Correlation and Prediction of Solubility and Entropy of Solution of Gases in Liquids by Means of Factor Analysis C. L. de Ligny’ and N. G. van der Veen Laboratory for Analytical Chemistry, University of Utrecht, Utrecht, The Netherlands
J. C. van Houwelingen Institute for &themtical Statistics, University of Utrecht, Utrecht, The Netherlands
Literature data on the entropy of solution and on log p/x of 20 gases in 39 solvents have been correlated by the equation: y = G, SIiG2S2,where y is the experimental datum, and G and S are adjustable parameters that depend only on the gas and on the solvent under consideration, respectively. The gas molecules range in size from He to C3Hs.The solvent molecules range in size from carbon disulfide to polyethylene of molecular weight lo5, and in polarity from hexane to methanol. A (generally applicable) mathematical procedure has been devised for the estimation of Gand S in cases where a large proportion of all possible yvalues is missing. The standard deviations of the correlations are 0.9 cal mol-’ deg-’ and 0.14, respectively. The equation allows the calculation of all missing data on the investigated gases and solvents.
Introduction The solubility of gases in liquids is both practically important in the most diverse fields of science and technology and theoretically interesting, e.g., because the gas molecules can be considered as probes measuring the intermolecular force field in the solution. A satisfactory theory for solutions of gases in liquids is much harder to design than for solutions of solids or liquids in liquids, the latter components differing from one another far less in significant properties such as molar volumes and intermolecular attractive forces. In our opinion, the best theory for the solubility of gases in liquids has been given by Pierotti (1963,1965). He considered the solution process to consist of two steps: (1)the creation of a cavity in the solvent of a suitable size to accommodate the solute molecule, and (2) the introduction into the cavity of a solute molecule which interacts with the solvent. The thermodynamic data for cavity formation are known functions of the temperature, the molar volume of the solvent, and the hard sphere diameters (rl and 6 2 of the solvent and the solute molecules, respectively. Pierotti assumed that the interaction entropy is zero and that the interaction free enthalpy is a known function of the molar volume of the solvent and the hard sphere diameters, force constants d k , polarizabilities, and dipole moments of the solvent and solute molecules. In practical applications of this theory, q,q / k and tz/k are treated as adjustable parameters (e.g., Pierotti, 1963, 1965; de Ligny and van der Veen, 1972,1975;Field et al., 1974).I.e., the solubility and the standard partial molar entropy of solution are considered to be known functions of these parameters, of the following type
and
pered by the following facts: (1) Log p / x is found as the difq / k , e&). (2) ference of two large quantities,f, (al)andfi (q, The value of q is fixed by the solubility of a hypothetical hard-sphere solute of u2 = 2.58 A, estimated by extrapolation of the solubilities of the noble gases. (3) Equation 2 is not very flexible, as it contains only one parameter which is fixed by the solubility of a hard-sphere solute at a particular temperature. As a result of these undesirable properties of eq 1and 2 it is found (de Ligny and van der Veen, 1975) that: (1)tz/k is a function of the chemical type of the solvent; (2) it is virtually impossible to predict the solubilities of heavy gases in polar solvents like acetone, dimethyl sulfoxide or alcohols; (3) eq 2 predicts a slight increase of the entropy of solution with increasing values of U Z , whereas a considerable decrease is actually found. In view of these failures of eq 1and 2, we thought it worthwhile to investigate the merits of another equation for the correlation and prediction of solubilities and entropies of solution of gases in liquids. Theory The data that we want to correlate are of the type yg,, where g denotes the gas and s the solvent under consideration. Recently, the following equation has been often used for the correlation of this type of data (e.g., Harman, 1970; Weiner and Parcher, 1972; Huber et al., 1973) n YgJ
- jC = l GjSj
(3)
where both G and S are adjustable parameters, depending only on the identity of g (the gas, in the present case) and of s (the solvent), respectively. The number of terms n on the right-hand side of eq 3 is chosen as large as appears necessary to reduce the residuals
jzl n
It testifies to the physical soundness of Pierotti’s theory that plausible values of the parameters ,ul,q / k , and tp/k are found. However, whereas this theory is probably the best way for the physical interpretation of gas solubilities, its usefulness for the correlation and prediction of gas solubilities is ham336 Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
