Thermodynamic representation of high-pressure vapour-liquid equilibria

Chemical l3ngineering Science, 1963, Vol. 18, pp. 613-630. Pergamon Press Ltd., oxford.

Printed in

Great Britain.

Thermodynamic representation of high-pressure vaponr-liqnid equilibria J. M. PRAUSMTZ Cryogenic Engineering Laboratory, National Bureau of Standards, Boulder, Colorado and apartment of Chemical Engineering, University of California, Berkeley 4, California (Received 10 January 1963; in revised form 1 April 1963) Abstract-The purpose of thermodynamic analysis of high-pressure equilibrium dam is outlined and techniques for performing such analysis along an isotherm are proposed. In the liquid phase, the activity coefficient of the heavy component is referred to the pure component but the activity coefficient for the light component is referred to the infinitely dilute solution; thus both activity coefficients approach unity as the mole fraction of the light component approaches zero. By adjusting all activity coefficients to correspond to a fixed pressure, they must satisfy the isothermal, isobaric Gibbs-Duhem equation without corrections for volume change on mixing; as a result it is possible to use well-known integrated forms of the Gibbs-Duhem equation to give analytical representation of the adjusted activity coefficients as a function of composition. It is shown that the general treatment proposed here reduces to the Krichevsky-Kasamovsky equation for very dilute solutions. The proposed analysis is applied to phase equilibrium data for the butane-carbon dioxide system up to the critical composition and it is shown that at 160°F the activity coefficients are well represented by the two-constant Van Laar equation; at 280°F a one-constant equation is sticient. A comparison is made between the observed adjusted activity coefhcient for butane and that calculated by various theories of solutions due to LONCXJET-HIGGINS,to Paroooor~~ and to Scorr. Only the last of these gives rough numerical agreement with the experimental results but it cannot account for the sharp curvature which is observed iu the vicinity of the critical point when the adjusted activity coefficient is plotted against composition. On the basis of the thermodynamic analysis, semi-empirical techniques are proposed for estimating high-pressure vapour-liquid equilibrium data in simple mixtures from a minimum of experimental data; these techniques are illustrated for the systems nitrogen-methane and methaneethylene.

been customary for many years to subject vapour-liquid equilibrium data at low pressures to thermodynamic analysis but this custom has not as yet been extended to vapour-liquid data at high pressures. Although the chemical literature is rich in experimental studies of the phase behaviour of binary mixtures at high pressures, very little attention has been given to the problem of how the experimental results may be meaningfully reduced with the aid of suitable thermodynamic functions. While numerous textbooks on chemical thermodynamics as well as recent articles in the chemical engineering literature deal generously with various techniques and their refinements for treating lowpressure equilibrium data, these sources tend inevitably to avoid discussion of how high-pressure equilibria are to be treated quantitatively. Even the tie new second edition of a classic text in thermodynamics [l] has very little to say on this subject. The aim of this paper is to show how thermoIT HAS

dynamic analysis can be quantitatively applied to the reduction of high-pressure vapour-liquid equilibrium data for simple systems including data near or at the critical point, to discuss briefly the relationship bettieen high-pressure equilibria and some contemporary theories of solutions, and to suggest some possible techniques for estimating highpressure phase behaviour from a minimum amount of experimental information. The discussion is limited to binary systems but some suggestions for the treatment of multicomponent systems are made in Appendix II. USESOF THERMODYNAMICANALYSIS Thermodynamics is a tool for understanding and thus the final aim of thermodynamic analysis of solution data is to make comprehensible, to order, categorize and interpret the phase behaviour of analysis mixtures. At present, thermodynamic

613

J.

M, PRAUSNITZ

serves two purposes, the ultimate and the proximate. The ultimate purpose is to establish relationships between macroscopic properties and intermolecular forces; at present such relationships are known only for a limited number of situations which can be successfully treated by the methods of statistical mechanics. The proximate purpose is to relate experimental results to a small number of well-behaved mathematical functions which can be expressed analytically with a minimum of constants, which permit smoothing and interpolation of data, which enable the data to be tested for thermodynamic consistency, and which facilitate meaningful correlation and. interpretation of phase behaviour. The statistical problems in formulating a theory of multi-component liquid systems at conditions up to the critical point are so formidable that the ultimate purpose of thermodynamic analysis for this case is not likely to be achieved for a long time. However, the time is well ripe for attaining the proximate purpose and there are two important engineering reasons for doing so. The first reason is due to the fact that it is extremely difficult to interpret or correlate raw equilibrium data. For example, in a two-phase system a raw equilibrium datum, e.g. a K value, depends on the non-idealities of two phases; each of these non-idealities, in turn, depends on the three primary variables, temperature, pressure and composition. Thus in a binary system, the K value depends on six determining factors: the variation of chemical potential (or fugacity) with temperature, pressure and composition in each phase. Hence it is essentially impossible to correlate K data directly unless one restricts one’s attention to a class of highly similar chemical substances. However, by introducing thermodynamic analysis the various contributions to the raw datum point are first separated from one another and then, by considering each of these contributions separately, interpretation and correlation may be achieved. It is worth while to remember that thermodynamic analysis of equilibrium data is completely analogous to partial differentiation of a function of many variables. The second reason is related to the fact that the number of binary fluid systems which are or may rather soon become of interest in technological

processes is already very large and if one thinks of the possible number of ternary, quaternary . . . etc. fluid systems which may conceivably find their place in chemical technology the number becomes essentially infinite. To investigate experimentally even a small fraction of these systems under varying conditions of temperature and pressure represents an effort which is prohibitive in both cost and time. The only reasonable long-range approach to increasing our quantitative knowledge of phase behaviour for engineering purposes, therefore, is to study experimentally only a few representative systems; on the basis of such studies one must attempt to generalize the results in such a manner that reasonable predictions can be made for the behaviour of previously investigated systems under new conditions, or even for the behaviour of new systems which have not been investigated at all. It is in achieving this generalization that thermodynamic analysis plays its most useful role. ACTIVITYCOEFFICIENTS ANDREFERENCE STATES Consider the vapour-liquid equilibrium of a binary system at elevated pressure. Usually, one of the components is at a temperature above its critical ; the temperature T of the system is such that

T < Tf T > T;

(1) Gw

where superscript c stands for critical and where subscript 1 designates the heavy, and subscript 2 the light component. The equations of equilibrium state that for each component the fugacity is the same in both phases. The tist task of thermodynamic analysis is to relate the fugacity of a component in a given phase to its concentration in that phase; for the vapour phase this is done through the fugacity coefficient +i: fi = 4iYiP

(i = 1,2)

(3)

where yi is the mole fraction of component i in the vapour phase at total pressure P. The fugacity t Equation (2) is not essential to the analysis proposed here. The analysis will be useful whenever the solution temperature T id close to or above the critical temperature Tf.

