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Second cross virial coefficients B12 for gas mixture (carbon dioxide+water) from 300 to 1000 K
A-182 J Chem.
Thermo&namics
1981,
13, 203-211
Second cross virial coefficients for the gas mixture (carbon dioxide + water) from 3OOtolOOOK CECIL E. VANDERZEE”
and NORMAN
B,,
C. HAAS
Department of Chemistry, University qf‘ Nebraska, Lincoln, Nebraska 68.588, U.S.A. (Received 29 August 1979; in revisedform
I March
1980)
Values of the second cross virial coefficient B, z for the gas mixture (carbon dioxide + water) were evaluated from a review of (p. V, T) measurements reported in the literature. Values of B, 2 are presented for the temperature range 300 to 1000 K. The results permit evaluation of the fugacity coefficients of the mixture components over a substantial pressure range.
1. Introduction Recently we needed fugacity coefficients for CO,(g) and H,O(g) in mixtures at temperatures between 300 and 600 K at pressures ((1 -Y)CO,+YH,O) from 10’ to 4 x lo4 kPa. After a long wide-ranging search for useful information in we succeeded in locating two sets of (p, I/, T) this range of conditions, measurements(‘-3’ that yield the cross-interaction virial coefficients B,, for the gas mixture in the temperature range 323 to 453 K, thereby permitting evaluation of the fugacity coefficients from the equation of state. Both referees of this paper called our attention to the work of Coan and KingJ4’ who obtained values of B, 2 in the range 298.15 to 373.15 K from measurements leading to the fugacity coefficients of H,O(g) in equilibrium with saturated solutions of COZ in H,O(l) at pressures from 1.73 x lo” to 5.15 x lo3 kPa. At higher temperatures, Greenwood”*“’ reported fugacity coefficients in the region 723 to 1073 K at pressures of I x lo4 to 5 x lo4 kPa based on extensive measurements of compression factors for the gas mixture. Ryzhenko and Malinin”) reported fugacity coefficients of lower accuracy in the range 673 to 1023 K at 4 x lo4 to 20 x lo4 kPa, and Walter@) gave approximate values for COZ at a total pressure of 1 x 10’ kPa at 915 to 1040 K. The purpose of this paper is to.organize the scattered information” -‘) and preserve it in useful form. The low-temperature results”-4’ are compatible with one another and with those at the higher temperatures, so it is possible to provide a coherent set of parameters covering the whole temperature range. The results are superior to empirical estimates, but there is a clear need for more experimental work on this ” To whom inquiries should be addressed 002 l-9614/8 11030203 + 09 $0 1.00/O 13
i‘l 1981 Academic Press Inc. (London) Ltd.
204
C. E. VANDERZEE
AND
N. C. HAAS
important mixture, especially at the lower temperatures. The properties involved are related by the equation of state: pv, = RT( 1 +&/I/,
+ C,/V,2),
(1)
where Kl = V/h
+n,)
(2)
is the mean molar volume, and subscripts 1 and 2 refer to COZ and H,O respectively. The molar virial coefficients of the mixture are 4ll =(I -yJ2B,,
+2y(l
-Yv3,2+Y2~,2,
(3)
and Cm = (~-Y)~C~~~+~Y(~-Y)~C~~~+~Y~(~-Y)C~~~+Y~C~,,,
(4)
where y is the mole fraction of H,O. In the work reported at the lower temperatures and pressures,” - 3, the ratio C,,,/V~ is negligible, so the third virial coefficients do not become involved in interpreting the work of Gerryc2) and of Maass and Mennie.(3) For the conditions of Greenwood’s work (5*6) however, their contribution cannot be neglected. At these temperatures, 723 to 1073 K, we note from the compilation by Dymond and Smith”) that the third virial coefficients Cl,, and C,,, of the pure components are nearly equal, so we might anticipate that the interaction coefficients Cir, and Cr,, would nearly equal those for the pure components. With this useful approximation, C, = Clll = C,,, should be a good first approximation at the higher temperatures. Greenwood’s results are extensive enough to permit evaluation of C, at selected mole fractions and test the approximation. The fugacity coefficients for component 1 are given by Wlp:lU
-Y)PI
= WT/,H(~-Y)BI~
+yB12) - WT/,lW +(3/2~~)((1-~)2~,,,+2~(l-~)c,,2+~2~,22).
(5)
Because of the pressure range and large deviations from ideal-gas behavior in Greenwood’s work,‘5*6) the above form is needed rather than the “slightly imperfect gas mixture” relationoO’ written explicitly in terms of pressure. With the approximation C, = Clll = C1i2 = Ciz2 = C,,,, the factor {(1-y)2C,,,+ 2y(l -y)Cr i2 +y*C, 22) = C,. Equations (1) to (5) imply that no higher terms are needed to represent the mixture. The second and third virial coefficients for the pure components have been compiled by Dymond and Smith.“’ A later compilationor’ of the properties of CO, gives second virial coefficients and other properties for a wide range of conditions. The 1968 Steam Tables”” summarize the properties of H,O. A more recent study of B,, for Hz0 is that of O’Connell and Prausnitz, (’ 3, who surveyed the literature up to 1967, and who also represented Bz2 for Hz0 by functions based on molecular models. Their’13’ calculated results agree well with the experimental results from 298 to 1000 K. There is considerable scatter in some of the experimental values for H,O at low and intermediate temperatureso4* ’ 5, so the successful representation of the experimental work by molecular models indicates that the O’Connell and Prausnitz values”3’ of B,, are to be preferred.
