shown that the three phases in the case of a binary system must have different compositions. Therefore, during the course of the transition, vaporization must be incongruent as the composition of the system changes from that of the first condensed phase to that of the second. During such a phase transition, unusual and counterintuitive effects have been observed.2-6 For example, it has been observed that (1) the partial pressures of two vapor species change in opposite directions when the temperature is changed;2 ( 2 ) the vapor pressure increases when the temperature is decreased and vice versa;3(3) the vapor pressure in an effusion experiment increases at constant temperat~re;~ (4) the vapor pressure in an effusion
* Author to whom correspondence @
should be addressed. Abstract published in Advance ACS Abstracts, March 15, 1995.
experiment decreases and then increases at constant temperat ~ r e (5) ; ~ at a given temperature, the vapor pressure can have three different stable values, two of which result from congruent vap~rization;~.~ (6) the phase transition occurs at different temperatures in the increasing-temperature and decreasingtemperature directions, i.e., a hysteresis is present in the phase transition at equilibrium.6 This paper elaborates the unusual effects described above and presents a thermodynamic theory to explain them. Equations are derived to predict the effects in terms of chemical and thermodynamic properties of the condensed and vapor phases involved. The theory describes the hysteretic structure of the phase diagram in the vicinity of a condensed-phase transition during dynamic vaporization and the trajectory of a binary system through such a transition.
Effusion Methods In the Knudsen effusion method? the vapor from a condensed sample in a cell heated in vacuum effuses through a small orifice in the cell. The vapor pressure, PK,is calculated from the rate of mass loss through the orifice, dgldt, the area of the orifice, A,, the temperature of the cell, T, and the assigned molecular weight of the effusing vapor, M*,by
PK* = [dgldt][2~tRTlM*]’/~lA, in which the symbols PK* and M* emphasize that the value obtained for the vapor pressure depends on knowledge of the chemical composition of the vapor. If the vapor is composed of more than one gaseous species, then eq 1 must be written for each species, i:
PK(i)= [dg(i)ldt][2~tRT/~(i)l”~IA Equation 1 can be used even in the case of a vapor with N species if the molecular weight is expressed as the appropriate average, M K , derived from eqs 1 and 2:
0022-36S419S/2099-4779$09.00/0 0 1995 American Chemical Society
4780 J. Phys. Chem., Vol. 99, No. 13, 1995
Edwards and Franzen
N
(3) i= 1
wherefii) is the mass fraction of species i in the vapor. That is, if M* = MK the vapor pressure is correctly expressed. Differing molecular weights of vapor species can have a profound effect on the chemical equilibrium within an effusion cell. If the chemical properties of a binary substance require it to vaporize congruently, then the elemental composition of the effusing vapor (that leaving the cell) must be equal to that of the condensed phase. For the congruent vaporization reaction (4) the molar ratio of A(g) to B(g) in the effusing vapor is a/p. But since rate of effusion is inversely proportional to molecular weight, the ratio of partial pressures within the cell is
from which one can calculate the mole fraction, X L , of A(g) in the vapor within the cell:
chemical processes during vaporization. In the method, vapor in equilibrium with a condensed phase is carried by a carrier gas, usually at ambient pressure, to a remote location at room temperature where the vapor can be collected for analysis. The rate of flow of the carrier gas must be fast enough that vapor diffusion is unimportant but slow enough that the vapor remains effectively saturated. In the ideal case, the vapor pressure, PL is calculated from the rate of transport of mass from the vapor, dgldt, the volume rate of flow of carrier gas, dVldt, the temperature, and the assigned molecular weight of the vapor, M", by
P," = [(dg/dt)/(dV/dt)][RT/~]
(9)
As in the case of Knudsen effusion, knowledge of the chemical composition of the vapor is needed in order to obtain the vapor pressure. If the vapor is composed of N species, then the partial pressure of each species is given by P,(i) = { [dg(i)/dt]/[dV/dt]}[RT/M(i)] (10) and eq 9 must be written with the assigned molecular weight equal to the appropriate average, Mu, expressed by
