Ind. Eng. Chem. Process Des. Dev. 1980, 19, 123-128
X,Xoverdl = extent of solid reaction in bed, overall extent of reaction Greek Letters 8 = "effectiveness factor" defined by eq 7 and 8 tp = pellet porosity t, = bed void fraction k = dimensionless rate constant defined by eq 6 y = dimensionless radial coordinate within pellet 0 = dimensionless gaseous reactant concentration defined by eq 4 = dimensionless position of reactant front within grain ps = true molar density of solid u and 8 = reaction moduli defined by eq 2 and following eq 10 = dimensionless gaseous reactant concentration within pellet Literature Cited Bartlett, R. W., Krishnan. N. G., Van Hecke, M. C.. Chem. Eng. Sci., 28, 2179 11 973) - -,. Elliott, J., Gleiser, M., "Thermochemistry of Steelmaking", Vol. I, AddisonWesley, Reading, Mass., 1960. Evans, J. W., Ranade, M., Chem. Eng. Sci., in press (1979).
123
Evans, J. W., Song, S., Ind. Eng. Chem. Process Des. Dev., 13, 146 (1974). Evans, J. W., Song, S., Leon-Sucre, C. E., Met. Trans., 78, 55 (1976). Hockings, W., "Proceedings AIME Blast Furnace, Coke Oven and Raw Materials Conference", Vol. 19, p 170, 1960. McKewan, W., Trans. TMS-AIM€, 212, 791 (1958). McKewan, W., Trans. TMS-AIM€, 224, 2 (1962). Nelder, J . A., Mead, R., Comput. J.. 7, 308 (1967). Ranade, M., M.S. Thesis, University of Calif., Berkeley, 1978. Shehata, K.,Ezz, S., Trans. Inst. Min. Met., 82, C38 (1973). Slattery, J. C., Bird, R. B., AIChE J . , 4, 137 (1958). Sohn, H. Y., Szekely, J., Chem. Eng. Sci., 27, 763 (1972). Strangeway, P. K., Ross, H. V., Trans. Met. SOC.AIM€, 242, 1981 (1968). Szekely, J., Evans, J. W.. Chem. Eng. Sci., 26, 1901 (1971). Szekely, J., Evans, J. W., Sohn, H. Y., "Gas-Solid Reactions", Academic Press, New York, N.Y., 1976. Szekely, J., Propster, M., Chem. Eng. Sci., 30, 1049 (1975). Szekely, J., Stanek, V., Can. J . Chem. Eng., 50, 9 (1972). Themelis, N.,Gauvin. W., Can. Min. Metal. Bull., 55, 444 (1962). Turkdogan, E. T., Olsson, R. G., Vinters, J. V., Met. Trans., 2, 3189 (1971).
Receiued f o r reuieu January 11, 1979 Accepted July 6, 1979
I
This investigation was carried out with support from the National Science Foundation under Grant No. ENG75-00501.
Extension of the Hayden-O'Connell Correlation to the Second Virial Coefficients of Some Hydrogen-Bonding Mixtures Fred P. Stein and Edwin J. Miller" Air Products and Chemicals, Inc., Allentown, Pennsylvania
18 105
The solvation parameter utilized by the Hayden-O'Connell correlation to describe specific interactions between unlike molecules for the purposes of calculating second virial cross coefficients has been determined and correlated for five hydrogen-bonding binary systems involving amines with methanol. A significant improvement in the accuracy of the calculated second virial cross coefficients, relative to the suggested mixing rule for the solvation parameter, has resulted. These aminelmethanol interactions are strong enough to require the use of chemical theory in the calculation of fugacity coefficients at some conditions. A linear correlation has been shown to exist between the entropy change and the enthalpy change on hydrogen bonding and between the enthalpy change and the change in stretching frequency of the hydrogen bond for these aminelmethanol systems. Thus, a measurement of change in stretching frequency on hydrogen bonding from an infrared spectrum can lead to a prediction of a second virial cross coefficient.