Br2 FOR (CARBON DIOXIDE+
WATER)
205
2. Evaluation In a paper involving predictive calculations for polar gas mixtures, Stockmayer”’ cited values of the cross-interaction virial coefficient B, z based on a thesis by Gerry.(2’ The measurements by Gerry, (2) which have not been published elsewhere, were made at 323.15, 343.15, and 363.15 K in the pressure range 100 to 1500 kPa. The thesis reports a considerable number of measurements made with the original apparatus and also with an improved modification in which there was less corrosion of the pressure vessel and less adsorption on the walls. The procedure involved injecting measured amounts of water into the thermostatted pressure vessel which contained a known amount of CO, and measuring the resultant pressure change. In our notation, the results take the form p-p0
= (n2RT/~)(1+2B,,nl/‘V)+B,,RTn:/V2,
(6)
where I/ is the volume of the pressure vessel and p. is the initial pressure when n, = 0. At each temperature, several runs were made, with various amounts of CO,. Values of B,, were not explicitly tabulated by Gerry, but the calculation is direct from his parameters for equation (6). We re-evaluated the results, and obtained nearly the same values as those cited by Stockmayer (l) based directly on Gerry’s evaluation of 2B,,. The results obtained from series II of Gerry’s work are listed in table 1 with our recalculated values in parentheses. The results from his series I were considered faulty due to corrosion and adsorption problems, and are not included in table 1. For that series, he found B,, values of - 120, - 123, and - 112 cm3 mol-’ at 323.15, 343.15. TABLE 1. Virial coefficients derived from measurements by Gerry”’ and by Maass and Menniec3’ on {(l -y)CO~+.vH,O}(g). Values of B,, are based on literature values of Et, and B,, given in columns 5 and 6. Subscript 1 denotes CO,, 2 denotes H,O T
-~
-42 -. .__~
K
cm3 mol. ’
323.15 343.15 363.15
149 (154)” 124 (125)” lll(ll1)”
-4, ____
.-
cm3 mol. ’
--B,,
-41
cm3mol
’
Gerryo’
323.45 354.95 367.35 371.55 380.95 399.15 421.75 453.75 473.05
990 794
662,795 Maass and Menniet3’ 102
126 140, 97b 151,113b 137, 87b 110,111* 51
75 81 58 60 46 23
550 490 475 425 352 285 215 170
cm3 mol-’ Ref. 11 103.5 89.7 78.0 Ref.11 103.3 75.8 73.6 69.1 60.9 52.2 41.7 34.3
_- -B*,
cm3mol
’
Ref. 13 800 635 510 Ref.13 555 488 462 423 356 293 226 200
Estimated random uncertainties in authors’ measured quantities: Gerry, B,,. +8 cm3 mol-’ : Maass and Mennie. B l,r kl0 cm3 mol-‘; B 22, k20 cm3 mol-‘; B,,, ,30 cm3 mol-‘. y Parentheses enclose results of our recalculations of author’s results. h Entries refer to two separate series of measurements.
206
C. E. VANDERZEE
AND
N. C. HAAS
and 363.15 K respectively. The results of the two series do agree well at the two higher temperatures. We also evaluated B,, from Gerry’s results, for comparison with other values.“) His values agree fairly well for the two series, but are about 25 per cent more negative than those cited in other sources. (12--r4) Equation (6) apportions the total non-ideality of the gas mixture between B,, and B,,, and it is quite possible that some systematic errors give B,, too large a share, leading to values of B,, which are less negative than the true values. We did not attempt to recalculate the results in terms of literature values of B,,, except to note that B,, would be made more negative by about 10 per cent, and other drifts in the data would not be removed. Maass and Menniet3) made (p, V’, T) measurements (gas densities) on the pure gases as well as on the gas mixtures (carbon dioxide + water) and (ammonia + water) to study “aberrations from the ideal-gas laws.” Their measurements on the pure gases yield second virial coefficients B which agree reasonably well with other values reported for these gases”. ’ ‘3 r3) and which indicate the quality of their measurements. For the gas mixtures, their results are presented in a form designed to display deviations from Dalton’s law. However, by reference to the other tables in their paper and to the related illustrative computations, it is possible to retrace their calculations and regain instrumental quantities such as the volume of the thermostatted gas bulb and the fraction of the gas sample within it, and then to use equations (1) and (2) to evaluate B, from their results, which cover the temperature range 367 to 453 K at pressures from 80 to 102 kPa. Their results consist of three runs, one yielding two values which are much too small, and two runs yielding five values each. The latter two runs are reasonably consistent. One of those was made with the ratio n1/n2 = 0.9936, and the other with n,/n2 = 1.016. Judging by the scatter in their results for B, r and B,, for the pure gases and by the indicated limits of resolution for the reported results, their values of B, are uncertain by about 15 cm3 mol-‘, so the derived values of B, Z will be uncertain by about 30 cm3 mol - ’ from random sources of error. Table 1 includes values of B, r, B,,, and B, 2 derived from their results, with our estimates of random uncertainties. Their values of B,, and B,, agree well with literature values,“‘, 13’ especially for BZ2, which are based on more abundant results than for B, r. Their values for B,, are close to the O’Connell and Prausnitz (l 3, selections at all temperatures. The agreement suggests that systematic errors in their results are relatively small, including effects from adsorption. Masses of the sample components may be the source of largest uncertainty. The results yield almost the same values of Blz based on B,, and B,, from their own measurements as those based on B,, and B,, from the literature. Coan and King (4’ delivered CO,(g) into H,O(l) through a fritted glass sparger (pore size 14 x 10M4 cm) at flow rates of 0.01 to 0.02 dm3 s-l Tand measured the mole fraction y of H,O(g) in the exiting vapor phase at Tand p. At equilibrium, the fugacity coefficient & for the H,O(g) is given by 42 = (r~xL&:l~~)ex~{
v;cL(p -P: W’-;
>
(7)
where & is the fugacity coefficient, p: the vapor pressure, and VT” the molar volume of pure Hz0 at temperature T. The mole fraction xL of Hz0 in the liquid phase was determined by evaluating the mole fraction (1 - xL) of CO, in solution from known
B,,
FOR
(CARBON
DIOXIDE+
207
WATER)
