K H = Henry's law constant, in units of atm K H + = reference solubility, eq 15, in units of atmosphere K H * = K H at zero temperature coefficient, eq 16, in units of atm k = Boltzmann's constant rn = molecular mass N = number of molecules ' n = slope of the temperature coefficient of gas solubility, eq 11 PA = partial pressure of component A p*O = vapor pressure of pure component A Q = defined by eq 14 R = gasconstant S A = defined by eq 13 S B = defined b y e q 1 2 T = absolute temperature TI = temperature reduced by the critical temperature V I = volume of liquid phase u = molar volume uf = free volume for a single solute molecule X A = mole fraction of solute A in the liquid phase 2 = number of B molecules surrounding a single A molecule ZB = coordination number in pure solvent B Greek Letters cy = (U*/UB)" 6 = solubility parameter a t 25 "C, in units of (cal/cm3)1/2 c,u =Lennard-Jones (6,12) pair-potential parameters = pair-potential depth between A and B molecules { = constantineq3 9 = constantineq2 X = molecular shape parameter (A = 1 to 1.695, X = 1 for spheres)
p =
density
4 = (7r&/6)u3p/m, molecular packing density, i.e., bulkmean particle volume fraction Subscripts A = component A, solute B = component B, solvent Superscripts c = critical point property 0 = purecomponent L i t e r a t u r e Cited Battino, R., Clever, H. L.. Cbem. Rev., 66, 395 (1966). Davidson, N., "Statistical Mechanics," p 479, McGraw-Hili, New York, N.Y., 1962. Dyrnond, J., Hildebrand, J. H., lnd. Eng. Cbem., Fundam., 6, 130 (1967). Fleury, D., Hayduk, W., Can. J. Cbem. Eng., 53, 195 (1975). Gotoh, K., Nature (London) Pbys. Sci., 231, 108 (1971a). Gotoh, K., Nature (London) Pbys. Sci., 232, 64 (1971b). Gotoh, K., Nature(London) Pbys. Sci., 234, 138 (1971~). Gotoh, K.,Nature (London) Pbys. Sci., 239, 154 (1972). Gotoh, K., Ind. Eng. Cbem., Fundam.. 13, 287 (1974). Hayduk, W., Cheng, S. C., Can. J. Cbem. Eng., 48, 93 (1970). Hayduk, W., Buckley, W. D., Can. J. Cbem. Eng., 49,667 (1971). Hayduk, W., Castaneda, R., Can. J. Cbem. Eng., 51, 353 (1973). Hayduk, W., Laudie, H., A./.Cb. E. J., 19, 1233 (1973). Jolley, J. E., Hildebrand, J. H., J. Am. Cbem. SOC.,60, 1050 (1958). Jonah, D. A., King, M. B., Proc. Roy. Soc., London, Ser. A, 323, 361 (1971). Miller. R. C., J. Cbem. Pbys., 55, 1613 (1971). Munn, R. J.. Trans. FaradaySoc., 57, 187 (1961). Pierotti, R . A., J. Pbys. Cbem., 67, 1840 (1963). Snider, N. S., Herrington, T. M., J. Cbem. Pbys., 47, 2248 (1987). Staveley, L. A. K., J. Cbem. Pbys., 53, 3136 (1970).
Received for review October 6, 1975 Accepted June 7,1976
Vapor Equilibrium in Associating Systems (Water-Formic Acid-Propionic Acid) Abraham Tamir' and Jaime Wisniak Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sbeva, Israel
Experiments were conducted for determination of the vapor-liquid equilibria in the ternary system water-formic acid-propionic acid at atmospheric pressure. A thermodynamic model which accounts for nonideal gas behavior and association effects is used for computing the liquid overall activity coefficients. These were correlated using the Redlich-Kister equation and the effect of the ternary constant was found insignificant. The boiling points revealed a ternary saddle-point azeotrope which boils at 107.50 O C and contains 42 mol YO water, 54 mol % formic acid, and 4 mol % propionic acid.