Introduction An interest in vapor-liquid equilibria requires a description of the vapor-phase nonidealities, usually in terms of the fugacity coefficient. If the virial equation is used to describe the vapor phase, then interest focuses on the second virial coefficients and, in particular, on the second virial cross coefficient between unlike species. If specific interactions between the molecules such as hydrogen bonding are involved, vapor-phase nonidealities manifest themselves at much lower pressures than one would expect from experience with hydrocarbons. Our interest in amine/alcohol mixtures and the availability of experimental P-V-T measurements for the vapor phase of some of these mixtures and one amine/water mixture made it possible to apply the Hayden-O'Connell (1975) correlation. Their correlation was attractive for this problem because they concluded that it was as good as any other available (Black, 1958; Kreglewski, 1969; Nothnagel et al., 1973; O'Connell and Prausnitz, 1967; Tsonopoulos, 1974) for simple substances and was often significantly more accurate for complex systems. The correlation has seen significant use recently in phase equilibrium studies 0019-7882/80/1119-0123$01 OO/O
on complex mixtures (Fredenslund et al., 1977; Anderson and Prausnitz, 1978). However, the suggested mixing rule, in the absence of experimental data, for the term that controls the size of the solvation term in the second virial cross coefficient proved to be unsatisfactory for amine/ alcohol mixtures. For example, the experimental second virial cross coefficient (Millen and Mines, 1974) for trimethylamine/methyl alcohol at 35 "C is -7200 L/kg-mol; the correlation with the suggested mixing rule qI2 = 0 gives B,, = -410 L/kg-mol. Clearly, the solvation effects must be considered. Accordingly, the solvation term in the correlation was fit to the available experimental data for the amine/methyl alcohol and amine/ water data, and subsequently, the solvation term was correlated with properties of the pure materials making predictions possible for similar pairs for which experimental data are not available. Theoretical Background Hayden and O'Connell (1975) presented a generalized method for predicting second virial coefficients which sums the contributions from free, bound-metastable, and chemically interacting pairs. The required information to use 1979
American Chemical Society
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Table I.
Available Experimental Data system
no. of temp range, temp points “C
diethylamine (DEA)/methanol
5 5 25 3
25-45 25-45 25-45 9-43 25-45
triethylamine (TEA)/methanol diethylamine (DEA)/water
4 3
25-45 25-45
methylamine (MMA)/methanol dimethylamine (DMA)/methanol trimethylamine (TMA)/methanol
5
the correlation for a pure material is critical temperature, critical pressure, dipole moment, and radius of gyration, all independently determined quantities. In addition, for chemically associating substances a parameter q was fit to the data to close the difference (in some instances substantial) between the experimental data and the nonassociating part of the correlation. The chemical contribution is (Hayden and O’Connell eq 29) Bchem= bo expIq(650/(c/k + 300) - 4.27)) X I1 - exp(l500q/‘11) (1) where bo and t / k are determined from the independent properties such as critical temperature etc. and the specific constants of the correlation, and q is fit to &hem. Carboxylic acids are handled as a special case with q = 4.5 and with different numerical constants used in eq 1. Although Hayden and O’Connell emphasized predictions for pure materials, they did address the problem of mixtures and presented a set of mixing rules. In the absence of mixture data, they suggest q12 = 0 unless the species are in the same group, in which case it is implied that the group q should be used. Examples of groups are hydroxyl, aldehyde, ketone, amine, etc. Each group has an q describing it, and each species (of a group) has its own (different) q. Thus, the general rule calls for q I 2 = 0 for amine/hydroxyl mixtures of interest here. Experimental D a t a Recent experimental data on hydrogen bonding in the vapor phase over a temperature range of 25 to 45 “C for a series of amines in methyl alcohol and for one amine in water permit an extension of the Hayden-O’Connell correlation to this type of interaction. Table I lists the available data. The experimental data were all converted in the original articles to chemical equilibrium constants for formation of the complexes, K12. From these equilibrium constants for hydrogen-bond formation, second virial cross coefficients were calculated from the chemical theory of gas imperfections (Prausnitz, 1969). The position of the constant 2 was misprinted therein; it was intended as shown in eq 2 according to