Henry’s-law constants at T. It is necessary to assume 74 = 1 for the activity coefficient of H,O in the liquid phase, and also that 7: = 1 for CO,(g) in using Henry’s law. Coan and Kingc4) gave references up to 1941 for their Henry’s-law constants, and cite no values or references forV[, the molar volume of CO,(aq) that is needed to evaluate the effect of total pressure on the solubility of COz in H,O. We recalculated the results of Coan and KingJ4’ with VI = 34.2 cm3 mol- ’ for CO,(aq), and with Henry’s-law constants Ki = 0.3371, 0.195, 0.141, and 0.116 mol kg-’ MPa-’ at 298.15, 323.15, 348.15, and 373.15 K respectively. These constants were taken from recent evaluationso6* 1‘) and compilations.” *) The values at 348.15 and 373.15 K were based on reference 18 and are less certain than those .at the two lower temperatures. All values are compatible with Weiss’s relation”‘) for Ki as a function of T. Fortunately, errors in Ki and VI have relatively small influence on the values of xL. In treating Coan and King’s results, it is necessary to omit terms involving third virial coefficients in applying equations (1) (3), and (5) along with equation (7). We used selected literature values” “i3) for B,, and B,,, and preliminary values of Biz close to the derived values. The results of our recalculations are given in table 2 for each pressure to display any apparent trends with p. TABLE
2. Values of B,, calculated from results of Coan and King, W’ based on auxiliary quantities given in the discussion following equation (7) 298.15 K
P
kPa 2270 2980 3000 3730 3740
-42
cm3 mol-’ 223 211 220 202 209
348.15 K
323.15 K P kPa 1733 2550 2585 3640 3640 4630
-4, cm3 mol-’ 170 149 156 150 148 143
-- P kPa 2330 3740 3750 5130 5150
~. -4, cm3 mol-’ 149 107 109 116 112
373.15 K P
-42
kPa
cm3 mol ’
3680 3720 4480 4480 5150 5150
102 100 94 91 93 95
Greenwood’s first papert5) reports details of measurements of compression factors Z: 2 = pVJRT = 1+ B,,,/JI/, + C,/V,2,
(8)
for the pure components and for various mole fractions y for the gas mixture at 723 to 1023 K at pressures from 5 x lo3 to ((1 -Y)CO, +YH,O) 5 x lo4 kPa. He fitted the results by least squares to a polynomial in p and (1 -y), and tabulated them at selected temperatures, pressures, and mole fractions. He included for comparison the results for the pure components from the work of Holser and Kennedy.“‘) In a second paper, (6) the results were fitted to a more restricted polynomial, for which the various coefficients are tabulated. In each case, the equations represented the experimental results with a standard deviation of 0.5 per cent. At the above pressures, Use) the term C,/V,’ is not negligible, so at each composition we sought values of B, and C, that best represented Greenwood’s values of Z by
208
C. E. VANDERZEE
AND
N. C. HAAS
equation (8). At the lower temperatures, 723.15 to 923.15 K, the results could be well represented by the condition C, 1r = Czz2 = C, described earlier in connection with equations (4) and (5) and values of B,‘and B,, were reasonably invariant with pressure and composition. At the temperatures above this range, the results of Holser and Kennedy”” could still be well represented by C, I I = Cz12. Greenwood’s results for H,O could be represented by the same values for C,,,, but his results for pure CO, gave somewhat erratic pattern&with irregular drifts, as did B, and C, for the mixtures, especially at pressures less than 2 x lo4 kPa. Greenwood noticed this irregularity in his second paper where he evaluated activity coefficients for the gas mixtures. At the higher temperatures, we got somewhat more consistent results for B,, from the representation in reference 5 than from reference 6, and gave more weight to results at the higher pressures and at mole fractions 0.2 < (I -v) < 0.8. The results of our calculations are given in table 3, where we list constants for the pure components from both sources (5*6*19’ along with the values of B, 2 that were most consistent with Greenwood’s results. Our uncertainty estimates reflect the scatter of the results and the accuracy (1 per cent of Z) claimed by Greenwood. The values of C, = C,,, = C,,, reported in table 3 are slightly larger than those given by Dymond and Smith in their compilation. (9’ Examination of the results from Kell, McLaurin, and Whalley’9*‘5’ reveals that their values of Czz2 are low above 673 K and should be accompanied by an additional term from a fourth virial coefficient in the equation of state. The values of Vukalovich’9~20’ may be low for a similar reason. There may, of course, be some systematic errors in the values derived from Holser and Kennedy”‘) and from Greenwood.‘5.6’ For the pure components, the values of B, 1 and B,, in table 3 are in good agreement with values from other sources.(9* I1 - 13) We note that the results of Holser and Kennedy”” were included in the Steam-Table evaluation.” 2, The values of B,, given in table 3 are our choices of values that best represent B, by equation (3) at each temperature. Each represents a range of values obtained at various pressures and mole fractions. There was little difference between results based or other sources.” ’ - ’ 3, on Bll and B,, from Holser and Kennedy”” TABLE
3. Virial
coefficients
calculated from Greenwood’s’5~61 compression equations (1) to (6) and (8) of this paper Holser
T ii
C111(= Cm) cm6 mol-*
723.15 773.15 823.15 873.15 923.15 973.15 1023.15 1073.15 Uncertainty interval : * 100
1400 1200 1100 900 800 750 750 700
and Kennedy””
B 11 cm3 mol1 5.5 8 10.5 13 14 15 17.5
t2
’
factors.
Symbols
to
Greenwood’5,6’
___~.B 22 cm3 mol-’
~.- B 11 cm3 mol-’
B 22 cm3 mol-’
-60 -49 -43 -35.7 -29 - 24.7 -21 -18
1 5.2 8.7 13 12 14 15 17.5
-60 -49 -43 -34 - 28.5 -25 -19 -17
+2
conform
+3
+3
B 12 cm3 mol. -15 -10 -7 -6 -4.5 Ti +3 k5
‘7
t
LZ12 FOR
(CARBON
DIOXIDE+
WATER)
209
We used the results from table 3 with equation (5) to calculate activity coefficients for the gas mixture to compare with the values found by Greenwood in his second paper.(6) In general, the agreement was only fair, with differences frequently larger than his claimed accuracy (1 per cent) and the estimated uncertainties in table 3. The pattern of deviations could not be corrected by adjustments in B, i, Br 2, and B,, ; nor could it be reasonably attributed to variation in C,. The two computational routes are quite different, however, and systematic errors in the measurements or representation of the (p, V, T) surface can have different influences. Differences were usually larger at high mole fraction of H20. We also used the constants from table 3 to calculate fugacity coefficients to compare with the results of Ryzhenko and Malinin. (” Here the agreement was usually within 2 per cent, well within their claimed accuracy, and fairly consistent over most of the composition range. We did not attempt a comparison with Walter’s results,“’ nor with those of Franck and Tiidheidec2” which are in the range 4 x lo4 to 2 x 10’ kPa above 673 K.
