Ind. Eng. Chem. Fundam. 1984, 23, 126-128
128
p-L = difference between empty hydrate and liquid phase Subscripts 0,O = property at ice point and zero kPa pressure j = component identification m = cavity type m w = water R = reference hydrate Registry No. Cyclopropane hydrate, 26163-08-6.
Literature Cited Childs, W. C. J. Phys. Chem. 1964, 68, 1834. Deaton, W. M.; Frost, E. M. U S . Bur. Mlnes Monogr. 8 1948. Dharmawardhana, P. B.; Parrish, W. R.; Sloan, E. D. Ind. Eng. Chem. fundam. 1980, 19. 410. Dharmawardhana, P. B.; Parrish, W. R.; Sloan, E. D. Ind. Eng. Chem. Fundam. 1981, 20. 306.
Holder, G. D.; Corbln, G.; Papadopoulos, K. D. Ind. Eng. Chem. fundam. 1980, 19, 282. Parrish, W. R.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1972, 1 1 , 26. Sortland, L. D.; Robinson, D. 8.Can. J. Chem. Eng. 1984, 42, 38. van der Waals, J. H.; Platteeuw, J. C. Adv. Chem. Phys. 1959, 2, 1-55.
Chemical and Petroleum Engineering Gerald D. Holder* Department Shirish T.Malekar University o f Pittsburgh Pittsburgh, Pennsylvania 15217 Chemical and Petroleum Engineering E. Dendy Sloan Department Colorado School of Mines Golden, Colorado 80401 Received for review October 26, 1982 Accepted October 7, 1983
Thermodynamic Consistency of Vapor-Liquid Equilibrium Data for the Water-Formaldehyde System
A method for testing the thermodynamic consistency of vapor-liquid equilibrium data for the water-formaldehyde system is developed. The method has a sound thermodynamic basis since it is founded on the Gibbs-Konovaiow theorems. Application to two sets of experimental data taken from the literature Is given.
Introduction Gmehling and Onken (1977), in their comprehensive collection of vapor-liquid equilibrium data for binary and multicomponent mixtures at moderate pressures, usually give data correlation and consistency tests besides experimental results for each data set. However, they do not give them for the water-,xmaldehyde system. In fact, for this particular system they give only the experimental data as measured by several authors. Brandani et al. (1980) have pointed out that, because of strong associations, the usual equations (like those employed by Gmehling and Onken for the other systems) for the activity coefficients are inadequate in correlating the experimental results for the water-formaldehyde system. For this reason, they developed a thermodynamic model, based only on chemical forces, which gives an accurate description of the vapor-liquid equilibrium for this system in the ranges of temperature and composition where the liquid aqueous solutions of formaldehyde are stable. The reasons the data for the water-formaldehyde system were not examined for consistency lies mainly in the difficulties which arise in evaluating the fugacity of formaldehyde in its standard state to calculate its activity coefficient. These difficulties arise from the need to extrapolate the vapor pressure of formaldehyde above its normal boiling point (Tb= -22 “C),for which there is a lack of experimental data, and from the fact that the Poynting correction to the vapor pressure cannot be ignored in this case and involves a reasonable estimate of the molar volume of formaldehyde in a hypothetical state, which is a troublesome task. In any case, the integral consistency test would not be performed because the aqueous solutions of formaldehyde are stable in a limited range of composition, depending on temperature. In fact, for each temperature, above a certain value of the formaldehyde mole fraction there appears a precipitated phase, which is a liquid phase containing large
0 0
005
010
020
015
025
030
035
X!