It is these B12’sthat were considered as experimental data, hereinafter. Association P a r a m e t e r f o r P u r e Materials The parameters are available in the Hayden-O’Connell article and its supplementary material for all of the components involved here except triethylamine. For triethylamine, no radius of gyration was available. Consequently, the data of Sutton (1958) on the C-C and the C-N bond distances, 1.54 and 1.47 f 0.02 A, respectively, and on the C-C-N and the C-N-C bond angles, 113 f 3O for each, were used to compute a radius of gyration. The molecule was “formed” by starting with the pyramidal structure of ammonia as shown by Moelwyn-Hughes (1961)
reference Millen and Mines ( 1 9 7 4 ) Millen and Mines i 1 9 7 4 j Millen and Mines (1974) Fild et al. (137U) Tucker ( 1 9 6 9 ) Christian and Tucker (1Y70) Millen and Mines ( 1 9 7 4 ) Tucker ( 1 9 6 9 )
Table 11. Parameters €or Pure Materials component T,,K MMA DMA TMA MEA DEA TEA MeOH
H,O
430.1 437.3 433.3 456.0 496.0 535.2 513.2 647.0
P,, bars 74.6 53.1 40.7 56.2 37.1 30.4 79.5 221.2
D
R.G., A
q
1.31 1.03 0.612 1.22 0.92 0.66 1.70 1.35
1.662 2.264 2.736 2.309 3.161 3.951 1.536 0.615
0.15 0.17 0.057 0.31 0.24 0.0 1.40 1.66
p,
Table 111. Calculated B’s for Water at 423.1 K at Various n’s ~~
~~
rl
B , Limo1
0 0.3 1.o 1.4 1.7 2.1
-313 -308 -261 -253 (max) -267 -333
with all of the first carbons “below” the nitrogen. The second carbons were not all placed “below” the first carbons because these positions seemed unlikely due to hindrance. A radius of gyration was calculated for three of the second carbons “above” the first carbons for two “above” and one “below”, and for one “above” and two “below” with the hydrogens in two different positions. The average of six such calculations was 3.981 A with a standard deviation of 0.103. We independently selected the values for critical temperature, critical pressure, and dipole moment for each component and took the radius of gyration listed by Thompson (1966) (and as used by Hayden and O’Connell). It was desirable to have q’s that were completely consistent with these parameters, so the q for each component was fit to the experimental virial coefficient data listed by Hayden and O’Connell, except as noted. The parameters and results for 17 are listed in Table 11. The 7’s agree closely with those of Hayden and O‘Connell except as mentioned below. The q’s for tertiary amines are small because they have no active hydrogen for hydrogen bond formation. Only the last three entries in Table I1 require any comment. The q for TEA was fit to the data of Lambert and Strong (1950) as recorded by Dymond and Smith (1969). The best curve fit was for q = -0.17, which indicates that the nonassociating part of the correlation is (already) more negative than the experimental data. Since negative q’s are not allowed in the theory, q for triethylamine was set to zero. The indication above and the implication by Hayden and O’Connell (1975) that positive 7’s produce second virial coefficients from the correlation that are more negative than the nonassociating results ( q = 0), although true for triethylamine, are not true in all instances as demonstrated for water at 423 K in Table 111. The reason for this behavior is that the effective “nonpolar” parameters used in calculation of the free and bound-metastable
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
125
Table IV. Solvation Parameter for Mixtures
a
system
vI2
rms errora
av B, Limo1
MMAiMeOH DMAiMeOH TMA/MeOH (Millen) TMAiMeOH (Fild) DEAiMeOH (Tucker) TEAiMeOH DEA/H,O
2.266 2.267 2.409 2.352 2.119 2.417 1.919
310 250 170 100 190 280 100
-5160 -5710 -7 620 -9610 -4950 -6920 -2950
rms error
=
[(Bexptl- Bcor)2/N]’1 2 .