3. Discussion Errors in B, are magnified in the calculation of B12 by equation (3), so small systematic errors from adsorption, corrosion, or other sources can have quite significant effects. Greenwood’5’ pointed to determination of the sample volume as the most critical measurement in his work, and amount of sample substance as a major factor at low pressures. In Maass and Mennie’s workJ3’ determination of the mass of sample in the thermostatted volume is the limiting measurement. Adsorption may be a problem in the lower-temperature range, (2*3) even at relative humidities of 50 per cent or less, and adsorption of “wet” COZ may be different from that for dry C02, especially on the metal surfaces in Gerry’s work.‘2’ In the strategy of Coan and King, (4) the non-ideality of H,O(g) is mainly due to the B, 2 cross-interactions. Their results in table 2 show some small trends with pressure. These are in the direction to be expected from the necessary omission of third virial coefficients in the calculations. Another possible cause is failure of the approximation ~4 = 1. the trends would require $j < 1 at the higher pressures (higher mole fractions of CO;). Incomplete saturation of the vapor phase, leading to values of (1 -y) that are too small, is a systematic error difficult to eliminate in such measurements, and would lead to values of Bi2 less negative than equilibrium values. The values of B,, from Coan and King’s results, table 2, are in fairly close agreement with those of Gerry at the lower temperatures, but are slightly less negative at 348.15 and 373.15 K, except for the results at lower pressures. Those of Maass and Mennie are mostly more negative than Gerry’s results, and have greater scatter. The high-temperature values of Greenwood fix the location of the Br 2 curve in that region fairly well, leaving the region from 400 to 700 K to be delineated. Examination of the ratio r = (B 11- Br 2)/(B, 1 - B,,) shows that it increases from values near 0.075 at the lower temperatures (298.15 to 373.15 K) to 0.4 at 923 to 973 K, and probably reaches a limit of 0.5 at higher temperatures. This ratio might be expected to be a smooth continuously increasing quantity with temperature. In the
C. E. VANDERZEE
210
AND N. C. HAAS
region 400 to 725 K, Y appears to increase almost linearly with temperature, and this quantity provides a very useful purpose in essentially averaging the measured results below 400 K. One run by Maass and Mennie is fairly compatible with the smoothing function ; the other has B, z values which are clearly too negative, with large values of r. The results of Gerry and of Coan and King could be blended easily by the smoothing function. Once a curve for I was selected, we regenerated B,, by the relation : B12 = (1 -r)B,
1+rB2*,
(9)
which becomes a strong generating function once I is established. The selected values of B,, are given in table 4, together with our estimates of uncertainties in the values. These finally selected values differ only slightly from our original selections prior to inclusion of the results of Coan and King, but we can now reduce our uncertainty estimates slightly below 400 K. TABLE4.SelectedvaluesofthevirialcoefficientB,,forthe{(l -y)CO,+yH,O)(g),evaluatedasdescribedin the text. Estimated uncertainty in B,,: k 15 cm3 mol-’ at 300 K, + 10 cm3 mol-’ at 400 K, decreasing to f5cm3mol-‘at 750K T ii 298.15 300 325 350 375
B 12 crdmol-’ -206 -200 -152 -120 - 101
T ii
B 12 cm3 mol - ’
T K
400 425 450 475 500
-86 -74 -63 -55 -48
550 600 650 700 750
B 12 cm3mol-’ -31 -28 -22 -17 -12
-T K 800 850 900 1000 1100
B 12 cm’mol-’ -9 -7 -5 -1 f2
In estimating the uncertainties, we have tried to include a reasonable allowance for systematic errors, but note the strong agreement between two quite different procedures(2*4) in the range 298.15 to 373.15 K. There is still a clear need for additional measurements on this important system over much of the temperature range, and we hope that this paper will provoke such measurements. The values of B12 from table 4, together with Bi i and B,, from compilations,‘” -i3) permit calculation of the fugacity coefficients for the components of ((1 -y)CO,+yH,O}(g) over much of the temperature range from 300 to 1100 K by application of equation (5). At low pressures, the influence of the third virial coefficients C, will be negligible. Above 623 K, the values of C, given in table 3 will be suitable (we estimate C, = 1300 cm6 mole2 at 623 K from reference 9), and will lead to fugacity coefficients close to the measured values of Greenwood@’ and Ryzhenko”) for pressures up to 3 x 104 kPa. In the temperature range below 600 K, values of C,,, and C222 diverge rapidly/gi so estimates of C, below 600 K are far more difficult and uncertain, and uncertainties from this term will increase rapidly at pressures above 5000 kPa. New measurements needed in this region include both B, 2 and C, terms, and even higher terms at the high pressures. The intent of this paper was to assemble some useful thermodynamic information, needed for interpretation of some interesting equilibria involving (carbon
B, r FOR (CARBON
DIOXIDE
+ WATER)
211
dioxide + water) in a forthcoming paper, so we refrain from examining models for the B,, interaction. We recognize that the smoothing function, equation (9), might be regarded as an “empirical” model. Coan and Kingc4) discussed the possibility of part of the B,, interaction being chemical in origin, and obtained reasonable thermodynamic parameters for the chemical interaction. Referee no. 2 enclosed a printout comparing the fit of several model equations with the experimental B,, values as well as the smoothed values from table 4. The most successful correlations were the equations of Hayden and O’Conne11,(22’ O’Connell and Prausnitz,‘23’ Nothnagel et LI~.,‘~~) and Tsonopoulos,‘25’ which represented the smoothed results within the uncertainty estimates. Only Nothnagel’s function gave high values of B, Z in the region 450 to 700 K. We should prefer to see additional experimental results in the range 375 to 750 K before attempting to select a “best” model. We are very grateful to Referee no. 2 for enclosing the above comparison, Referees for calling our attention to the work of Coan and King.‘4’
and to both
REFERENCES 1. Stockmayer. W. H. J. Chem. Phys. 1941, 9, 863. 2. Gerry, H. T. Ph.D. Thesis, Massachusetts Institute of Technology. 1932. 3. Maass, 0.; Mennie, J. H. Proc. R. Sot. (London) 1926, 1lOA. 198. 4. Coan, C. R.: King, Jr., A. D. J. Am. C/rem. Sot. 1971.93, 1857. 5. Greenwood, H. J. Am. J. Sci. 19@,267A, 1191. 6. Greenwood, H. J. Am. J. Sci. 1973, 273, 561. 1971, 561. 7. Ryzhenko, B. N.; Malinin, S. D. Geokimiya 1971, (8). 899; Geochemistry Internarional 8. Walter, L. S. Am. J. Sci. 1%3, 261, 151. 9. Dymond, J. H.; Smith, E. B. The V&l CoefJicienrs of‘ Gases. Clarendon Press: Oxford. 1969. 10. Guggenheim, E. A. Thermodynamics. Interscience Publishers: New York. 1967. 11. Angus, S.; Armstrong, B.; deReuck, K. M. Carbon Dioxide: International Thermodynamic Tables qf the Fluid Stare Vol. 3. Pergamon Press: Oxford. 1976. 12. Keenan, J. H.; Keyes, F. G.; Hill, P. G.; Moore, J. G. Steam Tables. Wiley: New York. 1%9. 13. O’Connell, J. P.; Prausnitz, J. M. Ind. Eng. C/rem. Fundam. 1970,9, 579. 14. McCullough, J. P.; Pennington, R. E.; Waddington, G. J. Am. Chem. Sot. 1952, 73. 4439. 15. Kell, G. S. ; McLaurin, G. E.; Whalley, E. J. Chem. Phys. 1968, 48, 4805. 16. Berg, R. L. ; Vanderzee, C. E. J. Chem. Thermodynamics 1978, 10, I1 13. 17. Weiss, R. F. Marine Chem. 1974, 2, 203. 18. Gmelins Handbuch der Anorganische Chemie No. 14. Verlag Chemie, GmbH.: Weinheim, Germany. 1960, 1973.
19. Holser, W. T. ; Kennedy, G. C. Am. J. Sci. 1959,257, 71; Am. J. Sci. 1958, 256, 744. 20. Vukalovich, M. P.; Trakhtengert, M. S.; Spiridonov, G. A. Tepfoenerge&r 1%7, 14, 65. 2I. Franck, E. U.; Todheide, K. 2. Phys. Chem. (Frankfort am Main) 1959, 22, 232. 22. Hayden, J. G.; O’Connell, J. P. Ind. Eng. Chem.. Process Des. Dev. 1975, 14. 209. 23. O’Connell, J. P.; Prausnitz, J. M. Ind. Eng. Chem.. Process Des. Dev. 1967, 6, 245. 24. Nothnagel, K. H.; Abrams, D. S.; Prausnitz. J. M. Ind. Eng. Chem.. Process Des. Dev. 1973, 12. 25. 25. Tsonopoulos, C. AIChE J. 1974, 20, 263.