Introduction Studies of multicomponent vapor-liquid equilibria with particular emphasis on the phenomena of association among the species are very few. From spectroscopic and other studies, it is well stablished that monomers of carboxylic acids tend to form dimers and tetramers, whereas in a mixture of the acids, association between dissimilar monomers to form a heterodimer is also possible. The association between the monomers is attributed to strong hydrogen bonds which are the strongest in propionic acid, less strong in acetic acid, and the weakest in formic acid. As a result of such association the number of moles in the system changes, which explains deviations from ideal gas behavior in certain cases. 274
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
The knowledge of the true composition in the phases is essential in vapor-liquid equilibria since equilibrium in association systems exists simultaneously between the monomers, dimers, and heterodimers. This can be obtained from the equilibrium constants for the various association reactions. For binary mixtures of carboxylic acids, it is evident that only by inclusion of the association effect in the analysis is it possible to obtain liquid activity coefficients which are thermodynamically consistent. This has been shown by Marek and Standart (1954) for the system water-acetic acid, by Tetsuo and Yoshida (1963) for water-formic acid, water-acetic acid, water-propionic acid, by Sebastiani and Lacquaniti (1967) for water-acetic acid, and by Wisniak and Tamir (1975) for
carbon tetrachloride-acetic acid. In all cases, it was sufficient to consider only the formation of dimers and to apply the model first suggested by Marek and Standart (1954) to calculate the liquid activity coefficients. In some other studies on binary mixtures containing fatty acids which were reported by Conti et al. (1960) and Fernand and Rivenq (1960), no data for the activity coefficients are reported. An extension of the model proposed by Marek and Standart (1954) for binary mixtures containing two associating species which undergo dimerization and heterodimerization was made by Tamir and Wisniak (1975). I t was successfully applied to acetic acidpropionic acid with regard to consistency of the overall activity coefficients. Studies on multicomponent mixtures containing associating components are very few. Conti et al. (1960) studied the vapor-liquid equilibria of acetic acid-chloroform-water with partial miscibility, acetic acid-formic acid-water, and acetic acid-formic acid-chloroform which are completely miscible. Kushner et al. (1966-1968) studied the systems water-formic acid-propionic acid. Common to the above investigations is the lack of data concerning the liquid activity coefficients and hence no evaluation of the results was made with regard to their thermodynamic consistency. In addition, no attention was paid to association effects which certainly exist in such systems. The objectives of the present investigation are to study ternary mixtures with emphasis on the association phenomena. A thermodynamic model for the vapor-liquid equilibria in such systems is developed which takes into account dimerization between similar species and heterodimerization between dissimilar species. Although it is derived with attention paid to the features of the ternary system studied experimentally, water-formic acid-propionic acid, its extension to multicomponent mixtures is straightforward. Unlike the model suggested in the past by Tamir and Wisniak (1975,1976),the present model is suitable also when the gas phase does not behave as an ideal gas. Hence, by considering association effects and deviation from ideal gas behavior, the calculated liquid activity coefficients are expected to be more accurate.
Vapor-Liquid Equilibria Relationships In previous investigations, Tamir and Wisniak (1975) suggested a method for the calculation of liquid activity coefficients in the presence of association effects. Indeed it was found that only by the inclusion of this effect were the data thermodynamically consistent. The analysis is extended here to include deviations from ideal gas behavior in the vapor phase which previously was not accounted for. The resulting fugacity coefficients can be calculated due to data provided by Nothnagel e t al. (1973). The derivations are based on a ternary system which was studied here experimentally. However, the extension of the analysis to a multicomponent system is straightforward. The ternary system consists of three species A, B, C (water, formic acid, and propionic acid) where species A consists of only the monomer designated as A. Species B consists of the monomer B1 and the dimer B2 formed by the reaction B l + B1+ B2
(1)
Species C consists also of the monomer C1 and dimer C2 where
In addition it is assumed that heterodimerization occurs according to B 1 + C1== BC
The true total number of moles, n , in the gaseous mixture is therefore
n = nA1
+ nel + n B 2 + nel + nen+ n B C
(4)
The true mole fraction of the monomer of A is
and similarly for the other monomers, dimers, and the heterodimer. The stoichiometric number of moles of the species is
nA = n A I
(6)
nB = ~ B I 2 n R ~+ n B C
+
(7)
+ 2nc? + n B C
(8)
nc
=
ncl
where the measured stoichiometric mole fraction Y A is YA
=
nA
nA
+ nB + nC
(9)
and similarly for species B and C. The relationships between y c and the corresponding true mole fractions, ‘7, are
Y A , YB,
S’A
=
‘7A1
1 + VB2
(10)
+ ‘7Cz + VRC
The true total number of moles, n , depends on the equilibrium constants of reactions 1, 2, and 3, namely
where f is the fugacity of the species in the gaseous mixture a t its total pressure and temperature. It is more convenient to use the true fugacity coefficient, defined for B1, B? as
and similarly for the other species. T o calculate the fugacity coefficients we use the equation of state
p=- nRT u - nb where n b is the excluded volume due to the finite size of the molecules (a mixture of monomers, dimers, and heterodimers). I t is well known that at constant temperature
(%-i)P
d l n & = RT
dP
Calculating the partial molar volume of species i, yields
vi,by eq 15
According to Hirschfelder (1942), a t low densities it is reasonable to assume that the excluded volume for the dimer is the same as that for the monomer; therefore b g l = bB, bH and bc., = bC2 bc. Following Nothnagel et al. (19731,we assume that nb = nA,bA +
+ nHJbH
(nB1
(3) Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
275
Table I. Physical Constants of Compounds at 25 "C Compound
Density
to eq 26 yields that the vapor-liquid equilibrium relationship for monomer B1 reads
Refractive index
YBXBPB~'= EBIIBJ' Formic acid
1.2136 1.2139" 0.9879 0.9880"
Propionic acid
1.3690 1.3693" 1.3837 1.3843a
(30)
where
Prausnitz (1969).