contributions to B are also dependent on q through relations developed by Hayden and O’Connell. Although the chemical contribution (eq 1)becomes more negative with increasing q, the free and bound-metastable contributions become less negative with increasing q. Under certain conditions the latter terms dominate the effect of q on B. The q value for methyl alcohol differs from that of Hayden and O’Connell ( q = 1.63) primarily because different experimental data were used for the fit: Knoebel and Edmister (1968) (313 K point deleted), Kretschmer and Wiebe (1954), Kell and McLaurin (19691, and Kudchadker and Eubank (1970). For water the basic question is what experimental data should be fit, particularly at lower temperatures. Our value of = 1.66 was obtained by using the constants shown in Table I1 and the experimental data listed by Hayden and O’Connell (lowest experimental temperature 373 K, B = -454 L/mol); Hayden and O’Connell obtained q = 1.70 using different critical constants (the very best curve fit is q = -0.15, but as noted above, negative q’s are not permitted by the theory so another local minimum was sought in another region; it turned out to be q = 1.66); if the Dymond and Smith (1969) recommendation to use Kell’s (1968) data is adopted at the four lowest temperatures (lowest 423 K, B = -330 L/mol), the fit shows q = 2.11. These differences matter when the correlation is extrapolated to the mid-point temperature of interest for the mixture data, 308 K, where our values ( q = 1.66) give B = -1500 L/mol, Hayden and O’Connell’s values ( q = 1.70) give B = -1570 I,/mol, and the fit of Kell’s data ( q = 2.11) gives B = -3800 L/mol. The data used by Hayden and O’Connell were adopted because the extrapolated B’s with q = 1.66 were nearer to expectations. Solvation P a r a m e t e r f o r Mixtures The solvation parameter for mixtures q12 was fit to the experimental data referenced in Table I using the parameters for the pure constituents shown in Table 11. The results are shown in Table IV. Deviations from the correlation were considered at the temperature of the experiments except for the data of Fild et al. (1970) for which the equation of their line was used at 10, 25, 35, and 45 “C. The average B is shown in Table IV to give some additional feel for the rms error. Typically, the mixture data have been fit within about 4%. The Hayden-0’Connell correlation is quite successful in matching the temperature dependence of the data, demonstrating that it has a good form for such highly solvating mixtures. Figure 1 shows the fit for one trimethylamine/methanol data set. Virial coefficients for pure trimethylamine and methanol along with cross coefficients calculated assuming no solvation are also shown. It is apparent that the hydrogen bonding results in cross coefficients which are markedly greater in magnitude than the coefficients for either pure compound. An estimate of the variation in the experimental data can be obtained by inspection of the trimethylamine/
25
35
45
TEMPERATURE l°Cl
Figure 1. Second virial coefficients for trimethylamine/methanol. Table V. Experimental Data and Correlation €or TMA/MeOH Millen and Mines data ( 1 9 7 4 )
283.16 298.16 303.16 308.16 313.16 318.16
-10400 -8830 -7210 -6300 -5 350
-10610 -87 50 -7270 -6090 -5130
q 1 2 = 2.409 rms error = 1 7 0
Fild et al. data ( 1 9 7 0 )
-17740 -9550
-17660 -9610
-6550
-6660
-4600
-4740
q I 2 = 2.352 rms error = 100
methanol data which were obtained by two different experimenters. (Both were similar vapor-phase experiments). Table V shows the comparison. Fugacity Coefficients The effect of vapor phase nonideality on a vapor-liquid equilibrium K value is determined by the vapor fugacity coefficients. Fugacity coefficients have been calculated here for the trimethylamine/methanol system from the volume-explicit form of the virial equation (Hayden and O’Connell, 1975) truncated after the second virial term and from the chemical theory as developed by Nothnagel et al. (1973). The two methods are equivalent when the degree of solvation is small, but the virial theory breaks down a t high degrees of solvation. At conditions where it is valid, the virial approach is preferred for calculating fugacity coefficients, particularly of multicomponent mixtures, because of computational simplicity and because virial coefficients are more readily available from correlations than are equilibrium constants that are needed for the chemical theory. Figure 2 shows fugacity coefficients for the trimethylamine/methanol system for 100 “C and 3.5 bars, which is the vapor pressure of methanol (TMA is more volatile). Both methods predict similar results and a t these conditions one cannot choose between the two without data. The effect of solvation must not be ignored in phase equilibrium calculations, as illustrated by the fugacity coefficients for this case with qI2 = 0. Figure 3 shows fugacity coefficients for methanol infinitely dilute in trimethylamine gas a t 45 “C . Values are shown up to 4 bars, which is the vapor pressure of trimethylamine. The conditions of Figure 3 allow the greatest deviation from ideality for a binary because the less volatile
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21
-
VlRlAL EQUATION CHEMICAL T H E O R Y
0.2
0.6
0.4
0.8
1.0
L
I
7
18
2. TMA IMILLENI 3. TMAiFlLOl 4 MMA OMA 6 OEA 5
1
0
004
I
012
MOLE FRACTION TRIMETHYLAMINE
O'gl
0,71
0.6
0.2
I
1
020
I
024
028
I 012
Figure 4. Solvation parameter for amine/methanol mixtures vs. association parameter for amine.
