local mole fractions. One such ratio would be t h a t of the molecules in the cell with highest potential energy. Therefore the name, local effective mole fractions. Nomenclature = =
-
R T X, X,, 2
= = = = =
difference between parameters, A,, = gzl - g j l residual Gibbs energy in the N R T L equation excess Gibbs energy gas constant absolute temperature overall mole fraction of component i local mole fraction of component i in a cell lattice coordination number
GREEKLETTERS CY
y
= =
nonrandomness parameter in the N R T L equation activity coefficient of component i
References
Benedict. 11..Johnson. C. 4..Rubin. L. C.. Trans. Amer. Znst. Chem. ’Ens.’,41, 371 ’(1945): Delzenne, .4.O., J . Chem. Eng. Data, 3 , 224 (1958). Griswold, J., Andres, D., Arnett, E. F., Garland, F. M., Ind. Eng. Chew&.,32, 878 (1940). Griswold, J., Chu, P. L., Winsauer, W. O., i b i d . , 41, 2352 (1949). Griswold, J., Wong, S.Y., Chewa. Eng. Progr. Symp. Ser., 48, 1 (1952). Guggenheim, E. A., “Mxtures,” Clarendon Press, ($ford (1952). Hala, E., Wichterle, J., Polak, J., Boublik, J., Vapor-Liquid Equilibrium Data at Normal Pressures, Pergamon, Xew York, N.Y., 1968. Kincaid, J. F., Eyring, H., Stearn, A. E., Chem. Rev., 28, 301 (1941). Larson, C. D., Tasbios, I). P., Ind. Eng. Chem. Process Des. Develov.. 11. 35 11972). lIargulek, ’LI.,’ Sitz., Akad. W f s s . Wzen, Jlath. Saturwzss. Kl., 104, 1243 (1895). Marina, J . AI., Doctoral Dissertation, Xewark College of Engineering, Newark, N.J., 1971. Nurti, P. S., Van Winkle M., J . Chem. Eng. Data, 3 , 72 (1958). Kielsen, R. C., Weber, J. H., ibid.,4, 145 (1959).
Orye, R. V., Doctoral Dissertation, Univ. California, Berkeley, 1965. Perry, J. H., “Chemical Engineers Handbook,” 1IcGraw-Hill, New York, N.Y., 1963. Prausnitz, J. M.,“Molecular Thermodynamics of Fluid-Phase Equilibria,’] Prentice Hall, Englewood Cliffs, N.J., 1969. Prausnitz, J. LI.,Eckert, C. A., Orye, R. V., O’Connell, J. P., “Computer Calculations of Multicomponent Vapor-Liquid Equilibria,” Prentice Hall, Englewood Cliffs, K.J., 1967. Renon. H.. Doctoral Dissertation, Univ. California, Berkelev, 1966. Renon, H., Prausnitz, J. AI., A.1.Ch.E. J., 14, 135 (1968). Renon, H., Prausnitz, J . M., Ind. Eng. Chem. Process Des Develop., 8, 413 (1969). Scatchard, G., Ticknor, L. B., J . Amer. Chem. SOC.,74, 3724 (1952). Scott, R. L., J . Chem. Phys., 25, 193 (1956). Severns, W. H., Sesonske, A,, Perry, R. H., Pigford, K:. L., AIChE J., 1, 401 (1955). Steinhauser, H. H., White, R. R., Ind. Eng. Chem., 41, 2912 I.
119491 ~\ - -
I
Stockhardt, J. S., Hull, C. II.,ibid.,23, 1438 (1931). Tassios D. P., Preprint, A.1.Ch.E. National Meeting, Washington, b.C., 1969. Van Laar, J. J., Z. Physik Chem., 72, 723 (1910); 83, 599 (1913). Wickert. J. S . .Tamwlin. W.S.. Shank.’ R. L.. Chem. Ena. Prom.. Surn~.’Ser.-To. 2. 28. 92 119k2). Wilso< G. lI.,J . Arne;. Chem. Soc., 86, 127 (1964). Wohl, K., Trans. A.Z.Ch.E., 42, 215 (1946). RECEIVED for review hIarch 20, 1972 ACCEPTED July 12, 1972 Work sup orted by 1Ierck Sharp & Dohnie Research Laboratories, Ita!kvay, N. J. Figures 1 and 3-6 plus Table V will appear following these pages in the microfilm edition of this volume of the Journal. Table V gives the list of sources for data used in this presentation. Figures 1 and 3 and 4 show standard deviation vs. alpha. Figures 5 and 6 show predictions of liquidliquid equilibrium and vapor-liquid equilibrium from vapor-liquid equilibrium and mutual solubility data, respectively. Single copies of all 11 manuscript pages may be obtained from the Business Operat’ionsOffice, Books and Journalh Division, hmeriran Chemical Society, 1133 Sixteenth St., N.W., Washington, D. C. 20036. Refer to the following code number: PROC-7367. Itelnit by check or money order $4.00 for photocopy or $2.00 for microfiche.