rB
rB,
Since = 1 - X;, too, can be calculated with eq 6 . Momentum effusion methods,8typified by the torsion effusion method? involve measuring the rate of momentum transfer, i.e., recoil force, from an effusion cell containing a vapor in equilibrium with a condensed phase. The vapor pressure, PT, is calculated from the recoil force, FT, with
P, = 2FT/A
(7)
Vapor pressure from eq 7 is absolute, i.e., depends only on measurements of mass, distance, and time, and, therefore, is not dependent on knowledge of the chemical composition of the vapor. Equations 1 and 7 can be combined to obtain the apparent molecular weight of the vapor, Mapp,
Other applications of effusion methods include target collection,1° in which effusing vapor is condensed on a target and analyzed, and Knudsen-cell mass spectrometry," in which the vapor from an effusion cell is introduced into a mass spectrometer for analysis. Effusion cells are used as sources in a variety of molecular-beam applications,12inert-gas matrix syntheses, l 3 thin-film growth,14 etc. The effusion method has been found to be useful in the controlled preparation of novel refractory phases at high temperatures, typically 1500 K or above.15 In particular, the method has been used to prepare nonstoichiometric vanadium monosulfide,16 which exhibits a continuous symmetry breaking transition,17 and more recently a similar study with a similar result was carried out for defect zirconium monosulfide. In addition, the new stoichiometric materials TmsS11'~and Tml~S22~O and the very nearly stoichiometric Yg-xSe721 and L U ~ + ~were S ~ *prepared ~ by Knudsen vaporization. The preparation of these new materials depended fundamentally upon the very close control of change of stoichiometry afforded by the Knudsen effusion method. In this paper, for the first time, a detailed explanation of the phenomenology involved in such processes is provided.
Transpiration Method The transpiration method23has uses in CVD syntheses as well as in measurements of vaporization pressures and studies of
N
Me = {EHi)/M(i)]}-'
(11)
i= 1
That is, if M" = Mu the vapor pressure is correctly expressed. A treatment of the transpiration method is included here because, in addition to the method's experimental importance, the theoretical results developed below for transpiration aid in describing effects from vaporization by effusion. Treatment of vaporization by transpiration is useful, in particular, in gaining full understanding of congruent vaporization under the condition that the compositions of the condensed phase, the equilibrium vapor, and the vapor leaving the system are all the same. It will be seen, for instance, that the course of a condensed-phase transition in the increasing temperature direction during effusion carries the vaporizing system through a state identical to that of congruent vaporization of the high-temperature phase during transpiration.
Pressure- Composition Diagram Because the single control variable in most transpiration and effusion studies is the temperature, it is convenient to describe and examine the course of a dynamic vaporization experiment with isothermal pressure-composition phase diagrams of the vaporizing system at key temperatures covering the temperature range of the experiment. Such a diagram can be used to understand the course of vaporization at the temperature of the isotherm. Figure 1 shows eight such isothermal pressurecomposition diagrams, at temperatures T, through Th, for a binary, A-B, chemical system that undergoes a phase transition between condensed phases A and e within the range of the isotherms represented. Diagrams of this type have been described e l ~ e w h e r e . ~The , ~ ~vertical axis represents the total pressure of the vapor species exclusive of any inert carrier gas, and the horizontal axis represents the overall composition of the system. The temperature increases monotonically from Ta through Th. To facilitate comparison of the cases of effusion and transpiration, the diagrams have been constructed so that in both types of experiment two different congruent processes occur in different temperature regions and continuously transform from one to the other with changes in the state variables.