614

ThermodynamicIrepresentation

of high-pressure vapour-liquid equilibria

coefficient can be calculated from an equation of state for the vapour phase; this calculation has been amply treated in the literature [2-4] and need not be discussed here. For the liquid phase satisfactory equations of state are not available and thus the relationship between fugacity and composition is more conveniently given by the activity coefficient yi: fi

=

Y$i.fi

(i = 1,2)

properties of the light component but also depends strongly on the properties of the heavy component; therefore this reference fugacity for a given solute is also a function of the solvent. GIBBS-DUHEM EQUATION AND THE EFFECT OF PRESSUREON ACTIVITY COEFFICIENTS

The activity coefficients of all the components in a multi-component solution are not independent but are related by the Gibbs-Duhem equation which at constant temperature and pressure has the form 7 Xi d In yi = 0

(4)

where xi is the mole fraction of component i in the liquid phase and fr is some (arbitrary) reference fugacity. For the heavy component f: is usually taken to be the fugacity of pure liquid 1 at the temperature of the solution and at some specified pressure. However, for the light component the corresponding definition for f; is usually not possible unless one is willing to extrapolate the properties of pure liquid 2 beyond its critical temperature. For some applications such a hypothetical pure state can be useful, especially if thermodynamic analysis is only desired for a set of closely related systems, but for general analysis, utilizing on!y a minimum of assumptions, such a state is always unsatisfactory because it is usually not possible to specify a consistent set of properties for that state, and because there is no assurance that for this hypothetical pure state the desired relationship

Now, for a binary two-phase system the phase rule states that at constant temperature it is impossible to vary the composition without also varying the pressure. In thermodynamic analysis of lowpressure vapour-liquid equilibria the small effect of nressure on the activity coefficients is often neglected entirely or it is taken into account approximately. At high pressures, the effect of pressure on the activity coefficient is large and thus, if the Gibbs-Duhem equation at constant temperature and pressure is to be used, all isothermal activity coefficients must be corrected to the same pressure. If the standard state is defined at a fixed pressure, this correction is given by the equation

limit yZ = 1

a In yi

will always be attained, regardless of the nature of the other component. A more satisfactory procedure is to define the reference fugacity for the light component by fi f; z limit - = H x2+0 x2

pi

( ap 1./ii7

as x2 + 1

(5)

The reference fugacity defined by Equation (5) is generally known as the Henry’s law constant and is commonly given the symbol H. The advantage of using this reference fugacity is that it is unambiguously defined, being derived from real rather than imaginary physical data; hence, for a given situation there can never be any question about its numerical value. The disadvantage of this reference fugacity is that it depends not only on the

(7)

where hi is the partial molar volume of component i in the liquid phase at composition x and at temperature T. In order to satisfy equation (6) it is necessary to adjust the activity coefficient through equation (7) in such a way that it is a function only of composition, for if this is done two advantages are obtained. First, the Gibbs-Duhem equation and its various integrated solutions may be applied to these pressure-independent activity coefficients thus enabling us to subject them to a test for thermodynamic consistency and to express them analytically by simple mathematical functions. Second, by separating from each other the effects of composition and pressure on the activity coefficient, interpretation and correlation of the equilibrium data are

615

J. M. Pa~usmz

From equation (10) it again follows that

very much facilitated. Therefore it is useful to define the adjusted (pressure-independent) activity coefficient as follows: For the heavy component

fl

1

pure

s P

=~eexpY’P”’

yz*‘r-)+ 1 as

-

%iP

(8)

PRT

In equation (8) all quantities are evaluated at the temperature T of the solution. The fugacity fi is for component 1 at the composition x1 and at the total pressure P, but f&, is the fugacity of pure liquid component 1 at the (arbitrary) reference pressure P” and ~7,is the partial molar volume of component 1 at the composition x1. The adjusted activity coefficient 7: is independent of the total pressure of the solution for any isotherm; it depends only on the composition and always refers to the reference pressure P”. From equation (8) it follows that regardless of the choice of P” (9)

E

-

f2

x,&F’

p s

exp -

a, dP

po

RT

(10)

In equation (10) all’ quantities are evaluated at the temperature T of the solution. The fugacity fi is for component 2 at the composition X, and at the total pressure P while H is the standard state fugacity (Henry’s law constant) evaluated at the reference pressure P”$ The partial molar volume of component 2 is evaluated at the composition x,. The purpose of the asterisk * is to indicate that this activity coefficient for component 2, unlike that for component 1, does nor approach unity as the mole fraction of component 2 approaches one. The adjusted activity coefficient yz(‘) is also independent of the total pressure; for any isotherm it depends only on the composition and always refers to the reference pressure P”. $ The effect of pressure given by the equation alnH

(3

ap

on the Henry’s law constant

is

&a”

~=iiT

where 17s~ is the partial molar volume of component infinite dilution.

2 at

(x2 -+ 0)

(11)

regardless of the choice of P”. The normalization relations given by equations (9) and (11) are desirable boundary conditions for integration of the Gibbs-Duhem equation. Some further discussion on the Gibbs-Duhem equation for high-pressure equilibria is given in Appendix I. CHOICE OF REFERENCEPRESSURE

For the light component the definition of the adjusted, pressure-independent activity coefficient is y;(p”)

xi+1

The reference pressure P” is completely arbitrary; this arbitrariness, however, in no way influences the fact that the adjusted activity coefficients defined above must satisfy equation (6). The partial molar volumes in the integrals of equations (8) and (10) are for the liquid phase and thus, for physically meaningful results, the reference pressure P” should be such that the liquid phase can exist over the entire pressure range P + P”; this means that P” must be equal to or larger than the highest observed pressure along the isotherm under consideration. In engineering work, however, this requirement is not practical; in performing typical calculations it is easiest to choose as the reference pressure the system pressure when x2 = 0, i.e. the saturation (vapour) pressure Pi of the heavier component. This reference pressure is convenient because it is the one at which the Henry’s law constant is evaluated; it is a pressure which depends only on the properties of the solvent and not on those of the solute. Unfortunately, however, PS, is the minimum rather than the maximum pressure along a given isotherm and thus the integration path from P to P; is in a hypothetical region since the liquid phase of a given composition cannot exist at any pressure less than the total pressure of the two-phase system. For engineering purposes this is not a serious deficiency since the partial molar volume is almost never known as a function of pressure and thus, as a matter of necessity, the partial molar volumes in the integrals must be considered as functions of composition and temperature only. In making comparisons between observed activity coefficients and those predicted by theories of solution, hypothetical regions should be avoided but

616

Thermodynamic

representation

of high-pressure

for practical engineering purposes it is probably most convenient to use a reference pressure given by P” = p”1

(12)

The adjusted activity coefficients denned by equations (8) and (10) form the basis of the thermodynamic analysis of high-pressure vapour-liquid equilibrium data. The data should be plotted in the form of P - y and P - x isotherms. Using volumetric data for the vapour phase (or a suitable equation of state) the fugacity coefficients for the vapour phase should be computed yielding the fugacities as indicated by equation (3). From volumetric data for the liquid phase the partial molar volumes are then found. The activity coefficients for the liquid phase are then calculated by equations (8) and (10). These activity coefficients must satisfy the Gibbs-Duhem equation (equation 6) if the data are thermodynamically consistent. Further, these activity coefficients can then be expressed by simple analytical functions which are solutions to the Gibbs-Duhem differential equation. This procedure is illustrated a little later. First, however, it is instructive to consider a large simplification in the procedure outlined above which gives a very useful result applicable to those high-pressure systems where the liquid phase is always dilute with respect to the light component.

vapour-liquid

equilibria

which was fist derived by completely different arguments by KRICHEVSKY and KAMRNOVSKY[5]. The assumptions which lead to equation (14) clearly delineate the conditions under which the Krichevsky-Kasamovsky equation may be expected to be valid: the liquid phase must be dilute with respect to the light component and the solution must have low compressibility. Just what “dilute” means depends a little on the system involved, but in general x2 should not exceed a few one-hundredths. To illustrate the applicability and limitations of equation (14) consider the solubility data for nitrogen in liquid ammonia reported by WIEBE and GADDY[6] up to 1000 atm. A semilogarithmic plot of the ratio of fugacity to mole fraction versus total pressure is shown in Fig. 1 for two isotherms. The Krichevsky-Kasarnovsky equation gives an excellent representation of the data at 0°C but at high pressures there is a significant deviation for the data at 70°C. The reason for this behaviour becomes clear when we note that the equilibrium mole fraction of nitrogen at 1000 atm and 0°C is 0.0221 while that at 1000 atm and 70°C is 0.129. The range of data at O”C,therefore, represent a truly dilute solution throughout, but the range of data at 70°C do not. Thus equation (13) is a good assumption at 0°C up to 1000 atm but it is not a good assumption at 70°C for pressures exceeding a few hundred atmospheres.