Calculation of Solution Nonideality from Binary T-x Data Herbert E. Barrier' and Stanley B. Adler The JI. W . Kellogg Co., A Division of Pullman, Inc., 1300 Three Greenway Plaza, Houston, Tex. 77046
The calculation of equilibrium vapor compositions and liquid-phase activity coefficients from observed temperature-liquid composition ( T - x ) data is discussed. Reliable results can be obtained by numerically integrating the coexistence equation. A useful and simple procedure is also given for calculating the activity coefficient at infinite dilution from bubble point data. Validity of the calculations i s demonstrated for systems where experimental vapor compositions have been measured.
I n the customary method of measuring binary vapor-liquid equilibria, direct measurements of temperature, pressure, aiid composition of equilibrated vapor and liquid phases are made. address, Laboratory, Kennecott Copper Corp., 128 Spring St., Lexington, Mass. 02173. To whom correspondence should be sent.
Measurement of equilibrium vapor compositions, however, often is much more difficult and is subject t o greater uncertainties than the measurement of liquid compositions. It is therefore sometimes desirable t o find the vapor compositions by calculation from the properties of the liquid phase alone. Thus, numerous procedures for treating isothermal binary Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
71
180
250 0
OBSERVE0 IBACHMAN A N 0 SIMONS)
0
OBSERVED (MA"
et u / )
-CALCULATED 170
lL
u' 160 L
+ L
$
150
I-
140
I
130
I
0.2
0.4
,
I
06
0.8
I
I
MOLE FRACTION ACETONE
Figure 1 . Temperature-composition diagram for acetonetetrachloromethane, 14.7 psia
(P-x) data to yield vapor compositions by calculation have been discussed in the literature (for example, Barker, 1953; Ljunglin and Van Ness, 1962; Uackay and Wong, 1970; and Tao, 1961). Similar procedures for treating isobaric binary data have not been widely applied, although Ljunglin and Van Ness (1962) have shown good results for the ethanoltoluene system. The present work shows that integration of an isobaric form of the coexistence equation leads to vapor compositions and (liquid) activity coefficients which are sufficiently accurate for many practical applications. Too, the coexistence equation provides a useful and convenient expression for calculating limiting (infinite-dilution) activity coefficients from bubble point data.
0.2
0.4
00
06
MOLE FRACTION BENZENE
Figure 2. Temperature-composition diagram for benzene1 -butanol, 14.7 psia
=
612
exp (a
+ b/T)
(5)
where the constants a and b depend on the particular binary system in question. By use of this expression, Equation 3 becomes
Isobaric Form of Coexistence Equation
A useful form of the coexistence equation for a binary system a t constant pressure is given by Equation 6-51 in the monograph of Van Ness (1964) :
Taking the total differential of both sides of Equation 6 results in:
+ p ( l - 2yl)dT
YIV
dln= adyl YZV
(7)
where When vapor-phase and liquid-phase heats of mixing are neglected, fi' is given by the simple expression, fi' = (-xLi
- x2Lz)/RTZ
(2)
where L1and L2are the molal heats of vaporization of the pure components a t the equilibrium temperature. Equation 2 does not result from a complete abandonment of rigor, but is obtained by eliminating terms small in comparison to the latent heats of vaporization a t low pressures. The liquid heat of mixing, for example, is seldom more than a few percent of the latent heat terms. An appropriate approximation for the In ylv/y2v term may be obtained from the virial equation of state truncated to the second term. The resulting expression is given as Equation 6-46 by Van Sess (1964): In
YIV ~2~
=
612P ~
RT
(1 - 2 y l )
(3)
=
2 Biz
- Bii - Bm
(4)
The term, 6 1 2 , is a function of temperature alone for a given binary system. A useful interpolation formula for this quantity is 72
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
=
2P exp ( a RT
--
p=-
-k RTZ
+b/T)
exp ( a
+b/T)
(9)
Combining Equations 1 and 7 with some rearrangement gives the final form of the coexistence equation used in this work:
Equation 10 cannot be integrated analytically. Numerical integration, however, is straightforward with the aid of a digital computer. If the vapor phase is assumed to be ideal, a12, a and p are zero, and Equation 10 reduces to ~-
dT
where 612
CY
mu YI
Yl)
- xi
(11)
The contributions introduced by accounting for real gas behavior generally are small and often are overshadowed by uncertainties in the computation of the fi term. Nevertheless, all results reported below were obtained by retaining the real gas corrections, that is by integrating Equation 10.