Figure 1. Vapor-liquid equilibria for formaldehyde (1)-water (2) at 760 torr.
amounts of water, although the concentration of polyoxymethylene glycols therein considerably exceeds the concentration in the supernatant phase. Among the sets of data collected by Gmehling and Onken (1977) on the water-formaldehyde system there is the isobaric set of Olevsky and Golubev (1954) and that of Tsochev and Petrov (1973), both at 760 torr. For the two data sets, Figure 1 shows the behavior of vapor composition with liquid composition. While the data of Olevsky and Golubev (1954) clearly prove the presence of an “apparent” azeotrope, those of Tsochev and Petrov (1973) indicate that formaldehyde has the lowest volatility
0196-4313/84/1023-0126$01.50/00 1984 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
D =
(3)
50
in the whole range of composition. Moreover, both seta of data have a minimum on the boiling point curve: 97.4 "C (xF = 0.131) for the data of Olevsky and Golubev (1954); 97.2 "C ( x F = 0.2365) for those of Tsochev and Petrov (1973). Clearly, the two seta of data are contradictory and we do not have, at preaent, any criterion for establishing which one of them is thermodynamically inconsistent. The purpose of this paper is to develop a method for checking the thermodynamic inconsistency of the vapor-liquid equilibrium data of the water-formaldehyde system, at least approximately. Theory, Results, and Discussion In a liquid mixture of water and formaldehyde, as well as water and anhydrous formaldehyde, we also find the species WFi, which are formally constituted by i molecules of formaldehyde and one molecule of water (Staudinger, 1932). Because of the low volatility of polymeric hydrates of formaldehyde, it can be assumed that in the vapor phase only the methylene glycol, WF, is present besides water and anhydrous formaldehyde (Piret and Hall, 1949). A stable liquid solution of aqueous formaldehyde, in equilibrium with its vapor, constitutes a two-phase system with (c 2) components, where c represents the maximum value of i in WFi. In such a system, c chemical reactions (i = 1, 2, ..., C) F WFi.1 = WFi (1)
ni,
n i p > n; ',... . n;. . . . . ni6e.2niFz. n;,
c
0
0
0
c.1
0
0
0 " "
c.2
0
0
0
: I
0
n
0 " "
3
0
0
l
'
2
0
1
0
.... 0
1
1
0
o . . . .0
1'
0
0
0 " "
1
0
0
1
.n;
Du=
+
+
127
'."
0
.... 0
0
1
0
....
0
1
0
.... 0
,...
1
0
0
1
....
0
0
0
"
0
0
0
... 0
0
0
.... 0
0
0
1
....
0
0
0
0
....
0
0
0
O
(other than transfers between phases) can occur. For this reason, the system here considered is divariant f = 2 + (c+2-c)-2 = 2 (2) where f is the number of degrees of freedom. Therefore, we can apply the second Gibbs-Konovalow theorem, which states that "if, among the temperatures which maintain the system in equilibrium at a given fixed pressure, there is an extreme temperature, then the state corresponding to this temperature is indifferent" (Prigogine and Defay, 1954a). For a detailed discussion of the meaning of the indifferent states see Prigogine and Defay (1954b). In our case, the condition that the system is indifferent is expressed by eq 3, where D is a determinant of order (c 2); in fact, we have (2 + c ) components (columns), 2 phases, and c independent chemical reactions (2 c) rows. For convenience, in writing D, we referred to the new set of independent reactions W iF = WFI (i = 1, 2, ..., C) (4)
+
Dr
=
" "
+
+
(5b)
Applying the method of Laplace, for Dw, DF, and DWF
Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
128
nL
C
0
0
0
0
0
1
1
c.1
0
0
0
0
1
0
1
c.2
0
0
0
1
0
0
1
i
o
0
1
0
0
0
1
3
0
1
0
0
0
0
1
2
1
0
0
0
0
0
1
1
0
0
0
0
0
0
.nA,
l
D ’,
nip,., n L C ,nu:
nGF, n.;.,
n;
( 5(
are obtain respectively C
Dw = nk
+ CinbF, i=l
DF = nb
C
+ icnbp, =l
Adding and subtracting nhF to the right-hand side of eq 8, we get C
C
DWF
= nh
+ iC= l n h i- nZ( - rCinbFt =l
(9)
Introducing eq 6,7, and 9 into eq 5, after rearranging and dividing by the total number of moles in the vapor phase and in the liquid phase, we obtain C