J. Phys. Chem., Vol. 99, No. 13, 1995 1781
Condensed-Phase Transitions in Binary Systems
A
P
I I
I I
P
3-0 0
0
TI
For present purposes, the following conventions and restrictions will be placed on chemical systems to be represented by Figure 1: (i) We consider condensed phases that vaporize to give principally two vapor species, A(g) and B(g), and these species are taken to be the components on the composition axis in the phase diagrams in Figure 1. (ii) The molecular weight of A is greater than that of B. (iii) The gases are ideal. (iv) The condensed phases are “line compounds”, i.e., neither vapor species is significantly soluble in either condensed phase. (v) The amount of condensed phase is much larger than the amount of vapor, so that the composition of the system is effectively identical to that of the condensed phase. Under this restriction, the state of a system in Figure 1 will always be located on one of the condensed-phase lines or on the horizontal tie-line connecting the two condensed phases and the equilibrium vapor. (vi) The molar volume of the vapor is much larger than the molar volume of the condensed phases, so that the vapor pressure in a transpiration experiment is unaffected by the pressure of the inert carrier gas. Some features applying only to effusion experiments have been included as broken and dotted lines in Figure 1. The vertical broken lines, labeled il and e, schematically represent vapor compositions, X i , that might be calculated from eq 6 for congruent effusion of the corresponding condensed phases. An equilibrium between a congruently effusing condensed phase and its vapor within the effusion cell is represented by each horizontal broken tie-line in the diagrams. Such lines connect the composition of the condensed phase with the composition X i of the equilibrium vapor in order to represent the required condition of the chemical system at equilibrium in an effusion cell. Here it is seen pictorially that the composition of a congruently effusing condensed phase and that of its equilibrium vapor are different. At the lowest temperature, T,, the final result of a dynamic vaporization experiment, without regard to the composition of the system at the start, would be congruent loss of phase by vaporization from the system. At the highest temperature, Th, the final result would be congruent loss of phase il by vaporization from the system. These results occur because for sample compositions between those of iland e the composition of the vapor leaving the system lies outside the range between those of 1 and e.
At intermediate temperatures the case might be different. For instance, at temperature Tfthe final result in a transpiration experiment could be congruent vaporization of either il or e. At initial compositions to the left of the euatmotic point,25i.e., the metastable maximum vapor pressure between A and e, the final result would be congruent vaporization of A, and from the right of the euatmotic point the final result would be congruent vaporization of e. These results occur because for sample compositions between those of I and the composition of the vapor leaving the system is that at the euatmotic point. Owing to effects of differing molecular weights of the vapor species, as expressed by eq 6, at intermediate temperatures the outcomes from effusion and transpiration can be different, as will be described subsequently. The isotherms Tb and Td are at the extreme temperatures of the phase transition in an effusion experiment, and isotherms Te and Tg are at the extreme temperatures in a transpiration experiment. In a transpiration experiment with phase e, vaporization is congruent at temperatures up to and including Tg.Above Tgthe transition occurs; vaporization is incongruent as 2 is formed while three phases are present, say at Th. After e is exhausted, the vapor pressure decreases at constant temperature to the minimum at isotherm Th, the phase transition is complete, and il vaporizes congruently. With congruently vaporizing /z in a transpiration experiment, no phase transition occurs at T,. As the temperature is lowered, the onset of the transition to e is at Te. When the temperature is lowered below T,, vaporization is incongruent as e forms, say at Td. After is exhausted, the vapor pressure decreases at constant temperature to the minimum at Td, the phase transition is complete, and e vaporizes congruently. The hysteresis in the phase transition is seen from this description; the temperature of the transition g il is higher than that of the transition il g. A similar but somewhat more complicated set of events occurs during a condensed-phase transition in an effusion experiment. In an effusion experiment with phase Q, effusion is congruent at temperatures up to and including Td. Above Td the transition occurs; effusion is incongruent as il is formed while three phases are present, say at the isotherms at T, - Th. After is exhausted, the direction of the vapor pressure change at constant temperature depends on the temperature. At temperatures within T d < T 5 Te the vapor pressure increases
-
-
Edwards and Franzen
4702 J. Phys. Chem., Vol. 99, No. 13, 1995
n"
H"
Transpiration 4'
P
P
S
5
4
\
- Effusion - - - - - - - - Transpiration
A"
1IT Figure 2. Schematic plots of logarithm of vapor pressure vs inverse temperature in a system vaporizing by transpiration projected onto a plane of arbitrq composition. Straight line HE: congruent vaporization of 1. Straight line GA: congruent vaporization of e. Curved line H"A": the three-phase equilibrium 1 4- e 4- vapor.
1I T Figure 3. Schematic plots of logarithm of vapor pressure vs inverse temperature in a system vaporizing by transpiration (dashed straight lines and the curved line) or effusion (solid straight lines and the curved line) projected onto a plane of arbitrary composition. Straight line H'B: congruent effusion of 1.Straight line DA': congruent effusion vapor. of e. Curved line H"A": the three-phase equilibrium, 1 e iCompare with Figure 2.