A USEFULSIMPLIFICATION: THE KRICHEVSKY-KASARNOVSK~ EQUATION If we consider a liquid phase wherein the light component is dilute, we may suppose that each molecule of the light component is, at all times surrounded by molecules of the heavy component; in other words, if the solution is sufficiently dilute, molecules of light component are only very rarely close enough to interact. The thermodynamic equivalent of this statement is to say that over the concentration range under consideration, $‘W = 1 and ij2 = fi; (13)

-KAICHEVSKY-KASARNOVSKY -*-

If we further assume that the partial molar volume of component 2 does not depend on the pressure, then equations (lo), (12) and (13) give lnfZ=lnH+

iTP(P - Pg

RT

EQUATION

DATA OF WIEBE AND GADOY

t I

I

lo3 too 200 SUCCESS

(14)

I

I

I

AND FAILURE SOLUBILITY

617

I

700 Atm.

I

800

OF THE KRICHEVSKY-KASARNOVSKY OF NITROGEN IN LIQUID AMMONIA

FIG. 1.

x2

I

600 300 -iOO 500 TOTAL PRESSURE,

I

900

I

loo0

EQUATION.

J. M. PRAUSNITZ

One interesting feature of the KrichevskyKasarnovsky equation is that it predicts a maximum in the solubility as the pressure becomes very large. From equation (14) it can readily be shown that at constant temperature and at constant vapour composition a; - 62” =---

(15)

RT

where ti$ is the partial molar volume of the light component in the gas phase. According to equation (15) the solubility increases with pressure as long as zY;> fi; but decreases when fi$ < a$‘. This prediction of the Krichevsky-Kasarnovsky equation is verified by the data of BASSETand DODD [7] for the solubility of nitrogen in water at 18°C up to 4320 atm. Even at these very high pressures the mole fraction of nitrogen in water is small and thus equation (14) applies. The experimental results show a maximum in the solubility near 3000 atm. This pressure is in rather good agreement with that calculated by equation (15) using volumetric data for gaseous nitrogen [8]. The Krichevsky-Kasarnovsky equation has been used successfully to represent solubility data for hydrogen in hydrocarbons [9], for light hydrocarbons in water [lo, 1l] and for helium in liquefied nitrogen and methane [12]. ANALYSISOF THE BUTANE-CARBON DIOXIDE SYSTEMUP TO THE CRITICALPOINT To reduce experimental equilibrium data to the adjusted activity coefficients defined by equations (8) and (10) the following information is needed for each isotherm : (a) P - x - y data from x2 = 0 to x2 = xi (b) P - 2iL- x data for the liquid phase from x.2 = 0 to x2 = x; (c) P - ag - y data (or an equation of state) for the gaseous phase. Unfortunately there are very few systems for which all of the necessary experimental information has been reported. Item (a) is available for many binary mixtures; item (c) for some and item (b) for very few.

One binary system for which all of the required data have been reported is the butane-carbon dioxide system [13, 141. The critical temperatures for butane and carbon dioxide are 305.5 and 87.8”F, respectively, and the phase behaviour of this mixture has been reported for the range lOO-280°F. To illustrate the thermodynamic analysis of highpressure vapour-liquid equilibria, adjusted activity coefficients have been calculated for this system at 160°F and 280°F. These two temperatures were chosen because they correspond to rather different physical situations. At both temperatures carbon dioxide is well above its critical temperature and hence behaves as a typical gas. However, the behaviour of butane is not the same at these two temperatures; at 160°F butane is still a reasonably well-behaved liquid, remote from its critical temperature with a density only slightly less than that at its normal boiling point. Hence at this temperature the introduction of highly volatile carbon dioxide into only moderately volatile liquid butane very much “disturbs” the butane and causes it to exhibit large positive deviations from ideal behaviour. In other words, the activity coefficient of butane shows a large rise with increasing carbon dioxide concentration. Further, a rather large amount of carbon dioxide must be introduced into the butane before it is sufficiently “gas-like” to reach a critical point; in other words xz is large. On the other hand, at 280°F the behaviour is rather different. At this temperature butane is already a highly extended liquid whose volatility and molar volume are considerably larger than those prevailing in the “normal” liquid range at or below the atmospheric boiling point. At this temperature, the introduction of carbon dioxide is much less “disturbing” and the activity coefficient of butane shows only a modest rise with increasing carbon dioxide concentration. At this higher temperature liquid butane is already well on its way to possessing “gas-like” properties and therefore, since not very much of the volatile carbon dioxide is needed to reach the critical point, x; at 280°F is considerably smaller than that at 160°F. The adjusted activity coefficients for this system were calculated according to the method briefly outlined before. The fugacities of both components along the two-phase co-existence curve are

618

Thermodynamic

representation

of high-pressure

determined from the volumetric data for the gaseous phase. The partial molar volumes for the liquid phase cannot be determined exclusively from volumetric data along the two-phase co-existence curve because the pressure varies with the composition. The additional information which is needed is the variation of liquid volume with pressure at constant temperature and composition. The partial molar volumes are calculated from the rigorous expressions

and

with

where us is the liquid molar volume of the solution at saturation. Partial molar volumes for the butane-carbon dioxide system at 160°F are shown in Fig. 2. At low concentrations of carbon dioxide the partial molar volumes do not change very much with

2 -3 0

I 0.1

I 0.2 MOLE

0.3 FRACTION

I 0.4

, 0.5

CARBON

0.6

I

CRITICAL

0.7

0.8

DIOXIDE

vapour-liquid

equihbria

composition. However, the change becomes very large near the critical conditions. The qualitative behaviour illustrated by Fig. 2 can readily be understood if one recalls that the partial molar volume of component i represents the volume change which is experienced by the entire liquid mixture when a very small amount of pure component i is added at constant temperature and pressure. At compositions near the critical, the system moves rapidly from “gas-like” to “liquidlike” properties; it is already close to the transition from liquid to gas and is very sensitive to small changes in its constitution. Hence it is not at ah surprising that the addition of a small amount of highly volatile carbon dioxide shifts the behaviour of the system towards “gas-like” properties and thus fiz is large and positive. Similarly, the addition of low-volatile butane shifts the behaviour of the system towards “liquid-like” properties and thus ZJ is large and negative. The Henry’s law constant is easily calculated using its definition as given in equation (5). The adjusted activity coefficients were then calculated as indicated by equations (8), (10) and (12) with the necessary simplifying assumption that the partial molar volumes depend only on the composition and not on the pressure. 0 The results are shown in Figs. 3 and 4 and the adjusted activity coefficients for butane show the type of behaviour described earlier. The adjusted activity coefficients for carbon dioxide are a little more difficult to interpret because these coefficients were normalized not with respect to the pure component, but rather with respect to the infinitely dilute solution. For small x2, the adjusted activity coefficient is near unity as expected since under dilute conditions Henry’s law (or more accurately the Krichevsky-Kasarnovsky equation) should apply. As xt rises, the coefficient falls. This follows from the fact that for mixtures of nonpolar molecules the adjusted activity coefficient (and hence the non-ideality with respect to that component) is always a maximum when that component is present in vanishing concentration

since

it is under

these

conditions,

when

the

0 It is essentially impossible to avoid this simplifying assumption if equation (12) is used since P18 < P, and since the liquid phase cannot exist at pressures lower than the equilibrium (saturation) pressure P.

PARTIAL MOLAR VOLUMES lb4 THE SATURATED LIQUID PHASE OFTHE BUTANE-CARBON DIOXIDE SYSTEM AT 160 OF FIG. 2.