judgment would probably be better based on the magnitude of the second virial coefficient. However, under many conditions of interest the virial equation will be valid for these amine/methanol mixtures because of either low pressure, high temperature, or dilution with nonsolvating components. Correlation of Solvation Parameters A correlation of the solvation parameters shown in Table IV would be useful in the prediction of q12's for which experimental mixture data are not available. For the amine/methanol group, a correlation for q12 against 1 for the amine involved in the binary is shown in Figure 4. The equation of the line is
0.8
1
016
ll AMINE
Figure 2. Fugacity coefficients for trimethylamine/methanol mixtures at 100 "C and 3.5 bars.
0.3
1
I
008
---
\
V I R I A L EQUATION CHEMICAL THEORY
I
I
I
I
0
1
2
3
PRESSURE IBARSI
Figure 3. Fugacity coefficients for methanol infinitely dilute in trimethylamine at 45 "C.
component is dilute a t pressures up to the vapor pressure of the more volatile component. The effect of solvation is large; a t 4 bars the value calculated from the virial equation with 7 = 2.409 is a factor of 4.2 lower than that for 7 = 0, while the value from the chemical theory is a factor of 2.2 lower. For the calculations shown on the high-pressure side of Figure 3, the virial equation has been extended beyond its range of validity in regard to fraction solvated, and the chemical theory results are preferred at those conditions. However, if one chooses to extend subatmospheric experimental data with the chemical theory, the correlations for B12 presented hereinafter are valid and useful; one simply reverses the direction of the calculation in eq 2, obtains Klz from the correlated or predicted B12,and proceeds with chemical theory calculations for fugacity coefficient. (Calculations were also performed using the chemical theory with ideal gas behavior of the true species. For the pressures in Figure 3, the differences from the chemical theory shown were less than 0.5% .) The solvation parameter for the TMA/methanol system shown in Figures 2 and 3 is 2.409. Hayden and O'Connell (1975) recommend a shift to the chemical theory for 7's of 4.5 and greater (determined from carboxylic acids). As illustrated in Figure 3, the shift would have to occur a t lower 7's depending on conditions and compositions; the
7712
= 2.415
-
0.380111, - 3.49477,'
The fact that the curve has a negative slope is not altogether surprising. Tertiary amines are unable to form hydrogen bonds with themselves and therefore have the lowest 7, values. However, the nitrogen in the tertiary amines has the strongest basic character because of the inductive effect of the alkyl groups, resulting in strong hydrogen bonding with alcohols and therefore large q12 values. These effects place the tertiary amines at the upper left of Figure 4. However, the relative position of the primary and secondary amine points cannot be explained as easily. It would be desirable to assign just one vl2 to all amine/ methanol mixtures. The average here is v12 = 2.30. However, the Hayden-O'Connell correlation is too sensitive to 77 in this range to permit such a simple correlation of q12. For example, using v12 = 2.30 for the diethylamine/methanol system gives an rms error (as defined in Table IV) of 1610 L/mol compared to 190 for the best-fit 7712 of 2.119. A comparison between the vI2 from the correlating line in Figure 4 and the q12 points for the trimethylaminel methanol mixture a t 77, = 0.057 shows the sensitivity of the Hayden-O'Connell correlation to 11 and the difference between the two sets of experimental data. Since the unlike bond in the diethylamine/water interaction is similar to the bond between unlike pairs of the amines and methanol, it would be desirable to include dimethylamine/water in the correlation for v12. Such a correlation is shown in Figure 5. (The equation of the line is q12 = 3.892 - 1.047(7, + 7 ~ ) ~Of) course, it is quite speculative since the extension to include hydrogen bonds between amines and water is based only on diethylamine,' water data.