Yg.8
-
GjSj = rg,s,n
(4)
to an acceptably small level. If data exist for each combination of g and s, the values of G and S can be found by factor analysis (Harman, 1970). The use of eq 3 has been restricted to this case. This means that
its utility for predictive purposes (by far its most interesting aspect) has been very limited, up till now. In the present case, data are missing for nearly 50% of all combinations of g and s. As the values of G and S can now not be found by factor analysis, the method described below was devised. Starting with a set of r d d o m numbers as estimates SI’for the adjustable parameters S1of the investigated set of solvents, estimates GI’ for the adjustable parameters G1 of the investigated set of gases were calculated by regression analysis of the data on each individual gas. I.e., eq 3 was reduced to Y s = GlS1’
(5)
where y s and SI’ are variables and G I is the regression coefficient. Using these values of GI’ as variables, better estimates S1/’were calculated by regression analysis of the data on each individual solvent. In this case, eq 3 reduces to Yg = Gl/Sl
(6)
where y g and G are variables and S1 is the regression coefficient. This procedure is reiterated until convergence has acbeen reached to estimates GI* and SI*of G1 and SI, counting for as much of the variance of yg,s as possible, i.e., with the following property (y,,,
- Gg,1Ss,1)2=
rZg s,l = minimal
(7)
g,s
g.s
Then the first set of residuals is calculated Yg.s
-~
1 * ~= 1 r*g , s , l =
2
~
j
~
J=2
j
(8)
and the above described procedure is repeated to extract the second factor, i.e., to calculate a set of estimates of Gz and one of S2 with the following property (rg,s,l
- Gg,2Ss,d2= C r2g,s,2= minimal
gas
(9)
g,s
From the second set of residuals ‘g,S,1
- G2*Sz*
n
= r&’,S,2 =
J=3
(10)
the third factor is extracted, and so on. These calculations are performed by the program MISFAC. There is a chance that this iteration procedure leads to a local, and not to the absolute, minimum. T o eliminate this risk, it should be verified that different sets of random numbers S,’ lead to identical sets of values of G,*SJ*. Of course, different sets of GI* values may differ by constant factors q , and the corresponding sets of SJ* values by factors l/q. To avoid this,ambiguity, an appropriate scaling condition is incorporated in the iteration procedure. In the application of this program to very incomplete data sets, like the present ones, some difficulties arise. (1) Convergence becomes increasingly slower in the extraction of higher factors (in the present cases, 10 iterations sufficed for the extraction of the first factor, whereas up to 50 iterations were required for the extraction of the second factor). (2) With increasing values of j , the risk increases that the 1 GJSJI values, predicted by eq 3, are incredibly large for some of the combinations of g and s for which no experimental data yg,s are available. In other words, the risk increases t h a t improbable values of Yg,,are predicted. If this occurs, the j t h factor should not be taken into account. However, the judgment whether or not predicted values Yg,sare trustworthy is a subjective one, in which chemical intuition comes into play. Therefore it is a difficult decision to make, except in very clear-cut cases. I t can be made objective by performing preliminary calculations on part of the available data. Then, the predicted values Yg,, can be compared with the experimental data yg,sthat were not included in the calculations and from the differences Yg,,-
yg,sthe standard deviation of the prediction can be calculated
for various numbers of factors n. The value of n yielding the smallest standard deviation should be preferred. The n factors, extracted by MISFAC, are not completely orthogonal. Therefore, if n > 1, a further reduction of Zg,s r2g,s,n,calculated from the analogon of eq 9 can be achieved as follows. Starting with the MISFAC estimates sj* as variables, estimates GJ* are calculated by regression analysis of the data on each individual gas, using the equation n
GjSj*
ys =
(11)
j= 1
Using these values of G,* as variables, better estimates Si** are calculated by regression analysis of the data on each individual solvent, using the equation n
yg =
j= 1
Gj*Sj
(12)
This procedure is reiterated until convergence has been reached to estimates of G, and Sj with the following propettY
These calculations are performed by the program NEWFAC. After extraction of a limited number of factors by MISFAC, or its above mentioned extension, the set of experimental data yg,scan be completed with predicted values Yg,sand a factor analysis program can be applied to the completed data set to extract further factors.