619

J.M. hAUSNITZ

1.2, 1.0

I

c

-

I

A general technique for obtaining analytic expressions for activity coefficients of liquid mixtures has been given by W~HL [15] ; the main idea of this technique is to define an excess free energy and then to expand this excess free energy in a series of algebraic functions of the mole fractions. The same technique can be used for solutions of gases in liquids but the physical meaning of the excess free energy is now quite different since the standard state for the light component refers to the intkitely dilute solution rather than to the pure liquid as in WOHL’Scase. As a result, the nature of the algebraic expansion for gas-liquid solutions is also different from that for liquid-liquid solutions. The excess free energy GE at constant temperature T and at constant reference pressure P” is defined by GE = G(mixture) - G(idea1 mixture) (18) G(mixture) = x1 In y\pO)~,+ x2 In yf(%, +

I!

I

I

CALCULATED FROM VAN LAAR EDUATION WITH A=27.7, 8=2.66 .

CALCULATED FROM DATA OF OLD?., REAMER, SAGE AND LACEY

/

/

n,RT

1.0

0

w4 2!!z + RT + RT

I 0.1

O.E1

I 0.2

0.3

0.4

0.5

0.6

0.7

0.8

x2 ADJUSTED ACTIVITY COEFFICIENTS FOR BUTANE (1) AND CARBON-DIOXIDE (2) AT 160s AND 120.6 PSIA

I -

0.6

-

t

v)

E

0.4-

l

FIG.3.

molecules of that component are completely surrounded by molecules of the other component, that it feels most “uncomfortable”. As x2 rises, the carbon dioxide molecules in the liquid phase begin to interact more and more with other carbon dioxide molecules and less and less with butane molecules. Therefore, the “discomfort” (and hence the adjusted activity coefficient) of the carbon dioxide molecule falls.

I

1

CALCULATED FROM EOUATIONS (28) 8 (29) WITH C =36

H= 2090

PSIA

-

ANALYTIC REPRESENTATIONOF ACTIVITYCOEFFICIFINTS

The adjusted’activity coefficients given by equations (8) and (10) were defined in such a manner that they must satisfy the Gibbs-Duhem equation (equation 6). Therefore, a suitable, integrated form of the Gibbs-Duhem equation should be useful for analytic representation of these coefficients as determined from the experimental data. 620

X2 ADJUSTED ACTIVITY COEFFICIENTS BUTANE ( I) AND CARBON DIOXIDE(21 AND 436 PSIA

FIG.4.

,

ii

CALCULATED FROM DATA OF OLDS, REAMER, SAGE AND LACEY

-0.4 -0.6 G.Q:z’-0.8

(19)

FOR AT 2BO“F

Thermodynamicrepresentation of high-pressurevapour-liquid equilibria G(ideal mixture) n,RT

The adjusted activity coefficients can be directly found from equation (22) by using the familiar relations

= x1 In x1 + x2 In x2 + -w4 35 + RT + RT

(20)

where & = chemical potential of component 1 as a pure liquid P2m= chemical potential of component 2 in infinitely dilute solution nT = total number of moles in the solution. Substitution of equations (19) and (20) into equation (18) gives GE nTRT

= xi In y$‘“)+ x2 In #”

(21)

As given in equation (21), the excess free energy vanishes when x2 = 0 but not when x1 = 0. This follows because the ideal mixture to which the excess free energy refers is not one where both components obey Raoult’s law but rather one where only the heavy component obeys Raoult’s law while the- light one obeys Henry’s law. In view of this definition of ideality, deviations from idealbehaviour are due not to interactions between molecules of component 1 and molecules of component 2 but rather to interactions of molecules of component 2 with each other; the ideal solution is the infinitely dilute mixture tihere molecules of component 2 are completely surrounded by molecules of component 1. As the mole fraction of the solute increases, molecules of solute begin to interact with each other and it is this interaction which is primarily responsible for non-ideality. According to this picture, it is suitable to expand the excess free energy as follows : GE -= n,RT

-a,,(w,

+

42X2)ZZ

-

A

- a222(41x, + q2x2)z: + higher terms (22) = “effective” volume of component 1; where ql = “effective” volume of component 2; 92 22 = “effective” volume fraction of component 2 given by q2~2lh9, a22

=

4222

=

+

92x2);

interaction coefficient between two molecules of component 2; interaction coefficient between three molecules of component 2.

= ln’y$p)

(23)

1 aGE -= In yz*(‘) RT ( an, 1 T.P,nl

(24)

To illustrate, consider the case where only interactions between two molecules are considered (i.e. c1222and higher interaction coefficients are set equal to zero). Then substitution yields for component 1 the familiar Van Laar relation A

In y\‘“) = Cl +

W~>h/~z112

(25)

with A=

a22q1

B = %a For component 2, the activity coefficient takes the considerably less familiar form 1 [l + @/A)(xz/x~)]~

- ’

(26)

(If it is assumed that qr = q2 then A = B and equations (25) and (26) become the two-sufbx Margules equations.) In deriving equations (25) and (26) not only have all interaction coefficients beyond the fist been neglected but, what is perhaps more serious, the effective volumes ql and q2 have been assumed to be invariant with composition, which is another way of saying that the structure of the solution does not change much with mole fraction. Constant values of q1 and q2 are not bad approximations for mixtures of liquids but this approximation probably constitutes a large oversimplification for mixtures of liquids and gases. Nevertheless, equations (24) and (25) have a great deal of empirical value even if the physical signiticance of the constants is not completely clear. One useful advantage of the expansion given in equation (22) is its simple extension to multicomponent mixtures ; this is briefly discussed in Appendix II.

621

J. M. PRAUSNITZ ACTIVITYCOEFFICIENT EQUATIONS FORTHE BUTANE-CARBON DIOXIDESYSTEM The adjusted activity coefficients of butane at 160°F show very large deviations from ideal behaviour but they can be fitted with the Van Laar equation (equation 25). The best constants are obtained by rearranging equation (25) into a linear form (27)

J[l

By plotting the logarithms of the adjusted activity coefficients of butane to the minus one-half power against the ratio of the mole fractions, the values of A and B can be obtained from the slope and intercept. The adjusted activity coefficients for carbon dioxide are then given by equation (26). Fig. 3 shows that the Van Laar equations give an excellent representation of the data for both components. Since the data conform to a set of equations which satisfy the Gibbs-Duhem equation, the data are necessarily thermodynamically consistent. At 280°F the adjusted activity coefficients of butane are not large. In this case it is better to try and represent first the adjusted activity coefficients of carbon dioxide which exhibit a much greater variation with composition. These coefficients could not be fitted to the Van Laar equation; this failure is not too surprising since the assumptions on which this equation is based are not at all valid at this temperature where the heavy component is itself not far from its critical temperature. Since very little is known about the structure of liquids well above their normal boiling points it is difficult to construct a suitable expansion for the excess free energy. However, a good fit of the data is obtained with the simple empirical expression ln ~z*(~l”)= - Cxg

(28) The corresponding expression for the adjusted activity coefficient of component 1 is found by substituting equation (28) into equation (6) and integrating; it is

1

In ~$~l”)= 2C 2x, - -x: - In x1 - 3 2 2 [

(29)

As shown in Fig. 4, equations (28) and (29) give a very good representation of the experimental data.