+
to that of the broken tie-line between A and its equilibrium vapor. At temperatures greater than Te the vapor pressure decreases at constant temperature to the minimum at the composition of A and then increases to the pressure of the broken tie-line between ,Iand its equilibrium vapor. At the minimum pressure the state of the system within the effusion cell is identical to that of congruently vaporizing I under transpiration conditions, i.e., the compositions of the condensed phase and its equilibrium vapor are the same. After the increase, the phase transition is complete, and A effuses congruently. With congruently effusing A in an effusion experiment, no phase transition occurs at Td. As the temperature is lowered, the onset of the transition to e is at Tb. When the temperature is lowered below Tb, say to Ta, effusion is incongruent as e forms. After A is exhausted, the vapor pressure decreases at constant temperature to the pressure of the broken tie-line at Ta, the phase transition is complete, and e effuses congruently. Again, the hysteresis in the phase transition is seen from this description; the temperature of the transition e ,Iis higher than that of the transition 11 8. But both transition temperatures are different from those in a transpiration experiment.
-
1249.0 K
2,wl
-
....
2.03
l
Pressure-Temperature Diagram With understanding gained from the pressure-composition isotherms in Figure 1, the trajectory of a binary system through a condensed-phase transition can be traced on a pressuretemperature diagram. Figures 2 and 3 show schematically such trajectories for transpiration and effusion experiments, respectively; both figures contain exaggerations of scale for the purpose of illustrating events during phase transitions, but the features in them are accurate. To see the features placed on accurate scales for reasonable sets of parameters, see Figures 4
1239.0 K
1239.5 K
8.062
'
l
8.064
'
l
8.066
'
l
8.068
'
1
8.070
'
1
8.072
'
1
8.074
104m
Figure 4. Plot of calculated results for logarithm of vapor pressure vs inverse temperature for the phases A and e vaporizing congruently by transpiration and effusion and for the three-phase equilibrium, 1+ e vapor. Enthalpy of transition = 20 kJ/mol. See text for other
+
parameters. and 5 . In Figures 2-5 the vertical (in the plane) axis is the logarithm (In) of the sum of the partial pressures of the vapor species, and the horizontal axis is the inverse of the temperature.
J. Phys. Chem., Vol. 99, No. 13, 1995 4783
Condensed-Phase Transitions in Binary Systems
,
2.125
1241.OK
,
’
1239.0K
1237.0K
8.25 Pa
-
2.1w
2.075
-
2.050
.7.75 Pa
3
2.025
-
2.ooo
.7.25 Pa
8.050
8.060
8.070
8.080
8.090
10%
Figure 5. Plot of calculated results for logarithm of vapor pressure vs
inverse temperature for the phases 1 and e vaporizing congruently by transpiration and effusion and for the three-phase equilibrium, 1 8 vapor. Enthalpy of transition = 8 kJ/mol. See text for other parameters.
+
+
The composition axis is vertical to the plane. In Figures 2 and 3 the solid diagonal straight lines represent equilibrium between a condensed phase and its vapor, e at lower temperatures and A at higher temperatures, while vaporization is congruent. The solid curved line represents equilibrium among e, A, and vapor (three phases, 3-P) during a transition while vaporization is incongruent. That the plot of In P vs 1/T of such a three-phase equilibrium is distinctly curved where the vapor and condensedphase compositions change rapidly with temperature is shown by equations derived subsequently and has been observed previously.26 Application of the phase rule yields a variance of unity in the case of incongruent vaporization and in both cases of congruency. In the case of incongruence, there are three phases, A, e, and vapor, at equilibrium; in the cases of congruency there are two phases with an additional required relationship between the compositions of the condensed and vapor phases. Therefore, the pressure is a function of the temperature only, and three curves of pressure vs temperature (or better, In P vs llT) result, corresponding to congruent vaporization of A, congruent vaporization of e, and incongruent vaporization of A and e. These P(T) lines correspond to different sample compositions, respectively, that of A, X(A), and that of e, X ( g ) , on the straight lines and a continuous range of compositions including that from X(A) to X ( e ) on the curved line. Thus Figures 2 and 3 are projections along the composition axis onto an arbitrary plane of fixed composition; under the conditions of very nearly reversible vaporization, the lines intersect only at isolated points, the A v line with the A e v (3-P)curve at E in transpiration and B in effusion, the e v line with the 3-P curve at G in transpiration and D in effusion, and the A v line and the e v line not at all. The straight lines relating to transpiration in Figures 2 and 3, HE and GA, cross at point X but do not intersect, because they lie at different compositions, viz., those of the condensed phases. By the same token, straight line H’B and curved line H”A” in Figure 3 cross at point Y at different compositions but do not intersect. It has been observed that the phase transitions discussed here, when viewed as functions