Data Preliminary calculations were performed on part of the available literature data, for reasons set forth above. The data on the entropy of solution comprised 261 data on the gases 1-5,8-12, and 14-19, shown in Table 11, column 1, dissolved in the solvents 1-7, 10-23, 25-30, 32 and 35-39, shown in columns 6 and 11,and in the solvent water. The data on log p / x comprised 370 data on the gases 1-20, and on 2,2-dimethylpropane and 3,3-diethylpentane, dissolved in the solvents 1-23,25-33 and 35-39, and in the solvents water and N-methylformamide. The data were taken from de Ligny et al. (1971,1972,1975),Wilhelm and Battino (1973),and Glew and Moelwyn-Hughes (1973). Data on CO2 in aromatic or polar solvents were not used, in view of the enhanced interaction of COPwith aromatic solvents (Field et al., 1974) and the probable formation of charge transfer complexes of COYwith polar solvents (de Ligny and van der Veen, 1975). For reasons given below the calculations were repeated after deletion of the data on the solvents water and N-methylformamide, and on the solutes 2,2-dimethylpropane and 3,3-diethylpentane. The final calculations were made on 289 data on @, for the gases 1-5,&12, and 14-19, dissolved in the solvents 1-7, 10-30,32 and 35-39, and on 408 data on log p l x , for the gases 1-20, dissolved in the solvents 1-39. The additional data were taken from Archer and Hildebrand (1963), Choi et al. (1970), Cook et al. (1957),Davidson et al. (1952),Hayduk and Buckley (1971),Hayduk and Castaneda (1973), Hayduk et al. (1972), Hildebrand and Lamoreaux (1974), Hildebrand et al. (1970), Jadot (1972), Jolley and Hildebrand (1958), Kobatake and Hildebrand(l961),Kuntz and Mains (1964), Lawrence et al. (1946),Lenoir et al. (1971),Michaels and Bixler (1961),Nelson and Hildebrand (1923), Powell (1972), Spencer and Voigt (1968), Stepanova (1970), Thomson and Gjaldbaek (1963), Washburn (19281,Wilhelm and Battino (1971), and Yeh and Peterson (1963). To the best of our knowledge, we used all accurate literature data, except data on solvents in which Ind. Eng. Chem., Fundam., Vol. 15, No. 4. 1976 337
Table I. Results of the Analysis of the Data on AS" and Log p/x by MISFAC and Factor Analysis
do, cal mol-' deg-'
log PIX
Factor Factor Factor Factor Factor Factor 1 2 3 1 2 3
Calculation procedure
Results of the Preliminary Calculations MISFAC
Large predicted values of I GjSj 1 where experimental data yg,+are missing? no no no Standard deviation of residuals 0.81 0.59 0.31 193 159 331 Number of degrees of freedom (number of data - number of solvents X number 227 of factors 1 1 Rank correlation coefficient of the G values and the atomic numbers A of the 0.60 noble gases (Dixon and Massey, 1957) Factor analysis after completion of the data set by MISFAC (2 factors) Standard deviation of residuals 1.01 0.75 0.59 0.31 Rank correlation coefficient of G and A 1 0.60 0.10 1 Results Obtained after Deletion of the Data on the Solvents Water and N-Methylformamide
no 0.13 292
253
0.94 0.10 0.94 0.94
0.13
MISFAC
Large predicted values of I GjSj I where experimental data yg,+are missing? Standard deviation of residuals Number of degrees of freedom Rank correlation coefficient of G and A Factor analysis after completion of the data set by MISFAC (1factor) Standard deviation of residuals Rank correlation coefficient of G and A Results of the Final Calculations
no 0.75 216 1
0.97 1
no 0.56 183 0.60 0.69 0.60
no 0.13 303
no
1
1
0.56
0.13
0.20
1
0.11 0.09 0.77 0.26
187
no 0.16 369
no 0.11 330 291
1
1
0.16
0.13 0.11 0.77 0.43
150
0.08
266
229
MISFAC
Large predicted values of IG,Sj 1 where experimental data yg,+are missing? Standard deviation of residuals Number of degrees of freedom Rank correlation coefficient of G and A Factor analvsis after comdetion of the data set bv MISFAC (1factor) Standarddeviation of residuals Rank correlation coefficient of G and A solubilities of less than four gases have been measured, data on gases, the solubilities of which have been measured in less than three solvents of differing polarities, and a few data that were excluded from the calculations for reasons given below.