uW(zF
C
+ CiZWF,) - uF(zW + CZWFi) + i=l
i=l
C
uWF(ZF
C
+ L=l C i Z W F , - ZW - C Z W F , ) i=l
= O (10)
where u and z indicate the true mole fractions in the vapor and in the liquid phase, respectively. Remembering that UWF
= 1 - U W - UF
(11)
from eq 10, by simple algebra, we obtain
Taking account of the definition of molar fractions, eq 15 finally gives YF = X F (16) i.e., the state corresponding to the minimum on the boiling point curve is an “apparent” azeotrope. We can conclude, therefore, that the vapor-liquid equilibrium data for the water-formaldehyde system of Tsochev and Petrov (1973) are thermodynamically inconsistent because they do not follow the Gibbs-Konovalow theorem, as they do not present an “apparent” azeotrope. As suggested by our experience, they probably adopted for the analysis of vapor-phase samples a technique which caused a systematic loss of formaldehyde, likely due to the formation of paraformaldehyde in some part of the withdrawal circuit. Since the data of Olevsky and Golubev (1954) include an “apparent” azeotrope, they are undoubtedly more reliable than those of Tsochev and Petrov (1973), even though we cannot say anyting about their accuracy. Moreover, the result of this work confirms the reliability of the thermodynamic model developed by Brandani et al. (1980). This model, in fast, predicts, according to the Gibbs-Konovalowtheorem, “apparent” azeotropes below and above atmospheric pressure. It is interesting to note that this model gives the poorest predictions of vapor composition for the experimental data of Tsochev and Petrov (1973),while the predicted vapor mole fractions are within 0.020 mole fraction of the experimental data for all the other sets reported by Gmehling and Onken (1977). This result can now be explained on the basis of the thermodynamic inconsistency of the data of Tsochev and Petrov (1973), as demonstrated in this work. Conclusions A method has been presented for testing the thermodynamic consistency of the vapor-liquid equilibrium data for the water-formaldehyde system. This method, which is based on the Gibbs-Konovalow theorems, can be applied for temperatures greater than 65 OC, which is the temperature predicted by the model of Brandani et al. (1980) at which “apparent” azeotropes appear for the waterformaldehyde system. In any case, we must point out that the method developed in this work can be used only as a rule of thumb, i.e., for rejecting data which are very wrong like those of Tsochev and Petrov (1973). Acknowledgment The authors are indebted to the Italian Consiglio Nazionale delle Richerche for financial aid. Registry No. Formaldehyde, 50-00-0.
Literature Cited
As shown by Brandani et al. (1980), from a material balance around the two phases, we have 1-uF 1 - u W
YW
=-
YF
(13)
where x and y indicate the “apparent” (stoichiometric) mole fractions in the liquid and in the vapor phase, respectively. Substitution of eq 13 and 14 into eq 1 2 gives
Brandani, V.; Di Giacomo, G.; Foscolo, P. U. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 179. Gmehllng, J.; Onken, U. “Vapor-Liquid Equilibrium Data Collection”; DECHEMA: Frankfurt, 1977; Vol. 1. Olevsky, V. M.; Golubev, I. F. T r . G k p . Vyp. 1954, 4 , 36. Piret, E. L.; Hail. M. W. I n d . €ng. Chem. 1949. 4 1 , 1277. Prigogine, I.; Defay, R. “Chemical Thermodynamics”, Longman: London, 1954; Chapter XXIX, p 480. Prigoglne, I . ; Defay, R. “Chemical Thermodynamics”; Londman: London, 1954 Chapter XXIX, p 470. Staudinger, H. “Dle Hochmolekularen Org. Verbindungen”, Springer: Berlin, 1932; p 232. Tsochev, V.; Petrov, P. Z . Phys. Chem. (Le@@) 1973. 252, 337.
Istituto di Chimica Applicata e Industriale Facolta d i Ingegneria Universitci de L’Aquila 67100 L’Aquila, Italy
Vincenzo Brandani* Gabriele Di Giacomo
Received for review February 22, 1983 Accepted August 9, 1983