Thermo&namics
1981,
13, 203-211
Second cross virial coefficients for the gas mixture (carbon dioxide + water) from 3OOtolOOOK CECIL E. VANDERZEE”
and NORMAN
B,,
C. HAAS
Department of Chemistry, University qf‘ Nebraska, Lincoln, Nebraska 68.588, U.S.A. (Received 29 August 1979; in revisedform
I March
1980)
Values of the second cross virial coefficient B, z for the gas mixture (carbon dioxide + water) were evaluated from a review of (p. V, T) measurements reported in the literature. Values of B, 2 are presented for the temperature range 300 to 1000 K. The results permit evaluation of the fugacity coefficients of the mixture components over a substantial pressure range.
1. Introduction Recently we needed fugacity coefficients for CO,(g) and H,O(g) in mixtures at temperatures between 300 and 600 K at pressures ((1 -Y)CO,+YH,O) from 10’ to 4 x lo4 kPa. After a long wide-ranging search for useful information in we succeeded in locating two sets of (p, I/, T) this range of conditions, measurements(‘-3’ that yield the cross-interaction virial coefficients B,, for the gas mixture in the temperature range 323 to 453 K, thereby permitting evaluation of the fugacity coefficients from the equation of state. Both referees of this paper called our attention to the work of Coan and KingJ4’ who obtained values of B, 2 in the range 298.15 to 373.15 K from measurements leading to the fugacity coefficients of H,O(g) in equilibrium with saturated solutions of COZ in H,O(l) at pressures from 1.73 x lo” to 5.15 x lo3 kPa. At higher temperatures, Greenwood”*“’ reported fugacity coefficients in the region 723 to 1073 K at pressures of I x lo4 to 5 x lo4 kPa based on extensive measurements of compression factors for the gas mixture. Ryzhenko and Malinin”) reported fugacity coefficients of lower accuracy in the range 673 to 1023 K at 4 x lo4 to 20 x lo4 kPa, and Walter@) gave approximate values for COZ at a total pressure of 1 x 10’ kPa at 915 to 1040 K. The purpose of this paper is to.organize the scattered information” -‘) and preserve it in useful form. The low-temperature results”-4’ are compatible with one another and with those at the higher temperatures, so it is possible to provide a coherent set of parameters covering the whole temperature range. The results are superior to empirical estimates, but there is a clear need for more experimental work on this ” To whom inquiries should be addressed 002 l-9614/8 11030203 + 09 $0 1.00/O 13
i‘l 1981 Academic Press Inc. (London) Ltd.
204
C. E. VANDERZEE
AND
N. C. HAAS
important mixture, especially at the lower temperatures. The properties involved are related by the equation of state: pv, = RT( 1 +&/I/,
+ C,/V,2),
(1)
where Kl = V/h
+n,)
(2)
is the mean molar volume, and subscripts 1 and 2 refer to COZ and H,O respectively. The molar virial coefficients of the mixture are 4ll =(I -yJ2B,,
+2y(l
-Yv3,2+Y2~,2,
(3)
and Cm = (~-Y)~C~~~+~Y(~-Y)~C~~~+~Y~(~-Y)C~~~+Y~C~,,,
(4)
where y is the mole fraction of H,O. In the work reported at the lower temperatures and pressures,” - 3, the ratio C,,,/V~ is negligible, so the third virial coefficients do not become involved in interpreting the work of Gerryc2) and of Maass and Mennie.(3) For the conditions of Greenwood’s work (5*6) however, their contribution cannot be neglected. At these temperatures, 723 to 1073 K, we note from the compilation by Dymond and Smith”) that the third virial coefficients Cl,, and C,,, of the pure components are nearly equal, so we might anticipate that the interaction coefficients Cir, and Cr,, would nearly equal those for the pure components. With this useful approximation, C, = Clll = C,,, should be a good first approximation at the higher temperatures. Greenwood’s results are extensive enough to permit evaluation of C, at selected mole fractions and test the approximation. The fugacity coefficients for component 1 are given by Wlp:lU
-Y)PI
= WT/,H(~-Y)BI~
+yB12) - WT/,lW +(3/2~~)((1-~)2~,,,+2~(l-~)c,,2+~2~,22).
(5)
Because of the pressure range and large deviations from ideal-gas behavior in Greenwood’s work,‘5*6) the above form is needed rather than the “slightly imperfect gas mixture” relationoO’ written explicitly in terms of pressure. With the approximation C, = Clll = C1i2 = Ciz2 = C,,,, the factor {(1-y)2C,,,+ 2y(l -y)Cr i2 +y*C, 22) = C,. Equations (1) to (5) imply that no higher terms are needed to represent the mixture. The second and third virial coefficients for the pure components have been compiled by Dymond and Smith.“’ A later compilationor’ of the properties of CO, gives second virial coefficients and other properties for a wide range of conditions. The 1968 Steam Tables”” summarize the properties of H,O. A more recent study of B,, for Hz0 is that of O’Connell and Prausnitz, (’ 3, who surveyed the literature up to 1967, and who also represented Bz2 for Hz0 by functions based on molecular models. Their’13’ calculated results agree well with the experimental results from 298 to 1000 K. There is considerable scatter in some of the experimental values for H,O at low and intermediate temperatureso4* ’ 5, so the successful representation of the experimental work by molecular models indicates that the O’Connell and Prausnitz values”3’ of B,, are to be preferred.