Results The results are shown in Tables I and 11. They are commented upon below. Results of the Preliminary Calculations on AS". In the application of MISFAC the following difficulty arose. The residuals rl were still rather large for the solvents water, toluene, m-xylene, and especially for He and Ne in the solvent perfluorobenzene. The residuals r2 were acceptably small, but the predicted values I GzSz I, for combinations of g and s for which experimental data were missing, were incredibly large. Therefore, the AS" data on He and Ne in perfluorobenzene were deleted from all MISFAC calculations and included again in factor analysis. (This is the main cause why the standard deviations obtained by MISFAC are smaller than those obtained by factor analysis (Table I, columns 2 and 3).) From factor analysis it followed that the second factor is necessary to correlate the data on the solvents water and, especially, perfluorobenzene. The values of G3 showed no correlation whatsoever with the atomic numbers A of the noble gases (Table I, column 4) and presumably account merely for experimental error. Therefore, the third factor was neglected. From the differences of the calculated Yg,,values and experimental data yg,sthat had not been included in the calculations, standard deviations of 1.06 and 2.91 cal molF1 deg-l for mono- or diatomic and for polyatomic gases, respectively, were calculated. These figures are based on 15 and 11degrees of freedom, respectively. Data on the solvent carbon disulfide, 338
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
no 0.91 255
yes 0.70
1
0.10
1.05
0.84 0.60
1
221
0.67 0.70
1
which had not been included in the calculations, were fitted by the regression equation yg = aG1+ bG2
(14)
The standard deviation of the fit, based on 13 degrees of freedom, is 0.69 cal mol-I deg-I. These standard deviations represent the combined effects of experimental error and errors of fit by eq 3 and 14. The former error amounts to 1 cal mol-l deg-l even in the most recent work (Byme and BattiFo, 1975). I t can be concluded that eq 3 is able to predict AS" values of mono- or diatomic gases within experimental error, but that the accuracy of predicted data or, polyatomic gases is worse. Further, the accuracy of predicted data on the solvent water is poor: CF4, predicted -31, measured -37 cal molF1 deg-1; SFG,predicted -37, measured -41 cal mol-' deg-l. Results of the Preliminary Calculations on Log p/x. Though the values of G3 are well correlated with those of A , the third factor was neglected as it does not yield a substantial decrease of the residuals (Table I, column 7). From the differences of predicted data and experimental data that had not been included in the calculations, a standard deviation of 0.27, based on 45 degrees of freedom, was calculated. Data on the solvents carbon disulfide and olive oil, which had not been included in the calculations, were fitted by eq 14. The resultant standard deviation, based on 20 degrees of freedom, is 0.22. Data on the solute ferrocene (Alfenaar, 1966; de Ligny et al., 1968; Barraqu6 e t al., 1968; Jons and Gjaldbaek, 1974) were fitted by the regression equation
Y, = aS1
+ bS1
(15)
The resultant standard deviation, based on eight degrees of freedom, is 0.34. I t can be concluded that the accuracy of predicted data on very large solutes, like ferrocene, is worse than on small solutes. The accuracy of predicted data on the
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solvent water is poor again: Dz, predicted 9.9, measured 7.7; SFs, predicted 5.8, measured 8.2. From the results of the preliminary calculations it was decided to delete the data on the solvent water and the log p / x data on the solvents water and N-methylformamide. (Data on the latter solvent, determined by means of gas chromatography (de Ligny et al., 1971), might be less accurate than the other data, determined by classical methods.) Further, it was decided not to include the data on ferrocene in future calculations and to delete the data on 2,2-dimethylpropane and 3,3-diethylpentane. Results of the Calculations on A s o , Obtained after Deletion of the Data on the Solvent Water, and of the Final Calculations. Deletion of the data on the solvent water