Similarly for A and C
Applying eq 17 yields
YAXAPA,' = E A ~ A , ~
(30.1)
YCXCPC1° = Ecvc1P
(30.2)
where
From definitions of the true mole fractions, fugacity coefficients, and eq 3 we obtain vB2 = K B B P ~ B ~ B ~ ' (22.1)
w L= KccP&vc12
(22.2)
'IBC = KBcP(&&/&BC)IIBItlCi
(22.3)
The substitution of the above equations into eq 10, 11,and 12 yields the following equations for the unknowns T A ~ ,V B ~ ,
'7c1 '7.41
+
from
PB~' = PB'VB~'
(2 - ~ c ) K c c P $ c v c ~+' v c l [ l
x
(iB$C/$BC)]
1
K B , P ( ~ B ~ , / ; ~ B C ) ~=~ 0B , ~(23.1) ~I]
(2 - Y B K B R P ~ B O B ~ ' + v B I [ l + vcl(l - YB) x (KBCP($B'$C/$BC)]- yB[1 + KCCP&&l']
PHlo = PB' =0
(23.2)
+ ' I B ] ( ~- Y ~ K B C P
- s'C[l
f
KBBP6B1B12]= 0 (23.3)
The gaseous phase may be related to the liquid phase through equilibrium relationships for the monomers, dimers, and the heterodimer. For the monomer of B, for example, fRIL
= fB1'
(24)
=&~I~?IBI
(25)
XB
(27)
f~,'~
The calculation of f B 1 1 2 / Pwhere is the fugacity of hypothetical pure B1 as liquid at total pressure P and temperature of the solution is as follows: a t the vapor pressure PB]' of pure B1 and from eq 14 and 19
where here fBI1. is a t P B ~ ' By . using eq 16 and 28, one ultimately obtains that f B I L a t pressure P is given by
V H , ' is . the liquid molar volume of a hypothetical monomer B1 which will be assumed to be equal to the molar volume of . of eq 29 and 19 species B, namely, V B , =~ V B ~Substitution Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
(33)
2KBBPB' eXp -
i-l+\
A similar equation can be derived for component C. Substi, from eq 30 and 27 into eq tution of the values of V A , , T B ~ vc1 23 yields finally three equations for the unknowns YA, YB, and yc,. The solution of the following equations can be aided by an iterative technique that utilizes the Newton-Raphson procedure (Perry, 1973) to ensure fast convergence
K R B ~ B ( Y B X B P B , ~ / E B ) '+ K ~ ~ ~ ~ ( Y ~ x ~ P ~ ~ ~ + K H c ( ~ B ~ c / ~ B c ) ( Y B X B ~ B ~ ' / ~ B ) ( Y ( . X C ~ C ~ ' =/ ~ 0C ) ]
( 2 - Y R ) K B B ~ B ( Y B X B P B ~ '+ / E(BY )B' X B P B , " / E B ) [ ~ + (1 - YB)(YCxCPCl'/EC) + KBC($B?'C/$BC)I - Y B [ P+ K c c & ( ~ c ~ c P c ~ ' / E=c 0) ~ ](34.2)
where the overall activity coefficient of B is defined by YBI~RI
-
(34.1)
Noting eq 19, eq 25 may be written as
YR = -
[b?;O1) [ b"RPo1
+ ~KBBPB'eXp
YAXAPA~'IEA - YA[P
or alternatively with aid of eq 14 YBIEBlfBIL'
(32)
which is the mole fraction of the monomer in pure species B is obtained from eq 13,14, and 19. Hence PB~' reads
- Y;\[I + K B B P ~ B ~ B ~ * f KCcP6C'?Ci2 f
276
In the case of component A, which exist only as monomer,
PA,' = PA'. For pure B (monomer dimer) which exerts a vapor pressure of PB', the partial pressure of the monomer, P R ~ ' required , in the previous equations may be calculated
(2 - Y ~ ) K ~ ~ ~ C ( ~ C ~+ ~(YcXcpc,'/Ec)[l P C ~ ' / E ~ ) ' + (1- Y C ) ( Y B X B P B ~ ' /+E KBC(~B&/$BC)I B) -Yc[p + KBB($B(YBXBPB~'/EB)*] =0 (34.3)
Experimental Section Purity of Materials. Analytical grades of formic and propionic acids purchased from Merck, Fluka, and B.D.H. were used. Gas chromatograph analysis showed some water in the acids of the order of 0.025 mole fraction. However, since the experimental system investigated was water-formic acid-propionic acid, no attempt was made for further purification of the acids. Physical properties of the pure components appear in Table I where distilled water was used. Apparatus and Procedure. An all-glass modified Dvorak and Boublik recirculation still (Boublikova, 1969) was used in the equilibrium determinations. The experimental features have been previously described by Wisniak and Tamir (1975).