A similar expression results for Component 2 at infinite dilution,
,052
0
0.2
0.4
0.8
0.6
MOLE FRACTION n - O C T A N E
Figure 3. Temperature-composition diagram p-cresol, 14.7 psia
for n-octane-
Use of the Runge-Kutta method to integrate Equation 10 requires a n initial value of the function dyl/dT (i.e., its value at x1 = yl = 0 ) . Substitution of z1 = yl = 0 into Equation 10 results in a n expression which is indeterminate. The required initial slope follows from Equation 10 b y the application of L’Hospital’s rule:
The quantity Qo is merely the value of
at x1 = 0; that is,
Qo = -L2/RT2
(13)
The quantity (dzl/dT)O can be approximated by differentiating the experimental T-x data a t the limit of z1= 0. Activity Coefficients at Infinite Dilution
The coexistence equation leads directly to a convenient expression for the (liquid) activity coefficient of a component at infinite dilution. Since the activity coefficient of component 1is given b y 71 = Pl#J1y,/floxl
the infinite-dilution value
(21+
In a nonazeotropic system, one of the last two equations will usually yield a more reliable result than the other because the numerator in the bracketed term will be the sum of two positive quantities for one component, but equal to a difference of quantities for the second component. (In a n azeotropic system, the condition described may occur for both or for neither component, depending on the type of azeotrope.) Thus, if (dzl/dT)Ois negative] (dzz/dT)”is positive. Furthermore, (dz2/dT)0is often of similar magnitude as L1/RT2, thereby contributing to the uncertainty in the difference calculation, Consequently, the activity coefficient of t h e light component a t infinite dilution usually can be calculated more reliably than the corresponding quantity for the heavy component. An expression for the Pl#~,O/jl~ term appearing in Equation 17 may be readily derived from equations given by Van Kess (1964). X direct combination of Equations 5-23 and 6-15 in the Van Ness (1964) monograph taken in the limit of yl + 0 yields
The above equation is based on the virial equation of state truncated a t the second virial coefficient, and is therefore restricted to low pressures. K i t h the exception of b12, calculation of the infinite-dilution activity coefficients from Equations 17 arid 18 requires only T-x mixture data; no assumptions need be made for the misture latent heat of vaporization. Furthermore, the & 2 term can be readily estimated for many systems. liquid-Phase Activity Coefficients
Vapor compositions calculated by means of the coexistence equation can be combined with the experimental T-xdata to generate liquid activity coefficients (Van Sess, 1964) :
(14)
0) is given by
where q0 is the vapor-phase fugacity coefficient evaluated a t = 0. The ratio yl/xl is indeterminate at z1 = 0, but with the help of L’Hospital’s rule it becomes
51
This equation is based on the use of the virial equation of state truncated at the second virial coefficient, and is therefore accurate only at low to moderate densities. Equation 19 takes into account corrections for real gas behavior, nonideal solution effects in the vapor phase, and deviation of vapor pressure from liquid fugacity. Computation Procedures
Combining Equations 12, 13, 15, and 16 results in
Van Kess (1970) has shown that the numerical integration
of the isothermal form of the coexistence equation must in all cases proceed in the direction of increasing pressure. For similar reasons, integration of the iqobaric coexistence equation must be carried out in descending temperature sequence. Converging solutions cannot be obtained by proceediiig in the direction of increasing temperature. Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
73
For systems forming minimum boiling azeotropes, the T-y relationships were generated by integrating from the pure component end points to the azeotropes. Computations were not made for masimum boiling azeotropes, but for such systems the integration should start a t the azeotrope and proceed toward the ends. The correlation of Pitzer and Curl (1957) was used to estimate second virial coefficients.The mixing term, J12, was calculated from Equation 4 with B12 evaluated from the Pitzer and Curl correlation using the mixing rules given by Prausnitz et al. (1967). The fit of 812 to Equation 5 was made over the temperature range of the T-x data set in question. Accurate latent heat data are essential for successful results. h reference latent heat was used for each component, and adjustments for temperature were made using the Watson correlaticlii in the form suggested by Reid and Sherwood (1966, p 148). The integration was carried out by means of the fourthorder Runge-Kutta technique. The increment size, AT, was established by trial. I n most cases reduction of AT from 2-1°F did not affect calculated vapor compositions in the fourth decimal place; consequently, the latter increment size was used for most systems investigated. Some 10-20 graphically smoothed T-x data points were used for each system, and values of x were interpolated a t the required temperatures using the three-point Lagrange interpolation formula. Estimates of the initial slopes, dxl 'dT, required for initiating the Runge-Kutta integration were evaluated by differentiating the three-point Lagrange interpolation formula, and applying the result t o the first three data points. An analogous procedure was used to estimate the terminal slopes of the x-T curve, required in Equation 18.