appears to have only a slight favorable influence on the residuals, obtained by MISFAC (Table I, columns 2 and 3). In the final MISFAC calculations, very large IGzSz1 values were predicted for some combinations of g and s for which experimental data yg,s are missing (in one extreme case amounting to 45 cal mol-' deg-l). So, the data set was completed by 1factor MISFAC prior to factor analysis. The second factor resulting from factor analysis is necessary to correlate the data on the solvent perfluorobenzene. The third factor was neglected, because of the low correlation of G3 and A (Table I, column 4). From the differences of calculated data and experimental data that had not been included'in the calculations, standard deviations of 1.05 and 2.83 cal mol-l deg-l are calculated for mono- or diatomic and for polyatomic gases,respectively. (As argued below, the experimental value of ASo for CHI in nitrobenzene is probably in error. When this datum is omitted, a standard deviation of 2.24 cal mol-l deg-l is calculated, only twice the experimental error.) We think that these figures are also representative for the accuracy of predictions by the results of the final calculations that were performed on as large a data set as possible. In the final calculations on A s o four literature data were not taken into consideration as they were thought to be unreliable, viz., data on CzH4 and C ~ Hin G the solvents methanol and ethanol, ranging from -31 to -39 cal mol-l deg-l (Boyer and Bircher, 1960).The values of GI, Gz, SI, and Sz, resulting from the final calculations, are shown in Table 11. Results of the Calculations on Log p/x, Obtained after Deletion of the Data on the Solvents Water and N methylformamide and on the Solutes 2,2-Dimethylpropane and 3,3-Diethylpentane, and of the Final Calculations. Deletion of these data has a considerable favorable influence on the residuals, obtained by MISFAC (Table I, columns 5 and 6). The standard deviation of the residuals r2, obtained by factor analysis of the data set, after its completion by 1 factor MISFAC, is substantially larger than that obtained by MISFAC. This holds both for the results, obtained after deletion of the data on the solvents water and N-methylformamide and on the solutes 2,2-dimethylpropane and 3,3diethylpropane and for the results of the final calculations, that were performed on as large a data set as possible. It is caused by the constraint on the values of G2 and Sp, imposed in factor analysis by the completion of the data set by 1factor MISFAC, Le., the constraint that GzSz = 0 for combinations of g and s for which no experimental data are available. Whenever this phenomenon occurs, it is a warning that, in MISFAC, the value of lG2Szl may be large for some combinations of g and s for which experimental data are missing. In that case the second MISFAC factor should not be taken into consideration. In the present case, the occurrence of incredibly large values of 1 GzS21 is not immediately clear from the results, the largest value being equal to only 0.9. As argued in section 2, the best procedure is then to perform calculations on part of the available data prior to the final calculations, and
Aso
Ind. Eng. Chem., Fundam., Vol. 15, No. 4,
1976
339
Table 111. Comparison of Predicted Values of -AS" and Log p/x with Experimental Data (Byme and Battino, 1975; Fleury and Hayduk, 1975)
-Go,cal mol-' deg-1 Solvents m-Xylene
Carbon disulfide
Gases
Exptl
Calcd, eq 3
Calcd, Pierotti theory
Ar Kr CF4 SFfi C3HB C3H8
13.4
13.1
13.6
13.7 15.1
13.5 15.9
13.5 13.4
0.5
1.0
Standard deviation
to compare the experimental data yg,sthat were not included in the calculations with the predicted values Yg,,,calculated by 1 factor and 2 factor MISFAC, respectively. As a result, we found standard deviations of the predicted values of 0.20 and 0.23, respectively (45 degrees of freedom). so, 1factor MISFAC is preferable. We think that a standard deviation of 0.20 is also representative for the accuracy of predictions by the results of the final calculations. The values of G1, Gz, SI, and S Z ,resulting from the final calculations, are shown in Table 11. The third factor was neglected in view of the low correlation of G3 and A . (Table I, column 7).