Table 11. Ternary Vapor-Liquid Equilibrium Data for Water( 1)-Formic Acid(2)-Propionic Acid( 3) at 760 mmHg Temp, "C
Vapor compn.
Liquid compn.
Activity coeff.
Obsd
Calcd
XI
x2
X3
Y1
Y2
Y3
Y1
Y2
Y3
100.85 101.28 101.65 101.85 101.97 102.25 102.25 102.51 102.97 103.31 103.36 104.18 104.75 105.10 105.74 105.92 106.20 106.38 106.42 106.74 106.88 106.88 106.98 107.00 107.03 107.04 107.15 107.27 107.41 107.42 107.43 107.44 107.55 107.82 107.85 108.01 108.05 108.12 108.24 108.33 108.80 108.96 109.06 109.50 110.32 110.43 111.05
100.83 101.02 101.44 101.55 101.89 102.49 102.07 102.56 102.58 103.00 102.45 103.84 104.89 104.53 105.22 105.00 105.60 106.60 106.42 106.34 106.91 106.98 107.54 107.58 107.56 106.62 107.56 107.08 107.83 107.49 107.71 107.59 107.49 107.82 108.25 108.32 108.26 108.53 108.29 108.61 108.94 109.06 109.17 109.24 109.44 109.66 110.63
0.931 0.933 0.903 0.886 0.831 0.830 0.857 0.813 0.772 0.796 0.832 0.761 0.718 0.695 0.676 0.691 0.652 0.592 0.605 0.600 0.571 0.574 0.542 0.394 0.393 0.586 0.539 0.533 0.527 0.466 0.333 0.289 0.529 0.442 0.496 0.484 0.498 0.358 0.373 0.481 0.460 0.455 0.436 0.375 0.208 0.354 0.347
0.044 0.056 0.076 0.077 0.071 0.116 0.101 0.106 0.073 0.130 0.114 0.192 0.211 0.213 0.242 0.254 0.279 0.258 0.230 0.310 0.274 0.219 0.180 0.522 0.526 0.289 0.243 0.373 0.199 0.446 0.530 0.555 0.299 0.425 0.256 0.280 0.246 0.434 0.444 0.132 0.174 0.193 0.265 0.356 0.464 0.341 0.232
0.025 0.011 0.021 0.037 0.098 0.054 0.042 0.081 0.155 0.074 0.054 0.047 0.071 0.092 0.082 0.055 0.069 0.150 0.165 0.090 0.155 0.207 0.278 0.084 0.081 0.125 0.218 0.094 0.274 0.088 0.137 0.156 0.172 0.133 0.248 0.236 0.256 0.208 0.183 0.337 0.366 0.352 0.299 0.269 0.328 0.305 0.421
0.939 0.958 0.935 0.903 0.838 0.847 0.867 0.867 0.792 0.824 0.889 0.827 0.790 0.780 0.725 0.748 0.708 0.643 0.717 0.665 0.669 0.706 0.679 0.406 0.406 0.670 0.668 0.570 0.689 0.483 0.320 0.287 0.623 0.478 0.636 0.593 0.554 0.404 0.395 0.674 0.656 0.638 0.579 0.456 0.290 0.404 0.542
0.037 0.032 0.044 0.064 0.067 0.100 0.093 0.069 0.063 0.111 0.080 0.133 0.154 0.147 0.204 0.199 0.232 0.218 0.178 0.262 0.223 0.173 0.160 0.505 0.505 0.235 0.202 0.351 0.163 0.440 0.575 0.604 0.265 0.420 0.214 0.261 0.263 0.383 0.467 0.152 0.153 0.171 0.249 0.376 0.479 0.375 0.230
0.024 0.010 0.021 0.033 0.095 0.053 0.040 0.064 0.145 0.065 0.031 0.040 0.056 0.073 0.071 0.053 0.060 0.139 0.105 0.073 0.108 0.121 0.161 0.089 0.089 0.095 0.130 0.079 0.148 0.077 0.105 0.109 0.112 0.102 0.150 0.146 0.183 0.213 0.138 0.174 0.191 0.191 0.172 0.168 0.231 0.221 0.228
0.98655 0.98492 0.98471 ,96952 0.97658 0.97170 0.95718 1.00219 0.97584 0.95591 0.96595 0.96945 0.96113 0.99732 0.93495 0.92926 0.93502 0.95765 1.01464 0.95059 1.00238 1.04129 1.07061 0.96954 0.97093 0.97089 1.05299 0.93108 1.09522 0.92814 0.92724 0.97374 0.99958 0.95896 1.07636 1.03763 0.95820 1.03380 0.96158 1.14480 1.15470 1.13570 1.09170 1.03096 1.24299 0.96158 1.21972
1.40181 1.00389 0.93872 1.24826 1.25088 1.16901 1.29028 0.89377 1.05097 1.09317 0.99209 0.88084 0.86872 0.81607 0.93325 0.88336 0.90266 0.86509 0.83211 0.87931 0.84040 0.83171 0.91200 0.89712 0.88976 0.83925 0.84931 0.92067 0.83904 0.92982 0.97638 0.97304 0.87943 0.91996 0.82442 0.89868 1.00920 0.79301 0.94513 0.82744 0.85206 0.84835 0.87759 0.93780 0.86735 0.94190 0.87011