Since these equations satisfy the Gibbs-Duhem equation, the experimental data may be judged as thermodynamically consistent. CALCULATION OF THEACTIVITYCOEFFICIENT FORBUTANEFROMSOLUTIONTHEORY During the last dozen years or so a lot of attention has been given to constructing a molecular theory of solutions of nonpolar liquids and thus it would appear instructive to compare briefly some of the theoretical equations to the results of the thermodynamic analysis of the experimental data for the butane-carbon dioxide system. The theory of conformal solutions [16] and the corresponding states theory of SCOTT [17] are, in principle, not restricted to liquids remote from the critical conditions but the cell theories of PRIGOGINE [18] are limited to liquids in their “normal” liquid range, near or below the atmospheric boiling point. Thus the work of Prigogine is not strictly applicable to the problem being considered here. Nevertheless it is of interest to see just how the theoretical and experimental results compare; we would certainly not expect them to agree well with one another but by noting the form and magnitude of the disagreement new insight may be obtained for eventual extensions and for possible improvement of the theoretical formulations. At 160”F, butane still has most of the properties of a normal liquid but at 280°F butane has already lost most of its liquid characteristics. Hence the comparison between theory and experiment is here limited to the results obtained at 160°F. The experimental adjusted activity coefficients for butane shown in Fig. 3 cannot be directly compared to those calculated theoretically because the coefficients shown in Fig. 3 refer to the pressure P; which is always lower than the saturation pressure of the liquid solution. Thus the adjusted activity coefficient yip”) refers to a state which is physically unattainable. For a realistic comparison between theory and experiment, it is more meaningful to compare the theoretical results to adjusted activity coefficients which are defined at a pressure at which the liquid phase can physically exist. The lowest possible pressure at which the liquid phase can exist for all compositions up to the critical, is the

622

Thermodynamic

representation

of high-pressure

critical pressure of the isotherm. Therefore the comparison below is made with experimental activity coefficients which have been adjusted to the critical pressure. This choice of reference pressure, although reasonable, is still arbitrary provided only that the liquid phase can always exist at this pressure. For our very limited purpose here this choice is not important. The theory of conformal solutions is a rigorous first-order perturbation treatment for very simple fluids which is correct only to the first-order differences in the characteristic intermolecular energy and size parameters of the two components. It is therefore valid only for components which are very much alike in their thermodynamic properties. If the characteristic intermolecular energy for each component is proportional to its critical temperature, and the characteristic size is proportional to the cube root of the critical volume, then the differences in these parameters, relative to those for butane, are Relative dilferences in characteristic energies

Tf- T, = 0.284 Tf

= -

(30)

Relative difference in characteristic sizes =

m1’3 - ow3 =

o*283

(31)

(vy3 The conformal

II III iY

solution

theory

requires

that

CELL THEORY FOR SAME-SIZE MOLECULES CORRESPONDING STATES (THREE-LIQUID1 CORRQPONDING STATES (TWO -LIOUIDl

THEORETICAL COEFFICIENTS DIOXIDE (2) COEFFICIENTS

AND EXPERIMENTAL ADJUSTED ACTIVITY FOR BUTANE IN THE .BUTANE (1 .)-CARBON SYSTEM AT 160°F. ADJUSTED ACTIVITY REFER TOTHE CRITICAL PRESSURE ll84psia

FIG. 5.

vapour-liquid

equilbria

these relative differences should be very small compared to unity. This requirement is not satisfied for the butane-carbon dioxide system and thus, since second- and higher-order differences are not considered by this theory, we would expect that the calculated activity coefficients for butane will be considerably lower than those which have been observed. This is indeed the case; in Fig. 5 the curve labelled I represents activity coefficients calculated by the theory of conformal solutions and it can be seen that the calculated results are much lower than the experimental ones which are shown by the dotted line. There are several cell theories of Prigogine and if we take the most general]/ which permits lattice deformations due to small differences in molecular sizes of the components, we find that the nonideality of the solution is tremendously overestimated; the calculated activity coefficients for butane are very large and, in fact, predict that butane and carbon dioxide split into two liquid phases long before the critical composition is reached. These very large activity coefficients are not shown in Fig. 5. The serious failure of the cell theory is probably due to the fact that the lattice concept is no longer applicable to butane at 160°F which is already 129°F above the atmospheric boiling point. The liquid butane is already sufficiently expanded at 160°F to allow the entry of carbon dioxide molecules without disturbing the liquid structure as drastically as is assumed a theory based on the lattice structure of liquids. An earlier cell theory of PRIGOGINE ignores the effect of molecular size differences completely and ascribes all deviations from non-ideality to differences in the characteristic intermolecular energy. We would expect that this theory would tend to underestimate the non-ideality as indeed it does; the results of this theoretical calculation are given in Fig. 5 by the curve marked II which lies significantly below the experimental line. The corresponding states theory of SCOTT is probably the most useful general theory for mixtures of simple non-polar liquids. Like the theory 11 Equation (10.4.17) in Ref. [18] cannot be used here because it is restricted to relative differences in characteristic sizes no greater than 0.1. The more general relation, equation (10.4.1), must be used.

623

J. M.

hAUSNIT2

of conformal solutions it is a perturbation treatment on the thermodynamic properties of a pure fluid but, unlike the theory of conformal solutions, it considers second-order terms in the differences of the characteristic molecular parameters and, in principle at least, it could be extended to consider still higher order terms. Scorr writes the free energy of a liquid (relative to ideal gas at the same temperature and pressure) as a universal function of the reduced temperature and reduced pressure; the reducing parameters are the characteristic intermolecular energy and molecular size. The free energy of a liquid mixture is then given by the same universal function using reducing parameters which are functions of composition. In this way the excess free energy and the activity coefficients can be found. The basic, problem is to decide on how the characteristic molecular parameters vary with the mole fraction. SCOTT proposes three possible formulas for this variation. The fist of these considers that the liquid phase consists of uniform cells all of the same size (“single-liquid solution”). This model will certainly greatly overestimate the activity coefficients for the case under consideration here where there is a large difference in the size of the molecules. The second formula considers the liquid to consist of two kinds of cells, one for butane molecules and one for carbon dioxide molecules (“two-liquid solution”), while the third formula considers the interactions in the liquid phase to consist of three kinds, the l-l, 1-2, and 2-2 interactions, just as we have in the case of the second virial coefficient of a binary gas mixture (“three-liquid solution”). The “two-liquid solution” relations are known to give fairly good results for mixtures of liquids near or below their atmospheric boiling points while the “three-liquid solution,” which is strictly valid only for dilute gas mixtures, gives excess free energies which are too low for liquid solutions. For the case considered here, where butane is already above its atmospheric boiling point but still far from being a dilute gas, the experimental activity coefficients should fall between those calculated by the “two-liquid” and the “three-liquid” solutions and this is indeed the case; in Fig. 5 curve III gives the “three-liquid solution” theoretical results and curve IV gives the “two-liquid solution” theoretical results.

Despite the fact that the experimental results show at least rough agreement with those predicted by Scorr’s theory, one cannot conclude that the present state of solution theory is even remotely adequate to predict the thermodynamic properties of solutions at large pressures where one component is above its critical temperature. While the magnitudes of the experimental adjusted activity coefficients are in the vicinity of those calculated by Scorr’s theory, the shape of the experimental curve is very much different from that predicted by theory. The experimental curve shows large curvature near the critical point where a very small change in composition has a very large effect on the thermodynamic properties; at present, the theory of solutions is not able to account for this effect. It is likely that large fluctuations in the vicinity of the critical point make a significant contribution to the excess free energy and, at present, theory is unable to account for these quantitatively. Since the statistical theory of the thermodynamic properties of fluids near the critical condition is still in its infancy, a rigorous theoretical treatment of liquid solutions at high pressures is not likely to become available in the near future; nevertheless, it is entirely possible that the existing theories for liquids remote from the critical conditions may be a good point of departure for a semi-empirical theory useful in the critical region. ESTIMATION OF HIGH PRESSURE EQUILIBRIA FROMVERYLIMITEDDATA While the present status of solution theory is not promising for the prediction of high-pressure vapour-liquid equilibria, the thermodynamic analysis presented earlier suggests some semi-empirical techniques for estimating such equilibria from a minimum of information, at least for rather simple, non-polar systems. We consider first the case where only the following experimental data for the mixture are available at the temperature T: the solubility .of the light component in the heavy component at low pressure (i.e. the Henry’s law constant), and the composition and pressure at the critical point. With this information, the adjusted activity coefficients of both components at the critical point can be estimated;

624

Thermodynamicrepresentation

of high-pressure

from these coefficients it is then possible to compute the constants in some analytical solution of the Gibbs-Duhem equation (such as those by Van Laar or Margules) which then give the activity coefficients at compositions other than the critical. Since only two constants can be calculated by this method it is necessary to represent the activity coefficients by an equation which contains one or two (but not more) empirical constants. The adjusted activity coefficients at the critical point are given by Yl

(PI%

_

&PCev

-f(pl.)