is probably risky and is not generally recommended when the pressure exceeds several atmospheres. Activity coefficients calculated by means of Equation 19 from the computed vapor compositions are shown as solid curves in Figures 6 and 7 . The limiting (infinite-dilution) activity coefficients shown were calculated from Equations 17 and 18. The plotted points in Figure 6 were calculated by reducing the observed x-y-T data (1lann et al., 1963), also using Equation 19. I n Figure 7, the plotted points are the activity coefficients obtained by Taylor and Wingard (1968) by reduction of their x-y-T data. (The calculated activity coefficient curves for 1-butanol and p-cresol are terminated in Figures 6 and 7 a t the liquid cornpodions corresponding to the last increments used in the numerical calculations. -41though these final increments were within 1°F of the pure component end points, they correspond to approximately 0.08 mole fraction unit on the liquid composition scale.) Calculated values of the activity coefficients a t finite concentrations are smooth and consistent with the infinite-dilution values computed from Equations 17 and 18. The quality of the calculated y-x relationship is obviously dependent upon the quality of the T-x data. Computed y values are particularly sensitive to the numerical T-x data (and their interpolation) when the slope of the 2'-x curve is very high. This may account for the maximum calculated for the activity coefficient of n-octane near the infinite dilution region (Figures 3 and 7). I n any w e n t , Figures 6 and 7 illustrate that good approximations of solution nonideality can be extracted from bubble point data by integrating the coexistence equation.
'-"I
0
OBSERVED (ROSANOFF AND E A S L E Y )
I
Typical Results
T-y curves were calculated for five binary mixtures by integrating Equation 10. I n general, computed and observed vapor compositions agreed within approximately 0.01 mole fraction (Figures 1-5) ; the maximum deviation was 0.02 mole fraction. The acetone-tetrachloromethane system (Bachman and Simons, 1952), Figure 1, forms a very shallow minimum boiling azeotrope a t 95.2 mol % acetone, and the calculations were terminated before reaching this composition. The good agreement in the benzene-1-butancl system (1Iann et al., 1963), Figure 2, and the n-octane-p-cresol system (Taylor and Kingard, 1968), Figure 3, may be somewhat surprising inasmuch as these systems have large heats of mixing while the calculations assume the heats of mixing are zero. We may speculate that partial cancellation of error occurs by simultaneously neglecting the vapor-phase and liquidphase heats of mixing. I n any event, it appears that Equation 2 is a n appropriate definition of the .Q term for most systems at low pressures. The ammonia-n-butane data (Figure 5, Kay and Fisch, 1958) differ markedly from other binary data examined in that the pressure is relatively high, 300 p i a . I n view of the high temperatures relative to the critical temperatures of ammonia and n-butane (270' and 305'F, respectively), the agreement of calculated and observed vapor compositions is better than might be expected. Two factors limit the use of Equation 10 a t high pressures: approximation of the D term by Equation 2 and use of the virial equation of state truncated to just two terms. The former is likely to be the more restrictive of these two limitations. -4lthough the predictions are satisfactory in this particular case, the integration procedure 74
Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 1 , 1973
, 0
0.2
0.4
0.6
0.8
I .o