5.18930 5.38470 5.23337 4.15034 3.73730 3.92733 4.00489 3.23132 3.21177 3.26160 2.48073 3.15836 2.67589 2.65543 2.65157 3.03925 2.59740 2.53755 1.87785 2.28460 1.94299 1.67890 1.60092 2.49965 2.59053 2.12129 1.64080 2.16612 1.49619 2.13000 1.72695 1.55576 1.72034 1.83846 1.58803 1.57082 1.75672 2.32358 1.71373 1.37222 1.34701 1.37721 1.40580 1.41367 1.46146 1.56137 1.22592
All analyses were carried out by gas chromatography on a Packard-Becker Model 417 apparatus provided with a thermal conductivity detector and an Autolab Minigrator type of electronic integrator. The column was 200 cm long and 0.32 cm in diameter and was packed with Chromosorb 101 and operated by programming from 140 to 175 "C with 10 "C/min. Injector temperature was 240 "C and the detector operated at 150 mA and 250 "C. Concentration measurements were generally accurate to f0.005 mole fraction unit. Boiling temperatures were measured with a Hewlett-Packard quartz thermometer, Model 2801 A with an accuracy of fO.OOO1 "C. The boiling temperatures were stable within f0.02 "C.
Results and Discussion Vapor-liquid equilibria measurements at 760 mmHg were made for the ternary system water-formic acid-propionic acid. The results for the boiling temperatures and concentrations are given in Table 11. T h e overall liquid activity coefficients, ?A, Yn, y c were calculated by applying the general
model previously described which accounts for deviations of the vapor phase from ideal gas behavior and association effects. The data required for computing the various fugacity coefficients were taken from Nothnagel (1973) and Bondi (1968). Vapor phase association effects were the dimerization between similar molecules of an acid and heterodimerization between molecules of both acids. The dimerization constants employed were: formic acid, Tetsuo (1963) log KBB= -10.743
3083 +T
(35)
3316 +T
(36)
propionic acid, Tetsuo (1963) log Kcc = -10.843
As no data are available for the heterodimerization constant, they were computed according to Prausnitz (1969) by Knc = 2-\/Ks~Kcc Ind. Eng. Chem., Fundam., Vol. 15,No. 4, 1976
(37) 277
( c )The Ability of Correlating Satisfactorily the Boiling Points Which Is Evaluated by
The vapor pressure of the pure components required in eq 30-33 were: water (noting that PA^" = PA"),Hala (1968) log PA" = 7.96681 -
R.M.S.D. = ~
1668.21 t 228.0
+
~
t
We used for this purpose the equation suggested by Wisniak and Tamir (1976) which reads
1563.28 247.06
(39)
+
T = XlTl"
propionic acid, TRC Tables (1961)
+
1617.06 log Pco = 7.54760 t 205.67
I
Yib
- 1%
?,a)
1=1
< 0.04
+ 5D,,)xi2
(41)
+
~ ~ ~ ) 1 x 2~ 3~ (44.1) ~ ~
The goodness of prediction of the observed values of the vapor composition and activity coefficients reported in Table I1 by the Redlich-Kister equation was as follows. The mean value of ( Y I . ~ , ~ , yl,cal)/yl,obs ~ based on 47 data points was 10.2%;for yz and y 3 the mean values were 12.3 and 21%. The worst values were 29.6,50, and 54%, respectively, where similar figures were obtained for the activity coefficients. 278
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
+ B(x1-
+ . . .]