Br2 FOR (CARBON DIOXIDE+
WATER)
205
2. Evaluation In a paper involving predictive calculations for polar gas mixtures, Stockmayer”’ cited values of the cross-interaction virial coefficient B, z based on a thesis by Gerry.(2’ The measurements by Gerry, (2) which have not been published elsewhere, were made at 323.15, 343.15, and 363.15 K in the pressure range 100 to 1500 kPa. The thesis reports a considerable number of measurements made with the original apparatus and also with an improved modification in which there was less corrosion of the pressure vessel and less adsorption on the walls. The procedure involved injecting measured amounts of water into the thermostatted pressure vessel which contained a known amount of CO, and measuring the resultant pressure change. In our notation, the results take the form p-p0
= (n2RT/~)(1+2B,,nl/‘V)+B,,RTn:/V2,
(6)
where I/ is the volume of the pressure vessel and p. is the initial pressure when n, = 0. At each temperature, several runs were made, with various amounts of CO,. Values of B,, were not explicitly tabulated by Gerry, but the calculation is direct from his parameters for equation (6). We re-evaluated the results, and obtained nearly the same values as those cited by Stockmayer (l) based directly on Gerry’s evaluation of 2B,,. The results obtained from series II of Gerry’s work are listed in table 1 with our recalculated values in parentheses. The results from his series I were considered faulty due to corrosion and adsorption problems, and are not included in table 1. For that series, he found B,, values of - 120, - 123, and - 112 cm3 mol-’ at 323.15, 343.15. TABLE 1. Virial coefficients derived from measurements by Gerry”’ and by Maass and Menniec3’ on {(l -y)CO~+.vH,O}(g). Values of B,, are based on literature values of Et, and B,, given in columns 5 and 6. Subscript 1 denotes CO,, 2 denotes H,O T
-~
-42 -. .__~
K
cm3 mol. ’
323.15 343.15 363.15
149 (154)” 124 (125)” lll(ll1)”
-4, ____
.-
cm3 mol. ’
--B,,
-41
cm3mol
’
Gerryo’
323.45 354.95 367.35 371.55 380.95 399.15 421.75 453.75 473.05
990 794
662,795 Maass and Menniet3’ 102
126 140, 97b 151,113b 137, 87b 110,111* 51
75 81 58 60 46 23
550 490 475 425 352 285 215 170
cm3 mol-’ Ref. 11 103.5 89.7 78.0 Ref.11 103.3 75.8 73.6 69.1 60.9 52.2 41.7 34.3
_- -B*,
cm3mol
’
Ref. 13 800 635 510 Ref.13 555 488 462 423 356 293 226 200
Estimated random uncertainties in authors’ measured quantities: Gerry, B,,. +8 cm3 mol-’ : Maass and Mennie. B l,r kl0 cm3 mol-‘; B 22, k20 cm3 mol-‘; B,,, ,30 cm3 mol-‘. y Parentheses enclose results of our recalculations of author’s results. h Entries refer to two separate series of measurements.
206
C. E. VANDERZEE
AND
N. C. HAAS
and 363.15 K respectively. The results of the two series do agree well at the two higher temperatures. We also evaluated B,, from Gerry’s results, for comparison with other values.“) His values agree fairly well for the two series, but are about 25 per cent more negative than those cited in other sources. (12--r4) Equation (6) apportions the total non-ideality of the gas mixture between B,, and B,,, and it is quite possible that some systematic errors give B,, too large a share, leading to values of B,, which are less negative than the true values. We did not attempt to recalculate the results in terms of literature values of B,,, except to note that B,, would be made more negative by about 10 per cent, and other drifts in the data would not be removed. Maass and Menniet3) made (p, V’, T) measurements (gas densities) on the pure gases as well as on the gas mixtures (carbon dioxide + water) and (ammonia + water) to study “aberrations from the ideal-gas laws.” Their measurements on the pure gases yield second virial coefficients B which agree reasonably well with other values reported for these gases”. ’ ‘3 r3) and which indicate the quality of their measurements. For the gas mixtures, their results are presented in a form designed to display deviations from Dalton’s law. However, by reference to the other tables in their paper and to the related illustrative computations, it is possible to retrace their calculations and regain instrumental quantities such as the volume of the thermostatted gas bulb and the fraction of the gas sample within it, and then to use equations (1) and (2) to evaluate B, from their results, which cover the temperature range 367 to 453 K at pressures from 80 to 102 kPa. Their results consist of three runs, one yielding two values which are much too small, and two runs yielding five values each. The latter two runs are reasonably consistent. One of those was made with the ratio n1/n2 = 0.9936, and the other with n,/n2 = 1.016. Judging by the scatter in their results for B, r and B,, for the pure gases and by the indicated limits of resolution for the reported results, their values of B, are uncertain by about 15 cm3 mol-‘, so the derived values of B, Z will be uncertain by about 30 cm3 mol - ’ from random sources of error. Table 1 includes values of B, r, B,,, and B, 2 derived from their results, with our estimates of random uncertainties. Their values of B,, and B,, agree well with literature values,“‘, 13’ especially for BZ2, which are based on more abundant results than for B, r. Their values for B,, are close to the O’Connell and Prausnitz (l 3, selections at all temperatures. The agreement suggests that systematic errors in their results are relatively small, including effects from adsorption. Masses of the sample components may be the source of largest uncertainty. The results yield almost the same values of Blz based on B,, and B,, from their own measurements as those based on B,, and B,, from the literature. Coan and King (4’ delivered CO,(g) into H,O(l) through a fritted glass sparger (pore size 14 x 10M4 cm) at flow rates of 0.01 to 0.02 dm3 s-l Tand measured the mole fraction y of H,O(g) in the exiting vapor phase at Tand p. At equilibrium, the fugacity coefficient & for the H,O(g) is given by 42 = (r~xL&:l~~)ex~{
v;cL(p -P: W’-;
>
(7)
where & is the fugacity coefficient, p: the vapor pressure, and VT” the molar volume of pure Hz0 at temperature T. The mole fraction xL of Hz0 in the liquid phase was determined by evaluating the mole fraction (1 - xL) of CO, in solution from known
B,,
FOR
(CARBON
DIOXIDE+
207
WATER)