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
24L
Table VI. Enthalpy and Entropy Change and Frequency Shift upon Complex Formation
1. TEAiMeOH 2. TMAiMeOH (MILLEN) 3. TMAiMeOH ( F l L D l 4. MMAiMeOH 5. OMAiMeOH 6. OEAIMeOH 7. OEAIHZO
23
MMA/MeOH DMA/MeOH TMAiMeOH (Millen) TMA/MeOH (Fild) DEA/MeOH TEA/MeOH DEA/H,O
16
11
18
19
-83.7 -91.2 -98.3 -107.1 -107.5 -106.4 -102.5
-23.4 -25.9 -28.9 -31.4' -30.6 -31.4 -27.7
AV,
cm-' 243 293 310 302 325 370
-29.7 from infrared measurements.
-- I 15
AS, AH, kJ/(mol K ) MJ/mol
system
a
14
127
I
I
I
20
'IA+'IH
Figure 5. Solvation parameters for amine/ hydroxyl interactions.
-
105
-
100
-
Y
-
I . 2
95
-
90
-
85
-
(0
< I
80
2" (em 1 )
Figure 7. Enthalpy of hydrogen bond in an amine/methanol complex vs. the change in stretching frequency for IR absorption.
I
I
I
I
I
22
24
26
28
30
32
- A H (MJlrnol)
Figure 6. Entropy vs. enthalpy change for complex formation of amines with methanol.
A H - A S Correlation A monotonic relationship between AH and A S for hydrogen bonding between amines and methanol might be expected in view of some observations by Joesten and Schaad (1974) on acid-base pairs. Such a correlation is shown in Figure 6 for the amine/methanol mixtures; the diethylamine/water point has also been displayed. (The equation of the line is A S = 2.97AH - 14.0.) All of the a s ' s in Figure 6 from Millen and Mines (1974) data have been recalculated; the As's in their Table I11 are erroneous, assuming that the AG's and AH'S, which cross check with other things, are correct. All other values were taken directly, after unit conversion, from the references listed in Table I. The recalculated and other values are tabulated in Table VI. The utility of the correlation shown in Figure 6 results from infrared experiments on the vapor phase which yield shifts in the hydrogen-bond stretching frequency, h v , that usually correlate (Joesten and Schaad, 1974) with the enthalpy of the hydrogen bond, AH, but not Gibbs energy. Millen and Mines (1974) have speculated on such a correlation for some amine/methanol complexes. The values in Table VI are shown in Figure 7. (The equation of the line is AH = -0.0669Au - 7.70.) Thus, a Au from an infrared experiment is used in conjunction with Figures 6 and 7 to compute a Gibbs energy change for complex formation; thence, an equilibrium
constant; thence, a second virial cross coefficient via eq 2. An q12 can be calculated from the virial coefficients if the Hayden-O'Connell correlation is to be used. Measurements of the shifts in hydrogen-bond stretching frequency have been made in the vapor phase for a number of amine/alcohol and ammonia/alcohol mixtures by Hussein and Millen (1974). The study does not include measurements of A H or A S for complex formation. Although Figures 6 and 7 are based only on amine/methand mixtures it is expected that worthwhile estimates of the cross virial coefficients could be made with this approach for such closely related mixtures. Millen and Mines (1977) have measured frequency shifts in the vapor phase for hydrogen-bonded complexes of water with several amines, pyridines, and ammonia. Aside from these two papers on Ne-H-0 hydrogen-bonded complexes, relatively little other such data on vapor-phase complexes exists. Additional data are needed to determine the possibility of generalizing relationships such as in Figures 6 and 7.