~ 3 )
z2) (46)
x, In
(Y,/x,)
(47)
"c
- 4(Ci, + 4Dij)Xj3 + 12Di,Xj2 (44)
+ 4(ci, - 4
xlxzxs[A
Here T o is the boiling point of the pure component in where 1 is the number of ( x , - x,) terms in the series, C k are binary constants, and the rest are ternary constants. We have found from many applications of the above equation that the variation of the local value of the R.M.S.D. (eq 45) along a set of data points is very sensitive to irregular values of the measured boiling point. Hence, the assessment of goodness of prediction of the boiling point by the above equation is recommended as an additional test for rejecting irregular data points. The various constants appearing in eq 43,44, and 46 were obtained by the Simplex method and are reported in Tables I11 and IV. For the binary systems water-formic acid and water-propionic acid we used the experimental results of Tetsuo and Yoshida (1963) where for formic acid-propionic acid the data of Tamir and Wisniak (1975) were used. The major conclusions drawn from these tables are the following. (a) The ternary system water-formic acid-propionic acid can be estimated solely from the data for the binary pairs because C1 is relatively small. (b) The term 1 ~ (eq ' 47) in the correlation of the boiling point (eq 46) may be neglected as concluded from the values of the R.M.S.D. in Table IV. This is true either for the binary or the ternary mixtures. This fact introduces a considerable simplification in computing the bubble points of mixtures since it becomes solely a function of the liquid composition. Bearing the last conclusion in mind, isothermals for the boiling points were computed by the aid of eq 46, and are reported in Figure 1. The behavior of these isothermals reveal a ternary saddle-point azeotrope. It is located as shown by the circle in Figure 1where "climbing uphill" from it leads quite quickly to a binary azeotrope which boils a t 107.6 "C and contains according to Tetsuo (1963) 42.4 mol % of water; the rest is formic acid. The determination of the ternary azeotropic point composition and boiling temperature was made by drawing the "suspected region" in spacial coordinates as shown in Figure 2 and with the aid of eq 46. According to the general theorem of Gibbs-Konovalov (Malesinsky, 1965), a ternary system exhibits azeotropic behavior a t constant pressure if
The equations for the other activity coefficients are obtained by cyclic rotation of the indices where C1 is a ternary constant. The binary constants are obtained from
In y, = (Bi, - 3CiJ
1+
Ck(x, - x,)~
1=1
+
+ 5Di,)xj2
k=O
3
In Y I = x z x : d ( B + ~ B I :-~ B23) + C12(2x1 - X Z ) + C13(2x1 - x3) + 2C23b3 - X d Diz(Xi - xd(3Xi - X ? ) + D13(xi - X3)(3xi - x 3 ) - 3Dz:r(X:j - X2)' f Cl(1 - 2 X 1 ) ] Xz2[B12 + c12(3Xi - X z ) + D12(Xi - Xd(5xi - xz)] + X3'[B12 + c13(3X1 - X d D13(X1 - x3)(5x1 - x3)I (43)
(B,j + 3Cij
2
w=C
where the data should obey an arbitrary limit for the value of the R.M.S.D. to indicate a good correlation. In the present case, a value of 0.05 was employed as a limit for correlating the data according to the Redlich-Kister equation (1948).
yl =
1
[xlx,
where
The summation is carried out over all experimental measurements corresponding t o the number of components, n , between two pairs of points, a, b. (b) The Ability of Correlating the Liquid Activity Coefficients by a Multicomponent Thermodynamic Equation Which Will Satisfactorily Predict the Vapor Composition. According to Ma et al. (1969), if this is achieved not all the data are required to satisfy the point-to-point test of McDermott-Ellis. Satisfactory prediction is quantitatively evaluated by the average root-mean-square deviation
In
J=1
+ ~ 2 T 2 "+ ~ 3 T 3 "+ w + C ( X -~ ~ 3 +) D ( z -~
In selecting the final 47 points reported in Table I1 we considered their behavior in regard to three tests. (a) The McDermott-Ellis Two-Point Test. This was applied first to reject points that were clearly inconsistent. Bearing in mind the accuracy of our concentration measurements (4=0.005mole fraction unit), it was decided according to McDermott (1964,1965) to accept the measurements which satisfy n
t
(40)
+
'%?E(xis + x,b)(log Yib - 1%
(45)
m
formic acid, TRC Tables (1961) log PB" = 7.3779 -
Z F(Tabs I - Teal)'
~
~
4
-dT = - - aT - 0 ax2 ax3
This condition is fulfilled in the present case as clearly reflected in Figure 2. The ternary saddle-point azeotrope was found to contain 42.0 mol % water, 54.0 mol % formic acid, and 4.0 mol % propionic acid at 107.50 T 0.02 "C. I t is instructive to compare our experimental results with those obtained by Kushner et al. (1967). In Table V we give some features relating to both investigations. The first point
Table 111. Redlich-Kister Correlation of Binary and Ternary Data, Eq 43 and 44 R .M .S.D .