Henry’s-law constants at T. It is necessary to assume 74 = 1 for the activity coefficient of H,O in the liquid phase, and also that 7: = 1 for CO,(g) in using Henry’s law. Coan and Kingc4) gave references up to 1941 for their Henry’s-law constants, and cite no values or references forV[, the molar volume of CO,(aq) that is needed to evaluate the effect of total pressure on the solubility of COz in H,O. We recalculated the results of Coan and KingJ4’ with VI = 34.2 cm3 mol- ’ for CO,(aq), and with Henry’s-law constants Ki = 0.3371, 0.195, 0.141, and 0.116 mol kg-’ MPa-’ at 298.15, 323.15, 348.15, and 373.15 K respectively. These constants were taken from recent evaluationso6* 1‘) and compilations.” *) The values at 348.15 and 373.15 K were based on reference 18 and are less certain than those .at the two lower temperatures. All values are compatible with Weiss’s relation”‘) for Ki as a function of T. Fortunately, errors in Ki and VI have relatively small influence on the values of xL. In treating Coan and King’s results, it is necessary to omit terms involving third virial coefficients in applying equations (1) (3), and (5) along with equation (7). We used selected literature values” “i3) for B,, and B,,, and preliminary values of Biz close to the derived values. The results of our recalculations are given in table 2 for each pressure to display any apparent trends with p. TABLE
2. Values of B,, calculated from results of Coan and King, W’ based on auxiliary quantities given in the discussion following equation (7) 298.15 K
P
kPa 2270 2980 3000 3730 3740
-42
cm3 mol-’ 223 211 220 202 209
348.15 K
323.15 K P kPa 1733 2550 2585 3640 3640 4630
-4, cm3 mol-’ 170 149 156 150 148 143
-- P kPa 2330 3740 3750 5130 5150
~. -4, cm3 mol-’ 149 107 109 116 112
373.15 K P
-42
kPa
cm3 mol ’
3680 3720 4480 4480 5150 5150
102 100 94 91 93 95
Greenwood’s first papert5) reports details of measurements of compression factors Z: 2 = pVJRT = 1+ B,,,/JI/, + C,/V,2,
(8)
for the pure components and for various mole fractions y for the gas mixture at 723 to 1023 K at pressures from 5 x lo3 to ((1 -Y)CO, +YH,O) 5 x lo4 kPa. He fitted the results by least squares to a polynomial in p and (1 -y), and tabulated them at selected temperatures, pressures, and mole fractions. He included for comparison the results for the pure components from the work of Holser and Kennedy.“‘) In a second paper, (6) the results were fitted to a more restricted polynomial, for which the various coefficients are tabulated. In each case, the equations represented the experimental results with a standard deviation of 0.5 per cent. At the above pressures, Use) the term C,/V,’ is not negligible, so at each composition we sought values of B, and C, that best represented Greenwood’s values of Z by
208
C. E. VANDERZEE
AND
N. C. HAAS
equation (8). At the lower temperatures, 723.15 to 923.15 K, the results could be well represented by the condition C, 1r = Czz2 = C, described earlier in connection with equations (4) and (5) and values of B,‘and B,, were reasonably invariant with pressure and composition. At the temperatures above this range, the results of Holser and Kennedy”” could still be well represented by C, I I = Cz12. Greenwood’s results for H,O could be represented by the same values for C,,,, but his results for pure CO, gave somewhat erratic pattern&with irregular drifts, as did B, and C, for the mixtures, especially at pressures less than 2 x lo4 kPa. Greenwood noticed this irregularity in his second paper where he evaluated activity coefficients for the gas mixtures. At the higher temperatures, we got somewhat more consistent results for B,, from the representation in reference 5 than from reference 6, and gave more weight to results at the higher pressures and at mole fractions 0.2 < (I -v) < 0.8. The results of our calculations are given in table 3, where we list constants for the pure components from both sources (5*6*19’ along with the values of B, 2 that were most consistent with Greenwood’s results. Our uncertainty estimates reflect the scatter of the results and the accuracy (1 per cent of Z) claimed by Greenwood. The values of C, = C,,, = C,,, reported in table 3 are slightly larger than those given by Dymond and Smith in their compilation. (9’ Examination of the results from Kell, McLaurin, and Whalley’9*‘5’ reveals that their values of Czz2 are low above 673 K and should be accompanied by an additional term from a fourth virial coefficient in the equation of state. The values of Vukalovich’9~20’ may be low for a similar reason. There may, of course, be some systematic errors in the values derived from Holser and Kennedy”‘) and from Greenwood.‘5.6’ For the pure components, the values of B, 1 and B,, in table 3 are in good agreement with values from other sources.(9* I1 - 13) We note that the results of Holser and Kennedy”” were included in the Steam-Table evaluation.” 2, The values of B,, given in table 3 are our choices of values that best represent B, by equation (3) at each temperature. Each represents a range of values obtained at various pressures and mole fractions. There was little difference between results based or other sources.” ’ - ’ 3, on Bll and B,, from Holser and Kennedy”” TABLE
3. Virial
coefficients
calculated from Greenwood’s’5~61 compression equations (1) to (6) and (8) of this paper Holser
T ii
C111(= Cm) cm6 mol-*
723.15 773.15 823.15 873.15 923.15 973.15 1023.15 1073.15 Uncertainty interval : * 100
1400 1200 1100 900 800 750 750 700
and Kennedy””
B 11 cm3 mol1 5.5 8 10.5 13 14 15 17.5
t2
’
factors.
Symbols
to
Greenwood’5,6’
___~.B 22 cm3 mol-’
~.- B 11 cm3 mol-’
B 22 cm3 mol-’
-60 -49 -43 -35.7 -29 - 24.7 -21 -18
1 5.2 8.7 13 12 14 15 17.5
-60 -49 -43 -34 - 28.5 -25 -19 -17
+2
conform
+3
+3
B 12 cm3 mol. -15 -10 -7 -6 -4.5 Ti +3 k5
‘7
t
LZ12 FOR
(CARBON
DIOXIDE+
WATER)
209
We used the results from table 3 with equation (5) to calculate activity coefficients for the gas mixture to compare with the values found by Greenwood in his second paper.(6) In general, the agreement was only fair, with differences frequently larger than his claimed accuracy (1 per cent) and the estimated uncertainties in table 3. The pattern of deviations could not be corrected by adjustments in B, i, Br 2, and B,, ; nor could it be reasonably attributed to variation in C,. The two computational routes are quite different, however, and systematic errors in the measurements or representation of the (p, V, T) surface can have different influences. Differences were usually larger at high mole fraction of H20. We also used the constants from table 3 to calculate fugacity coefficients to compare with the results of Ryzhenko and Malinin. (” Here the agreement was usually within 2 per cent, well within their claimed accuracy, and fairly consistent over most of the composition range. We did not attempt a comparison with Walter’s results,“’ nor with those of Franck and Tiidheidec2” which are in the range 4 x lo4 to 2 x 10’ kPa above 673 K.