&We

WC y~(P1’)*c= m

-

s

pc_iT,“dP

~18

RT

where m is an exponent greater than unity. In the absence of any pertinent data a reasonable approximation can be achieved by this (tentative) empirical rule : m = 4 for T < 0.85Tf

(30)

‘= i$ dP

y;w)~C

=

B

- ’I

0.85Tf < T c 0*95T,”

m = 2

for

T >0*95Tf

or

y~p’=)x, fg'::; Ph

exp

=

y2

If the critical composition is such that x2 < O-5 it is probably not worth while to use a two-parameter equation; rather, one should use the oneparameter equations (28) and (29). The constant C is found by ln yZ*(P1*),C = - C(x?J2 (34)

(36)

RT

(37)

,3,g

exp p42

i&dP

-

~11

*(PI’)X2#P19 y2

s ’

I

The total pressure P is found equations (36) and (37) to give

(33’

Once the constants A and B (or C) are known, the adjusted activity coefficients for both components can be calculated over the entire liquid composition range at constant temperature. It is .also necessary to make an estimate of the liquid-phase partial molar volumes of both components as a function of liquid composition; this

for

Yl =

1 [l + @/A)(x;/x;)]~

m = 3

The isothermal P - Y - x diagram can now be constructed as follows. At some liquid composition x calculate the adjusted activity coefficients and partial molar volumes of both components. The equilibrium vapour composition y is found by the equilibrium relations

(32)’ In

equilibria

estimate need not be of high accuracy. The partial molar volumes at the critical point can be calculated from an equation of state and 57, the partial molar volume of the light component at infinite dilution, can be estimated fairly well from existing tabulations and correlations [19, 201. A useful interpolation function for intermediate compositions is given by

(31) s ~15 RT Since the composition and pressure at the critical point are known, it is possible to compute the fugacity coefficients and the partial molar volumes for both components at the critical point from an equation of state for the vapour phase. The adjusted activity coefficients at the critical point can then be found subject to the previous assumption that the partial molar volumes are functions only of composition and not of pressure. For example, the Van Laar constants A and B may be found by simultaneous solution of the two equations exp _

vapour-liquid

by summing

x exp c ’ a,

(38)

Any two of the last three equations may be used to compute the two unknowns, yr (or y2) and P. The simultaneous solution of these equations is necessarily trial-and-error since the fugacity coefficients +1 and 92 depend on y and on P as given by a suitable equation of state for the vapour phase. The task of carrying out all of the calculations outlined above is formidable; it is a tedious job to do this by hand and if calculations of this type are

625

J.M. PRAUSNITZ to be performed repeatedly it is certainly worth while to use an electronic computer. However, in order to see if this technique is at all reasonable, calculations were carried out for two isotherms for the system nitrogen-methane. First, the molar volume at the critical point was calculated using the Benedict-Webb-Rubin equation with the constants reported by STOTLERand BENEDICT[21]. Calculation of the partial molar volumes and fugacity coefficients with the Benedict equation is very tedious and therefore, to simplify the effort, the partial molar volumes and fugacity coefficients at the critical point were calculated from expressions based on the Redlich-Kwong equation [22] using the critical volume of the mixture as determined by the Benedict equation; these expressions are given in Appendix III. At compositions other than the critical, the fugacity coefficients were computed as suggested by REDLICHet al. [4]. Fig. 6 shows the computed isotherms and compares them with the data of BLOOMERand PARENT [23]. Considering the approximations involved the agreement is most satisfying. As a second case it is tempting to try to predict the high-pressure vapour-liquid equilibria of a simple mixture for which no data at all are available, that is, for a case where only data for the pure components may be used. The procedure for this case is exactly the same as that described by equations (30) to (38) except that additional techniques must be called upon to estimate the Henry’s

law constant and the critical constants of the mixture. A simple system, for which such a daring calculation might be attempted, is the methaneethylene system at -78°C. The Henry’s law constant for this system was estimated first using the method suggested by UHLIG [24] and then by the regular solution correlation of PRAUSNITZand SHAIR[25]; in this latter method the Henry’s law constant for methane at -78°C was calculated in several higher olefins and by extrapolation the Henry’s law constant in ethylene was found. The two methods gave results which agreed with each other to within about five percent. The critical composition corresponding to - 78°C was found from the empirical quadratic equation T’( -78°C) = x,Tf + x2T, + + 2x,x,[2T,’

- T; - T;] (39)

where T; is the critical temperature of ethylene and T,Cis that of methane; Tk is the critical temperature of the equimolar solution which was estimated from the corresponding-states correlation of KING [26]. The critical pressure corresponding to -78°C was then found from an equation similar to equation (39) PC = x,p; + XZP; + 2x,x,[2Pi - P; - Pi] (40)

where Pf and Pi are the critical pressures of pure ethylene and methane respectively and where P$ is the critical pressure of the equimolar mixture which was also estimated from KING’S correlation. The 800 I I I I I I I I I Benedict-Webb-Rubin equation of state was used to compute the critical volume and this value was then used with the expressions given in Appendix III to find the partial molar volumes and the fugacity coefficients at the critical point. From the adjusted activity coefficients at the critical point the Van Laar constants were found and the partial molar volume of methane at infinite dilution was calculated from an expression derived by SMITH and WALKLEY[27] on the basis of the free-volume theory of liquids. The P - x - y diagram could -CALCULATED --DATA OF BLOOMER 100 then be constructed as outlined above. The calcuAND PARENT I I I I I I I I I 1 lated isotherm is shown in Fig. 7 together with the 01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 experimental results of GUTER et al. [28]. The MOL FRACTION NITROGEN agreement is surprisingly good in view of the rather VAPOR-I.IQUID EOUILIERIA FOR THE METHANE-NITROGEN SYSTEM crude method of calculation; one should certainly FIG.6. 626

Thermodynamicrepresentationof high-pressurevapour-liquid equilibria not conclude that the approximations which were made in this case will necessarily yield such good results for other systems also. In a calculation of this type the most sensitive parameter is the Henry’s law constant and especially at low temperatures it is often not simple to estimate this parameter with very good accuracy. CONCLUSION Thermodynamic analysis of high-pressurevapourliquid, equilibria is considerably more difficult than the corresponding analysis of low pressure equilibria because in the former case the effect of pressure on the thermodynamic properties of the liquid phase can no longer be neglected. The effect of pressure on the thermodynamic properties can be calculated only from the volumetric properties of both phases; therefore a complete thermodynamic representation of high-pressure equilibria at any fixed temperature requires not only information on the compositions of the two phases in equilibrium as a function of pressure, but also data on the densities and compressibilities of these phases. For the vapour phase the necessary volumetric information can often be calculated by an equation of state, at least for non-polar systems, but for the liquid phase only very rough approximations exist for the unfortunately common case where experimental volumetric data are lacking. Experimental studies of high pressure vapour-liquid equilibria