Ci,
Bii Water-formic acid Water-propionic acid Formic-uroDionic acids
-0.34522 1.35580 0.40364
Water-formic acid-propionic acids
0.11201 0.51097 -0.02460
Dii
Yl
Yi
0.05485 0.08022 0.07688
0.0433 0.1729 0.1243
0.0148 0.0653 0.0519
Overall y
R.M.S.D. y (eq 42)
0.5522 0.5505
0.04551 0.04551
C1 = 0.03911 c1=0
Table IV. Correlations of Boiling Points, Eq 46
co
System Water-formic acid with w without w Water-propionic acid with w without u: Formic-propionic acid with LO without UI Water-formic-propionic acids with u' without w
28.010 29.996
CI
C:,
-12.069 -12.214
-5.4562 1-5.7773
R.M.S.D.
C3
7.7532 7.8967
0.0648 0.0656
-59.759 -60.515
46.263 47.481
-62.272 -63.606
30.968 32.067
0.1281 0.1352
-28.842 -29.127 A 80.854 82.458
8.6556 8.9576 B 150.63 147.72
-8.5109 -8.6458 C -83.343 -80.007
12.380 12.342 C -6.2794 -9.2826
0.4208 0.4238 0.3893 0.3949
Table V. Comparison between t h e Present Work and the Work of Kushner (1967) Azeotropic point
Kushner (1967) Present work
No. of exptl points
R.M.S.D Y (eq 42)
R.M.S.D. T (eq 45, 47)
mol % water
mol % formic acid
BP, "C
20 47
0.0741 0.0455
0.190 0.395
37.9 42.0
57.4 54.0
107.20 107.50
FORMIC ~1006'C)
' ACID
Figure 1. Isothermals at 760 mmHg for the system water-formic acid-propionic acid calculated by eq 46 for w = 0. to emphasize in regard to the ternary data of Kushner (1967) that nothing has been mentioned concerning their thermodynamic consistency. Therefore, we first applied our model t o compute the liquid overall activity coefficients and found that only 15 out of the 20 data points of Kushner (1967) satisfy the McDermott-Ellis consistency test (eq 41) and hence may be considered as acceptable. Secondly, we correlated the ternary data by t h e Redlich-Kister eq 43. T o achieve this we used the data of Tetsuo and Yoshida (1963) and our own
'ACID X 2 # Figure 2. Determination of the ternary saddle point azeotrope for the system water-formic acid-propionic acid at 760 mmHg. (1975) to compute the binary constants. The latter were based on consistent data of the overall activity coefficients corrected for association effects. Finally, we correlated the boiling points of Kushner (1967) according t o eq 46 using the above binary data for computing the binary constants. Considering the above and the comparison in Table V, we conclude that our Ind. Eng. Chem.. Fundam.. Vol. 15, No. 4, 1976
279
ternary data are more accurate than those of Kushner (1967). This does not create a significant difference in the value of the azeotropic boiling point; however, the content of water as reported by Kushner (1967) is higher by 11%,as indicated in Table V. The differences between the results may also be appreciated by considering the dotted isothermals in Figure 1 based on the ternary data of Kushner (1967).