Prediction of Cross Coefficient for Et hylamine/Met hanol There are no experimental virial coefficient data available for the ethylamine/methanol system. An extrapolation of the correlation in Figure 4 gives 1.961 for q12, which when combined with the pure-component parameters of Table I1 predicts B12= -4580, -3410, and -2610 L/mol for 25, 35, and 45 " C , respectively. An interpolation in Figure 5 gives 2.101 for q12, which when combined with the pure-component parameters of Table I1 predicts BI2= -5750, -4180, and -3120 L/mol for 25, 35, and 45 "C, respectively.
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The path outlined a t the end of the previous section beginning with the change in stretching frequency for the hydrogen bond, Au = 250 (Hussein and Millen, 1974), and culminating in predicted Biz's leads to -7000, -5260, and -4020 L/mol for 25, 35, and 45 " C , respectively. The variation in these three sets of values shows the sensitivity of the Biz's to the methods of estimation discussed here. However, the use of any of these sets of values would result in a significant improvement compared to neglecting the effect of hydrogen bonding association. Among these three methods of prediction, the latter method is preferred because the approach is based on more-fundamental information. Conclusions The Hayden-O'Connell correlation has been used successfully to correlate the second virial cross coefficients for five amine/methanol systems and one amine/water system. The hydrogen-bonding solvation between these acid-base pairs is strong enough to require the use of chemical theory in the calculation of fugacity coefficients a t some VLE conditions. The Hayden-O'Connell correlation was extended to include a correlation for the solvation parameter v12, for hydrogen bonding in systems involving the five amines with methanol, against the association parameters of the pure amines v A (Figure 4). The extension of the amine/ methanol correlation to involve hydrogen bonding in amine/water systems (Figure 5 ) is speculative because only one experimental system (DEA/water) is available. Nevertheless, the inclusion of the hydrogen bond between amine and water by the simple extension of the correlation of Figure 4 to Figure 5 lends encouragement to the prospect that a broader correlation exists for amine/hydroxyl hydrogen bonds. The demonstration of a linear correlation between entropy change and enthalpy change for hydrogen-bond formation in amine/hydroxyl systems (Figure 6) and between enthalpy change and change in stretching frequency of the hydrogen bond (Figure 7 ) makes it possible to predict a second virial cross coefficient from an infrared spectrum measurement of change in stretching frequency of the hydrogen bond along with the chemical theory of gas imperfections. This procedure is of interest for similar systems where only measurements of an infrared spectrum exist and a second virial cross coefficient is required. Acknowledgment The authors are grateful to Air Products and Chemicals, Inc., for permission to publish this paper. Nomenclature B = second virial coefficient, L/mol
bo = equivalent hard-sphere volume of molecules, L/mol (from Hayden-O'Connell, 1975) AG = Gibbs energy change for formation of amine/methanol complex, MJ/mol A H = enthalpy change for formation of amine/methanol complex, MJ/mol K = chemical equilibrium constant for vapor-phase complex formation N = number of data points P = pressure, bar R = gas constant A S = entropy change for formation of amine/methanol complex, kJ/(mol K) T = absolute temperature, K mol = kg-mol Greek Letters = energy parameter divided by Boltzman constant, unitless
C/K
(from Hayden-O'Connell, 1975) association parameter for pure interactions, solvation parameters for unlike interactions (from Hayden-O'Connell, 1975) p = dipole moment, D Au = shift in hydrogen-bond stretching frequency, cm-' 7=
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6 6 3 1147AI ---, ." ,-,.
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I
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Received f o r review March 8, 1979 Accepted July 12, 1979