-

should always include measurements on liquidphase densities, especially near the critical point. Although thermodynamic analysis at high pressure is not as simple as that at low pressures, it is also not prohibitively difficult especially if one has the use of an electronic computer. The advantages of such analysis have been discussed earlier and need not be repeated here except as a reminder that progress in the understanding of equilibria and in providing generalizations and correlations for their prediction can only come about through thermodynamic interpretation. At present, the statistical theory of fluids is not capable of giving an adequate description of expanded liquids and dense gases but when that theory does reach the required stage of sophistication it will necessarily give results in terms of thermodynamic functions and not in terms of K values or other useful engineering terms. Thermodynamic analysis, therefore, is the essential step which links molecular physics with phase equilibrium problems in chemical engineering. Acknowledgement-The author is grateful to the donors of the Petroleum Research Fund for partial financial assistance and to A. E. SHERWOOD, W. T. ZIEGLER, 0. REDLICH and H. C. VAN NESS for their critical review of the manuscript. Much of this work was done while the author was a Guggenheim Fellow in residence at the Eidgeniissische Technische Hochschule in Zurich, Switzerland; he is grateful to the John Simon Guggenheim Memorial Foundation for its support, to the Swiss institution for its hospitality, and especially to Dr. N. IBL for helpful discussions. For many kind favours extended during his stay in Zurich the author also wishes to express his gratitude to Mr. and Mrs. H. MAMELOKand to Professor and Mrs. H. J. MULLER.

CALCULATED -

APPENDIX 1

DATA OF GUTER, NEWITT (t RUHEMANN

It is shown below that the activity coefficients defined by equations (8) and (10) satisfy equation (6). For a binary liquid system along an isotherm the Gibbs-Duhem equation is x1 dpi + x2 dp2 = vdP

01

Ii 0

0.1

VAPOR-LIQUID

0.2

I II I I 0.3 0.4 0.5 0.6 0.7 MOL FRACTION METHANE EQUILIBRIA

F$iW.i”g

I 0.8

)44lHANE-ETHYLENE

where v is the molar volume of the liquid mixture. The chemical potential pi at the composition x and at the total pressure P is given by

t-1 0.9 SYSTEM

(A.I.l)

1.0

PO’, x, P) = PI_(T,

P”>+ RT

hff’(T;;y ;!) lpurc

E

’

(A.I.2)

FIG. 7. 627

J. M. PRAUSNITZ

It is necessary to differentiate between twocases. In the first case (a) there are two light components and one heavy component; in the second case (b) there are two heavy components and one’ light component.

From equation (8) j-1 = x,ypf,(') pure exp Thus

~0, x, P) = P~~~~,,(T, P”>+ RT In y(lp)xl+

s

Case (a)

P

+

6, dP p”

(A-1.4)

Let subscripts 2 and 3 refer to the light eomponents. The Henry’s law constants are defined by

Similarly, for component 2, using equation (10) p2(T, x, P) = @(T,

P”) + RT In y;(“‘x2

+

H, ~limit f x1+0 0 x

P +

52

s PO

dP

(A.I.5)

dp, = RT d In $“)

(A.I.6)

G

+RTdlnx,+i&dP+ dP dxl

nTRTk,'G

(A.I.7)

+

-u23z2z3

When equations (A.I.6) and (A.I.7) are substituted into equation (A.I. 1) x1 d In r\‘“) + x2 d In $“)

-

-"222zz

-a223&3

-a333z~

g233z2z;

+

-

higher terms

(A.II.1)

Let subscript 2 refer to the light component. The Henry’s law constant is now a function of the molar ratio of component 1 to component 3 in the light-component-free solution :

(A.I.8) However, the Gibbs-Duhem equation says that for any extensive property X at constant temperature and pressure

f

H E limit m.*o0 x 2 with x1/x3 = constant. In view of this definition, deviations from ideality are due only to interactions of molecules of species 2 with each other. The excess free energy is given by

(A.I.9)

Since the volume is an extensive property, equation (A.I.9) shows that the bracketed term in equation (A.I.8) must vanish; thus equation(A.I.8) reduces to x1 d In rl’“’ + x2 d In $‘”

- u33z;

Case (b)

dP=O

x1 dX, + x2 dX, = 0

= --a22d

~lzXz+tl3X3)

-

+

+dx’j-;[x@)T,p+x,(~)T,j

3

x3-0

Non-ideality in this solution is due to interactions between molecules of species 2 with each other, molecules of 3 with each other, and molecules of species 2 with molecules of species 3. The excess free energy can be written

d In JJ\~) + RT d In xl + iJ dP +

dP dx,

H, 3 limit x2=0 0 x

x3=0

The differentials of the chemical potentials are

dell = RT

f

and 2

GE = bRT(!?,x, + !TzXz + W3)

= 0 (A.I.lO)

-u22zf

-

a222z3

+ higher terms (A.II.2) APPENDIX

11

The expansion for a binary solution given .in equation (22) can readily be extended to a solution containing more than two components. To illustrate, consider a ternary solution; extension to solutions containing still more components will then be obvious.

In cases 1 and 2 the activity coefficients can be found from the general relation for any component

628

1

--

RT

aGE

( 1 anl

= In rl’) [or In y;(‘)]

T.P. allothern

(AJI.3) i=

1,2,3....

Thermodynamic representationof high-pressurevapour-liquid equilibria

APPEND= III. CALCULATION OF FUGACITYC~EFFICIENT AND PARTIAL MOLAR VOLUMEFROM THE WI =&[f[(g),,,.,-$?j REDLICHKWONGEQUATION OFSTATEWITHMOLAR VOLUME,TEMPERATURE AND CQMPWTION AS INwhere IZ~is the number of moles of DEPENDENT VARIABLEs v = (n1 -t nz>v The Redlich-Kwong equation is and n1 RT a Yl=p=-(A.III.1) nl + n2 v - b T”‘v(v + b) When equations (A.III.l, 4, 5, The constants a and b, for a pure substance, are stituted into equation (A.III.7) the related to the critical constants by a= b =

Q.4278R2T5’2 C PC 0*0867RT,

(A.III.2)

2y1a1 + Q2Jha2) RT312b v-t-b - abl ln--+ RT3/2b2 [ V

(A.III.3)

+ yia,

b = y,b, + y,bz

(A.III.4) (A.III.5)

Fugacity coeficient The fugacity coefficient I$~ is defined by e,-$p

(A.III.6)

For a binary mixture the fugacity coefficient is related to the volumetric properties by

(A.III.7) species i, (A.III.8) (A.III.9) 8, 9) are subfinal result is

In+, =lns+&-

PC For a binary mixture the constants are given by a = yfa, + 2yly2J(ala2)

dV-l+$

v+b

In-

b v+b

v

+

1-

1ng

(A.III.lO)

Partial molar volume For calculation of a partial molar volume in a binary mixture using an equation of state which is explicit in the pressure, the most convenient expression is -(W%),,“, (A.III.ll) O1= (WW,“l,“l When equations (A.III.1, 4, 5, 8, 9) are substituted into equation (A.II1.U) the final result is

Al = CT”‘u(v + b)] -%Y lal +&&a,) - [&/(v + b)lI - CRTI(U - b)lU + (a/T1”)[(2v + b)/v2(v + b)2] - [RT/(v - b)2]

CW(v- WI>cA.III . I21

when fil is known, 6, is found by ” fi2=

Ylfil

(A.III.13)

Y2

&FFXENCES

and Bmwm, L., Thermodynamics (2nd Ed.) McGraw-Hill,New York 1961. A., Chem. Revs. 1949 44 141. PRAusNITz J. M., Amer. Inst. Chem. Engrs. J. 1959 5 3. REDLICH0.. Krm A. T. and TURNQUI~TC. E., Chem. Engng. Progr. Symp. Ser. 1952 No. 2,48 49. L~wrs G. N., RANDAU M., F’~ZER K. S.