Acknowledgment Thanks are due to Yehudit Reisner and Moshe Golden for their contribution in the experimental and numerical work. Nomenclature A, A1 = species A; monomer of A b = size parameter B, B1, B2 = species B formed by B1 Bs; monomer of B; dimer of B C, C1, Cp = species C formed by C1 C2; monomer of C; dimer of C BC = heterodimer formed by B1 + C1 D = parameters in eq 44,46 E = defined by eq 30 f = fugacity of a component in the gaseous mixture a t its total pressure and temperature f = fugacity of pure component KAA,K B B , Kcc, K B C = vapor phase equilibrium constant for the formation of AS, B2, C2, BC, respectively, mmHg-' rn = total number of experimental runs n = true total number of moles; number of species in eq 41, 42 P = total pressure, mmHg PIo = vapor pressure of the pure species i (i = A, B, C), mmHg PI = vapor pressure of the pure monomer of species i (i = A, B, C), mmHg P,20 = vapor pressure of pure dimer of species i (i = A, B, C), mmHg R = universal gas constant t , T = temperature OC, K u- = total volume V , V = partial molar volume, molar volume x,,y , = stoichiometric mole fraction of species i in the liquid phase; in the vapor phase (i = A, B, C) yjr,,..l = the calculated vapor composition of the ith component in the j t h experimental run.based on values of liquid activity coefficients which were computed from their multicomponent thermodynamic correlation ?A,, YB,, yc, = liquid activity coefficients for the monomer of A (or B, or C)
+ +
280
Ind. Eng. Chem., Fundam., Vol. 15,No. 4. 1976
yc = overall liquid activity coefficient = true mole fractions of species i in the liquid phase; in the vapor phase (i = A, B, C) = fugacity coefficient for a component in a mixture; defined for B in eq 14 and similarly defined for the other
?A, YB,
ti,.vI $
species
Subscripts and Superscripts A = water B = formic acid C = propionic acid i = species L, V = in the liquid phase; in the vapor phase cal = calculated obs = observed O = purespecies 1, 2 = monomer, dimer 1, 2, 3 = water, formic acid, propionic acid in eq 43-47 and in Tables 111, IV Literature Cited Bondi, A., "Physical Properties of Molecular Crystals, Liquids and Glasses", Wiley, New York, N.Y., 1968. Boublkikova, L., Lu, B. C. Y., J. Appl. Chem., 19, 89 (1969). Conti, J. J., Othmer, D. F., Gilmont. R., J. Chem. Eng. Data, 5, 301 (1960). Hala, E.,et al., "Vapor-Liquid Equilibrium Data at Normal Pressures", Pergamon Press, 1968. Hirschfelder, J. O., McClure, F. T., Weeks, I. F., J. Chem. Phys., I O , 201 (1942). Kushner, T. M., Tatsievskaya, G. I., Serafimov, L. A,, Zh. Fiz. Khim., 42, 2248 (1968). Kushner, T. M., Tatsievskaya, G. I., Serafimov, L. A,, Zh. Fiz. Khim., 41, 237 (1967). Kushner, T. M., Tatsievskaya, G. I., Irzun, V. A,, Volkova, L. V., Serafimov. L. A., Zh. Fiz. Khim., 40, 3010 (1966). Malesinsky, W.,"Azeotropy and Other Theoretical Problems of Vapor-Liquid Equilibrium", Interscience, London, New York. N.Y.. Sydney, 1965. Ma, K . T., McDermott, C., Ellis, S. R. M., Ind. Chem. Eng. Symp. Ser., 32, 104 (1969). Marek. J., Standart. G., Coll. Czech. Chem. Commun., 19, 1074 (1954). McDermott, C., Ellis, S. R. M., Chem. Eng. Sci., 20, 293 (1965). McDermott, C., Ph.D. Thesis, University of Birmingham, 1964. Nothnagel, K. H., Abrams, D. S., Prausnitz, J. M.. lnd. Eng. Chem., Process Des. Dev., 12, 25 (1973). Perry, J. H., "Chemical Engineers Handbook", 5th ed, pp 2-54, 1973. Prausnitz, J. M., "Molecular Thermodynamics of Fluid-Phase Equilibria", p 138, Prentice-Hail, Englewood Cliffs, N.J., 1969. Redlich, O., Kister, A. T., lnd. Eng. Chem., 40, 345 (1948). Rivenq, F., Bull. SOC.Chim. Fr., 1505-1507 (1960). Sebastiani. E., Lacquantini, L., Chem. Eng. Sci., 22, 1155 (1967). Tamir. A., Wisniak, J., Chem. Eng. Sci.. in press, 1976. Tamir. A.. Wisniak, J., Chem. Eng. Sci., 30, 335 (1975). Tetsuo, I., Fumitake. Y.. J. Chem. Eng. Data, 8, 315 (1963). "TRC Tables Selected Values of Properties of Chemical Compounds", Thermodynamic Research Center Data Project. College Station, Texas, 1961. Wisniak, J., Tamir, A,, Chem. Eng. Sci., in press, 1976. Wisniak, J., Tamir, A,, J. Chem. Eng. Data, 20, 168 (1975).
Received f o r review October 9, 1975 Accepted M a y 27,1976