+ + +
+
+
+
of temperature only, can be classified as topological fold catastrophe^.^ A pair of intersections, E and G or B and D, then, is a catastrophe set, i.e., the boundary of the onset of the fold catastrophe. Equations for the effusion catastrophe set have been derived.6 In Figure 2 for a transpiration experiment, phase A is in equilibrium with its vapor on line HE, and phase e is in equilibrium with its vapor on line GA. All three phases, A, e. and vapor are at equilibrium along the curved line H”A”. Similar interpretations apply to Figure 3 for an effusion experiment. For reference, lines HE and GA are included as dotted lines in Figure 3. Lines H’B and DA’ in Figure 3 lie at higher pressures at a given temperature than do lines HE and GA in Figure 2, as can be deduced by examining Figure 1, owing to effects from differing molecular weights of the vapor species, A(g) and B(g). The equation relating these pressures is complicated in the general case and is given e l ~ e w h e r e . ~ , * ~ In an effusion experiment, phase A is in equilibrium with its vapor on line H’B, phase e is in equilibrium with its vapor on line DA’, and all three phases, A, e, and vapor are at equilibrium along the curved line H”A”. The curved line H”A” is the same in both transpiration and effusion experiments; in a twocomponent system with three phases at equilibrium, the vapor pressure and compositions of all phases are fixed at a given temperature. Interpretations of stability relationships in the open areas in Figures 2 and 3 must be done with caution because of restriction v; the composition is always effectively that of the condensed phase. Therefore, for instance, an interpretation that below the segmented line HXA in Figure 2 only vapor is present would not be meaningful in the present context. The trajectory of the state of a binary system through a phase transition during a transpiration experiment can be traced on Figure 2. If an experiment is begun with phase e at low temperatures and the temperature is increased, the state will be on line GA until the temperature at point G, TG,is reached. If TGis never exceeded, the sample will be exhausted on line GA, so long as restriction v can be maintained. Above TG,phase A will begin to form, three phases will be present, and the state of the system will follow line H”A”. If, while three phases are present, the temperature is lowered below TG, the system follows line H”A”. But on line H’’A”, vaporization is always incongruent, the composition of the system changes with time, and eventually the state will move isothermally to line HE if the temperature is above TG and to line GA if the temperature is below TE. If the temperature is between TG and TE, the composition of the vapor is that at the euatmotic point, e.g., at Tfin Figure 1, and the line in Figure 2 to which the pressure drops at a given temperature depends on the composition of the sample when the isotherm is set. A similar sequence of events occurs if congruently vaporizing I. with its state on line HE is cooled to a temperature below TE. On HE the sample will be exhausted without intensive change so long as restriction v can be maintained. The trajectory of the state of a binary system through a phase transition during effusion can be traced on Figure 3. With phase e at low temperatures, as the temperature is increased, the system will be on line DA’ until the temperature at point D, TD, is reached. If TD is never exceeded, the sample will be exhausted on line DA’, so long as restriction v can be maintained. Above TD,phase A will begin to form, three phases will be present, and the system will follow line H”A”. If, while three phases are present, the temperature is lowered below TD, the system follows line H’’A”. But on line H”A”, vaporization is always incongruent, the composition of the system changes with time, and eventually the system will move to line H’B if
Edwards and Franzen
4784 J. Phys. Chem., Vol. 99, No. 13, 1995