3. Discussion Errors in B, are magnified in the calculation of B12 by equation (3), so small systematic errors from adsorption, corrosion, or other sources can have quite significant effects. Greenwood’5’ pointed to determination of the sample volume as the most critical measurement in his work, and amount of sample substance as a major factor at low pressures. In Maass and Mennie’s workJ3’ determination of the mass of sample in the thermostatted volume is the limiting measurement. Adsorption may be a problem in the lower-temperature range, (2*3) even at relative humidities of 50 per cent or less, and adsorption of “wet” COZ may be different from that for dry C02, especially on the metal surfaces in Gerry’s work.‘2’ In the strategy of Coan and King, (4) the non-ideality of H,O(g) is mainly due to the B, 2 cross-interactions. Their results in table 2 show some small trends with pressure. These are in the direction to be expected from the necessary omission of third virial coefficients in the calculations. Another possible cause is failure of the approximation ~4 = 1. the trends would require $j < 1 at the higher pressures (higher mole fractions of CO;). Incomplete saturation of the vapor phase, leading to values of (1 -y) that are too small, is a systematic error difficult to eliminate in such measurements, and would lead to values of Bi2 less negative than equilibrium values. The values of B,, from Coan and King’s results, table 2, are in fairly close agreement with those of Gerry at the lower temperatures, but are slightly less negative at 348.15 and 373.15 K, except for the results at lower pressures. Those of Maass and Mennie are mostly more negative than Gerry’s results, and have greater scatter. The high-temperature values of Greenwood fix the location of the Br 2 curve in that region fairly well, leaving the region from 400 to 700 K to be delineated. Examination of the ratio r = (B 11- Br 2)/(B, 1 - B,,) shows that it increases from values near 0.075 at the lower temperatures (298.15 to 373.15 K) to 0.4 at 923 to 973 K, and probably reaches a limit of 0.5 at higher temperatures. This ratio might be expected to be a smooth continuously increasing quantity with temperature. In the
C. E. VANDERZEE
210
AND N. C. HAAS
region 400 to 725 K, Y appears to increase almost linearly with temperature, and this quantity provides a very useful purpose in essentially averaging the measured results below 400 K. One run by Maass and Mennie is fairly compatible with the smoothing function ; the other has B, z values which are clearly too negative, with large values of r. The results of Gerry and of Coan and King could be blended easily by the smoothing function. Once a curve for I was selected, we regenerated B,, by the relation : B12 = (1 -r)B,
1+rB2*,
(9)
which becomes a strong generating function once I is established. The selected values of B,, are given in table 4, together with our estimates of uncertainties in the values. These finally selected values differ only slightly from our original selections prior to inclusion of the results of Coan and King, but we can now reduce our uncertainty estimates slightly below 400 K. TABLE4.SelectedvaluesofthevirialcoefficientB,,forthe{(l -y)CO,+yH,O)(g),evaluatedasdescribedin the text. Estimated uncertainty in B,,: k 15 cm3 mol-’ at 300 K, + 10 cm3 mol-’ at 400 K, decreasing to f5cm3mol-‘at 750K T ii 298.15 300 325 350 375
B 12 crdmol-’ -206 -200 -152 -120 - 101
T ii
B 12 cm3 mol - ’
T K
400 425 450 475 500
-86 -74 -63 -55 -48
550 600 650 700 750
B 12 cm3mol-’ -31 -28 -22 -17 -12
-T K 800 850 900 1000 1100
B 12 cm’mol-’ -9 -7 -5 -1 f2
In estimating the uncertainties, we have tried to include a reasonable allowance for systematic errors, but note the strong agreement between two quite different procedures(2*4) in the range 298.15 to 373.15 K. There is still a clear need for additional measurements on this important system over much of the temperature range, and we hope that this paper will provoke such measurements. The values of B12 from table 4, together with Bi i and B,, from compilations,‘” -i3) permit calculation of the fugacity coefficients for the components of ((1 -y)CO,+yH,O}(g) over much of the temperature range from 300 to 1100 K by application of equation (5). At low pressures, the influence of the third virial coefficients C, will be negligible. Above 623 K, the values of C, given in table 3 will be suitable (we estimate C, = 1300 cm6 mole2 at 623 K from reference 9), and will lead to fugacity coefficients close to the measured values of Greenwood@’ and Ryzhenko”) for pressures up to 3 x 104 kPa. In the temperature range below 600 K, values of C,,, and C222 diverge rapidly/gi so estimates of C, below 600 K are far more difficult and uncertain, and uncertainties from this term will increase rapidly at pressures above 5000 kPa. New measurements needed in this region include both B, 2 and C, terms, and even higher terms at the high pressures. The intent of this paper was to assemble some useful thermodynamic information, needed for interpretation of some interesting equilibria involving (carbon
B, r FOR (CARBON
DIOXIDE
+ WATER)
211
dioxide + water) in a forthcoming paper, so we refrain from examining models for the B,, interaction. We recognize that the smoothing function, equation (9), might be regarded as an “empirical” model. Coan and Kingc4) discussed the possibility of part of the B,, interaction being chemical in origin, and obtained reasonable thermodynamic parameters for the chemical interaction. Referee no. 2 enclosed a printout comparing the fit of several model equations with the experimental B,, values as well as the smoothed values from table 4. The most successful correlations were the equations of Hayden and O’Conne11,(22’ O’Connell and Prausnitz,‘23’ Nothnagel et LI~.,‘~~) and Tsonopoulos,‘25’ which represented the smoothed results within the uncertainty estimates. Only Nothnagel’s function gave high values of B, Z in the region 450 to 700 K. We should prefer to see additional experimental results in the range 375 to 750 K before attempting to select a “best” model. We are very grateful to Referee no. 2 for enclosing the above comparison, Referees for calling our attention to the work of Coan and King.‘4’
and to both
REFERENCES 1. Stockmayer. W. H. J. Chem. Phys. 1941, 9, 863. 2. Gerry, H. T. Ph.D. Thesis, Massachusetts Institute of Technology. 1932. 3. Maass, 0.; Mennie, J. H. Proc. R. Sot. (London) 1926, 1lOA. 198. 4. Coan, C. R.: King, Jr., A. D. J. Am. C/rem. Sot. 1971.93, 1857. 5. Greenwood, H. J. Am. J. Sci. 19@,267A, 1191. 6. Greenwood, H. J. Am. J. Sci. 1973, 273, 561. 1971, 561. 7. Ryzhenko, B. N.; Malinin, S. D. Geokimiya 1971, (8). 899; Geochemistry Internarional 8. Walter, L. S. Am. J. Sci. 1%3, 261, 151. 9. Dymond, J. H.; Smith, E. B. The V&l CoefJicienrs of‘ Gases. Clarendon Press: Oxford. 1969. 10. Guggenheim, E. A. Thermodynamics. Interscience Publishers: New York. 1967. 11. Angus, S.; Armstrong, B.; deReuck, K. M. Carbon Dioxide: International Thermodynamic Tables qf the Fluid Stare Vol. 3. Pergamon Press: Oxford. 1976. 12. Keenan, J. H.; Keyes, F. G.; Hill, P. G.; Moore, J. G. Steam Tables. Wiley: New York. 1%9. 13. O’Connell, J. P.; Prausnitz, J. M. Ind. Eng. C/rem. Fundam. 1970,9, 579. 14. McCullough, J. P.; Pennington, R. E.; Waddington, G. J. Am. Chem. Sot. 1952, 73. 4439. 15. Kell, G. S. ; McLaurin, G. E.; Whalley, E. J. Chem. Phys. 1968, 48, 4805. 16. Berg, R. L. ; Vanderzee, C. E. J. Chem. Thermodynamics 1978, 10, I1 13. 17. Weiss, R. F. Marine Chem. 1974, 2, 203. 18. Gmelins Handbuch der Anorganische Chemie No. 14. Verlag Chemie, GmbH.: Weinheim, Germany. 1960, 1973.
19. Holser, W. T. ; Kennedy, G. C. Am. J. Sci. 1959,257, 71; Am. J. Sci. 1958, 256, 744. 20. Vukalovich, M. P.; Trakhtengert, M. S.; Spiridonov, G. A. Tepfoenerge&r 1%7, 14, 65. 2I. Franck, E. U.; Todheide, K. 2. Phys. Chem. (Frankfort am Main) 1959, 22, 232. 22. Hayden, J. G.; O’Connell, J. P. Ind. Eng. Chem.. Process Des. Dev. 1975, 14. 209. 23. O’Connell, J. P.; Prausnitz, J. M. Ind. Eng. Chem.. Process Des. Dev. 1967, 6, 245. 24. Nothnagel, K. H.; Abrams, D. S.; Prausnitz. J. M. Ind. Eng. Chem.. Process Des. Dev. 1973, 12. 25. 25. Tsonopoulos, C. AIChE J. 1974, 20, 263.