BEATTTE J.

KRICHE~SKYI. R. and KAMRNOVSKY J. S., J. Amer. Chem. Sot. 1935 57 2168. WIEBE R. and GADDY V. L., J. Amer. Chem. Sot. 1937 59 1984. B-ET J. and DODDM., C. R. Acad. Sci., Paris 1936 203 775. KRI&EVSKY I. R., J. Amer. Chem. Sot. 1937 59 596. BENIUM A. L., KATZ D. L. and WILLIAMSR.B., Amer. Inst. Chem. Engrs. .I. 1957 3 236. KOBAYASHIR. and KATZ D. L., Zndustr. Engng. Chem. 1953 45 440. LELAND T. W., MCKE~A J. J. and Kom K. A., Industr. Engng. Chem. 1955 47 1265. GONIKBERGM. G. and FASTOVSKIV. G., Acta Physicochim. URSS 1940 13 399. OLDS R. H., REAMERH. H. SAGE B. H. and LACEY W. N., Zndustr. Engng. Chem. 1949 41475. SAGE B. H., WEBSTERD. C. and LAC~Y W. N., Industr. Engng. Chem. 1937 29 1188.

J. M. PRAUSNITZ [15] 1161 i17j j18f [19] [2OJ [21] 1221 [23j [24] [25] [26] [27] [28]

WOHL K., Trans. Amer. Inst. Chem. Engrs. 1946 42 215. ~NCXJET-HIWINS H. C., Proc. ROY. Sot. 1951 A205 247.

SCOTTR L., J. Chem. Phys. 1956 25 193. Pamoor~~ I., Molecular Theory of Solutions, North Holland Publistig Co. 1957. HUDEBRAND J. H. and Scorr R. L., Solubility of Non-EZectroZytes (2d Ed.) Reinhold, New York 1950. P~AUSNITZJ. M., Amer. Inst. Chem. Engrs. J. 1958 4 269. STATLERH. H. and BENEDICTM., Chem. Engng. Progr. Symp. Ser. 1953 No. 6,49 25. REDLICH0. and KWONG J. N. S., Chem. Revs. 1949 44 233. Bm~aa 0. T. and PARENTJ. D, Chem. Engng. Progr. Symp. Ser. 1953 No. 6,49 11. UHLIG H. H., J. Phys. Chem. 1937 41 1215. PRAUSNITZJ. M. and SHAIRF. H., Amer. Inst. Chem. Engrs. J. 1961 7 682. KING M. B., Trans. Faraday Sac. 1958 54 149. SMITHE. B. and WALKLEY J., J. Phys. Chem. 1962 66 597. GUTER M., NEWI~~ D. M. and RUHEMANN M., Proc. Roy. Sot. 1940 Al76 140. R&n&--L’auteur demontre l’int&% de l’analyse thermodynamique des dorm&s d’equilibre aux pressions &levees et propose des techniques pour l’analyse des isothennes. En phase liquide le coefficient d’activite du composant lourd est rapport& au composant pur, mais le coefficient d’activite du composant leger est rapport6 a la solution a dilution infinie. Les deux coefficients d’activite tendent done vers 1 lorsque la fraction molaire du composant leger tend vers zero. En ajustant tous les coefficients de sorte qu’ils correspondent a une meme pression ils doivent satisfaire l’equation de Gibbs-Duhem isotherme et isobare, sans les corrections pour les variations de volume. On peut done utiliser les formes int&r&es bien connues de l%quation de Gibbs-Duhem pour rep&enter de facon analytique la fonction: coefficients d’activite ajust&composition. Le traitement general present6 en cet article conduit B l’equation de Krichevsky-Kasamovsky pour les solutions tres dilu&. La methode est appliqu6e aux donnQs d%quilibre du systeme butane-anhydride carbonique allant jusqu’a la composition critique. A 160°F les coefficients d’activite peuvent &tre rep&e&s par l%quation de Van Laar aux deux coefficients; ii 280°F une equation a une constante suffit. Lc coefficient d’activite ajust du butane est compare aux valeurs calculees a l’aide des theories de LonguetHiggins, de Prigogine et de Scott. Cette demiere theorie est la seule a dormer des valeurs numeriques plus ou moins en accord avec les r&hats exp&imentaux mais elle n’explique pas l’allure de la courbe coefficient d’activite ajust&composition ii l’approche du point critique. En se basant sur l’analyse thermodynamique l’auteur propose des techniques semi-empiriques pour l’estimation des don&es d’equilibre vapeur-liquide de melanges simples a pression elevb a partir dun nombre r&duit de resultats exp&imentaux. Ces techniques sont illustr&s a l’aide des systemes azote-methane et m&hane&hylene. Znsannnenfassung-Die Ziele einer thermodynamischen Analyse von Gleichgewichtsmesswerten bei hohen Drucken werden skizziert und Methoden zur Durchfiihrung einer solchen Analyse kings einer Isotherme vorgeschlagen. In der fltissigen Phase wird der AktivitPtskoetBzient der schweren Komponente auf den reinen Stoff bezogen, der Aktivitatskoeffizient der leichten Komponente jedoch auf die unendlich verdtinnte Losung; auf diese Weise gehen beide Aktivitltskoetlixienten gegen Eins, wenn der Molenbruch der leichten Komponente gegen Null geht. Alle Aktivit&skoetIizienten werden auf einen festgelegten Druck umgerechnet und miissen dann der isothermen-isobaren GibbsDuhem’schen Gleichung ohne Korrektur ftir die Volumenlinderung beim Vermischen geniigen; es ist infolgedessen moglich, wohlbekannte integrierte Formen der Gibbs-Duhem’schen Gleichung zu bentitzen, urn eine analytische Darstelhmg der dem Bezugsdruck angepassten AktivitltskoefIizienten in Abhigigkeit von der Zusammensetzung zu erhalten. Es wird gezeigt, dam die hier vorgeschlagene allgemeine Methodik im Falle sehr verdiinnter Losungen auf die Gleichung von KrichevskyKasamovsky zurtickgeftlhrt werden kann. Die vorgeschlagene Analyse wird auf die Messwerte des Phasengleichgewichtes beim System Butan/Kohlens&urebis zur kritischen Zusammensetzung angewendet und es wird gezeigt. da& bei 160°F die Aktivitatskoeffizienten durch die zwei Konstanten enthaltende Van Laar’sche Gleichunn gut dargestellt werden kiinnen; bei 280°F gent&t eine Beziehung mit mu einer Konstante. Die ange: passten Aktivitatskoethzienten des Butans werden mit den gem& verschiedenen Theorien der L&ngen nach Longuet-Higgins, Prigogine und Scott berechneten Werten veralichen. Ledialich die Theorie - des letztgenannten Aujors ftihrt zu einer angenaherten Uebereinstimmung mit denexperimentellen Ergebnissen; aber such sie vermag nicht die scharfe Krtimmung der Kurve zu e&l&en, die man in der Gegend des kritischen Punk& beobachtet, wenn man die angepassten Aktivit&tskoe%rienten in Abhangigkeit von der Zusammensetzung auftragt. Auf der Grundlage der thermodynamischen Analyse werden halbempirische Methoden vorgeschlagen, die gestatten, das Verdampfungsgleichgewicht einfacher Gemische bei hohen Drucken, ausgehend von einem Minimum an experimentellen Daten, abzuschiitzen; diese Methoden werden anhand der Systeme Stickstoff/Methan und Methan-Aethylen illustriert.

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