the temperature is above TDand to line DA' if the temperature is below TB. If the temperature is between TDand TB,marked by vertical dotted lines in Figure 3, the composition of the effusing vapor is between that of A and that of e, e.g., at Tc in Figure 1, just as the composition at the euatmotic point is in < T the same range between TG and TE. Then in the range Td the line in Figure 3 to which the pressure drops at a given temperature depends on the composition of the sample when the isotherm is set. That is, the direction of isothermal pressure change between the vertical dotted lines might be either up, from line DB to line H'B, or down, from line DB to line DA'. A similar sequence of events occurs if a congruently vaporizing system with state on line H'B is cooled to a temperature below TB. On H'B the sample will be exhausted without intensive change so long as restriction v can be maintained. The direction of vapor-pressure change during isothermal transition from the 3-P equilibrium (line ."A") to a two-phase equilibrium can be different in an effusion experiment from that in a transpiration experiment. During transition from the 3-P equilibrium to the equilibrium between g and vapor on line DA', the vapor-pressure change is always negative, as in a transpiration experiment. However, transition to equilibrium between A and vapor at temperatures between TE and TB results in an increase in vapor pressure. Above TE,the vapor pressure f i s t decreases to that on line HE and then increases to that on line H'B. Trajectories with these characteristics in effusion experiments have been observed and r e p ~ r t e d . ~
Theory The temperatures of a condensed-phase transition in an effusion experiment (the effusion catastrophe set) have been calculated elsewhere.6 Here we derive equations for the transition temperatures in both transpiration and effusion experiments and derive a differential equation to express the slope of the vapor pressure as a function of temperature when both condensed phases and the vapor are at equilibrium. We integrate this differential equation for an ideal but reasonable case. We assign chemical formulas to phases A and in terms of the components of their congruent equilibrium vapor, A(g) and B(g). The assigned formula of A is A&,(c) and that of @ is A,Br(c). The congruent vaporization of each phase is represented by eq 4 with the appropriate respective stoichiometric coefficients. We adopt the conventions in writing the formulas that a b = 1 and e + f = 1, so that 1 mol of vapor is always produced in the congruent vaporization equations. As examples we note that under these conventions the formula of SrS(s), which vaporizes congruently to produce Sr(g) and s ~ ( g ) , would ~' be sro.667(S2)0.333(S), for which eq 4 would be
+
and that of InsSes(l), which vaporizes congruently to produce InzSe(g) and Se2(g),28would be (In~Se)o,sss(Se2)0,4~2(1), for which eq 4 would be ( 1 n 2 s e ) 0 , ~ 8 ~ ( s e 2 ) 0 , 4 ~ 2 ( 1= ) 0.5881n2Se(g)
+ 0.412Se2(g) (13)
This formula-writing conyention is adopted for present convenience and clearly carries no implications about structures in the condensed phase. The congruent-vaporization reactions, reaction 4, of phases ;3. and e are respectively A,Bb(A) = aA(g)
+ bB(g)
(14)
and their equilibrium constants are, respectively,
Introduction of the Gibbs-Helmholtz equation and algebraic rearrangement of these equations yields the forms
R In P = - W , ( A ) / T
+ AS",(A) - R ln[(XL)"(1 - x3b] (18)
R In P = - W , ( g ) / T
+ AS",(@)- R ln[(XL)'( 1 - G Y ] (19)
which express the vapor pressure as a function of temperature when A or g, respectively, vaporizes congruently. AWv(A), AS",@), AW,(e), and AS",@) are enthalpies and entropies, respectively, of vaporization of phases 13. and e by reactions 14 and 15. The value of X l is calculated in each case for the appropriate type of experiment, viz., from eq 6 for effusion and as the stoichiometric coefficients themselves (a or e ) for transpiration. Other chemical equations of interest in the present consideration are the two simultaneous incongruent vaporization reactions [a/(a - e)]A,BXc) = [e/(a- e)]A,B,(c)
+ B(g)
(20)
balanced so 1 mol of gas is produced in each, and the transition reaction
and its reverse, balanced with 1 mol of gas on each side. The last restriction is vital, because eq 22 is a composite of eqs 20 and 21, two independent net reactions; it is written and used here for algebraic convenience only. Without the last restriction, eq 22 would have no unique set of balancing coefficients. For a given isotherm in Figure 1, the equilibrium constants in eqs 16 and 17 are fixed, and thus these equations represent the vapor-composition lines on the isotherm:
Equation 23 gives the curve representing vapor in equilibrium with phase A, and eq 24 gives the same for phase g. These two curves intersect at the vapor pressure and vapor composition of the 3-P equilibrium of A, g, and vapor. Thus, if we equate the pressure from eqs 23 and 24, we obtain an equation expressing the pressure-temperature-composition surface of the 3-P equilibrium:
Kp(e)/Kp(A) = [(Xl)'(1 - x ~ m x l ) "