MOLE FRACTION CARBON DISULFIDE
Figure 4. Temperature-composition diagram for carbon disulfide-acetone, 14.7 psia
260 0
240
OBSERVED (KAY AND FlSCHl CALCULATED
220 LL
u-200 3
$
I80
w 160
c 140 I20 io0
0
02
04
06
08
3
M O L E FRACTION N H 3
Figure 5. Temperature-composition diagram for ammonian-butane, 300 psi0
+
6 0
B
50
f
40
*
w
-30
CALCULATED FROM COEXISTENCE EQUATION CALCULATED EQ (221 a m
i
P R
Y Y LL
s > 2 0
T V
t 5 0
L
y
1.0 - ”
1
0
0.2 0.4 0.6 0.8 MOLE FRACTION B E N Z E N E I N LIQUID
=
second virial coefficient, ft3/lb-mol
= fugacity, psia H = enthalpy, Btu/lb-mol AH = heat of mixing, Btu/lb-mol L = latent heat of vaporization, Btu/lb-mol
I.o
Figure 6. Activity coefficients in benzene-1 -butanol system, 14.7 psia
01
p y 6
Q FROM COEXISTENCE EQUATION X CALCULATED, EQUATIONS (22) A N D (231
=
GREEKLETTERS
@
- CALCULATED
absolute pressure, psia gas constant = 10.73 (ft3)(psia)/(oR)(Ib-mol) = 1.987 Btu/(OR) (lb-mol) = absolute temperature, OR = molal volume, ft3/lb-mol = mole fraction in liquid phase = mole fraction in vapor phase =
quantity defined by Equation 8 quantity defined by Equation 9 activity coefficient quantity defined by Equation 4 vapor-phase fugacity coefficient = f/(Py), d’imensionless = enthalpy term defined by Equation 2 =
= = = =
SUBSCRIPTS
i, ii
= component i ij(i # j) = a n interaction term characteristic of components i and j
SUPERSCRIPTS
L V !
= = -
-
0 o
0.2
04
0.6
0.8
MOLE FRACTION n - O C T A N E
Figure 7. Activity coefficients in n-octane-p-cresol system, 14.7 psia Conclusions
Vapor compositions in binary systems can be estimated within 0.01 or 0.02 mole fract,ion from observed low-pressure iosbaric T-2. dat,a by integration of the coexistence equation. Coiisequently, estimates of liquid-phase activity Coefficients sufficiently accurate for many purposes call be obtained in this manner. I n addition, the coexistence equation leads to a useful expression for calculating infinite dilution activity coefficients from terminal T-xboiling point measurements. These procedures are useful when T-2 measurements alone are available or when measured vapor compositions are suspect. Nomenclature
a
=
6
=
= =
liquid phase vapor phase pure saturated state infinite-dilution state standard-state value
empirical constant defined by Equation 5 empirical constant defined by Equatien 5
Literature Cited
Bachman, K. C., Simons, E. L., Znd. Eng. Chem., 44, 202 (1952). Barker, J. A,, Austral. J . Chem., 6,207 (1953). Kay, W. B., Fisch, H. A., AZChE J., 4, 293 (1958). Ljunglin, J. J., Van Ness, H. C., Chem. Eng. Sci., 17, 531 (1962). Jlackay, D., Wong, K. K., Can. J . Chem. Eng., 48, 127 (1970). Llann, R. S., Shemilt, L. W., Waldichuck, >I., J . Chem. Eng. Data, 8 , 502 (1963). Pitzer, K. S., Curl, R. F., J . Amer. Chem. SOC.,79, 2369 (1957). Prausnitz, J. ll., Eckert, C. A,, Orye, R.V., O’Connell, J. P., “ComDuter Calculations for Multicomaonent VaDor-Liauid Equilibria,” p 20, Prentice-Hall, Englewood Cliffs, N.J., 19’67. Reid, R. C., Sherwood, T. K., “The Properties of Gases and Liquids,” 2nd ed., p 148, McGraw-Hill, New York, N.Y., 1966. Rosanoff, LI. A., Easley, C. W., J . Amer. Chem. Soc., 31, 953 (1909). Tao, L. C., Znd. Eng. Chem., 53, 307 (1961). Taylor, Z . L., Wingard, R. E., J . Chem. Eng. Data, 13, 301 (1968). Torgerson, R. L., LIS thesis, Univ. of Delaware, June 1965. Van Ness, H. C., “Classical Thermodynamics of Non-Electrolyte Solutions,” p 136, Rlacmillan, New York, N.Y., 1964. Van Ness, H. C., AZChE J . , 16, 18 (1970). RECEIVED for review April 3, 1972 ACCEPTEDJune 7, 1972
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
75