Discussion and Conclusions By substitution of the values of G1, Gz, SI, and S Z from Table I1 in eq 3,289 literature data on @ and 408 literature data on log plx can be correlated with standard deviations of 0.84 cal mol-'l deg-l and 0.13, respectively (Table I, columns 3 and 5). As regards 2 out of the 289 residuals r2 were very large, viz., for CF4 in toluene (-3.31 cal mol-I deg-') and for CHI in nitrobenzene (-3.63 cal mol-l deg-I). If the values of r2 have a normal distribution, the chances that r2 3 3.31 and r2 3 3.63 are only 1:5000 and 1:16 000, respectively, so it is likely that the experimental data (Ezheleva and Zorin, 1962) are in error. With regard to log plx, only one very large residual rz was found, viz., for CH4 in carbon disulfide (0.63). The chance for such a large residual to be found being only 1:105,we conclude that the experimental value (Hildebrand and Lamoreaux, 1974) is probably in error. Far more important than the ability of eq 3 to correlate existing data is its ability to predict missing data on the investigated gases and solvents,Above, we estimated the accuracy of predicted values of ASoto be 1.05 and 2.24 cal mol-1 deg-l for mono- or diatomic and for polyatomic gases, respectively, and the accuracy of predicted vaiues of log p l x to be 0.20. After completion of the calculations two more papers (Byrne and Battino, 1975; Fleury and Hayduk, 1975) on the solubility of gases appeared. These data enable us to check the validity of our accuracy estimates (Table 111).The last line of this table shows that the accuracy of data, predicted by eq 3, is possibly even better than estimated above. This table shows also that Pierotti's theory is not able to account for the decrease of ASo with increasing size of the gas molecule. It does predict log p l x data in the concerned apolar solvent with the same accuracy as eq 3 does. However, it must be remembered that it is virtually impossible to predict gas solubilitiesin polar solvents by Pierotti's theory.
e,
Nomenclature A = atomicnumber a,b = regression coefficients 340
Log p l x
Ind. Eng. Chem.,Fundam., Vol. 15, No. 4, 1976
Exptl
Calcd, eq 3
Calcd, Pierotti theory
5.81 5.32 6.01 5.32 4.02 4.23
5.79 5.33 5.82 5.19 4.16 4.12
5.62 5.38 6.04 5.45
0.12
0.12
f = denotes a functional relationship G = adjustable parameter, depending on the gas under consideration k = Boltzmann's constant (erg deg-1) p = partial vapor pressure of a solute, mmHg r = residual, defined in eq 4 S = adjustable parameter, depending on the solvent under consideration A F = standard partial molar entropy of solution, cal mol-I deg-' x = mole fraction of a solute y = experimental datum Y = calculated datum Greek Letters = molecular energy parameter, ergs cr = molecular diameter, 8,
6
Subscripts c,i = denote cavity formation in the solvent and solute-solvent interaction, respectively .g,s = denote gas and solvent, respectively Literature Cited Alfenaar, M., Dissertation, Utrecht, 1966. Archer, G., Hildebrand, J. H., J. Phys. Chem., 67, 1830 (1963). Barraque, Chr., Vedel, J., Tremillion, B., Bull. Soc. Chim., 3421 (1968). Boyer, F. L., Bircher, L. J., J. Phys. Chem., 64, 1330 (1960). Byrne, J. E., Battino, R., J. Chem. Thermodyn., 7, 515 (1975). Choi, D. S., Jhon, M. S.,Eyring, H., J. Chem. Phys., 53, 2608 (1970). Cook. M. W., Hanson, D. N., Alder, B. J., J. Chem. Phys., 26,748 (1957). Davidson, D.,Eggleton, P.,Foggie, P., Quart. J. Exptl. Physiol. 37, 91 (1952). de Ligny, C. L.. Alfenaar, M., van der Veen, N. G., Rec. Trav. Chim., 87, 587 ( 1968). de Ligny, C. L.,Denessen, H. J. M., Alfenaar, M., Rec. Trav. Chim., 90, 1265 (1971). de Ligny, C. L.. van der Veen, N. G., Chem. Eng. Sci., 27, 391 (1972). de Ligny, c. L., van der Veen, N. G., J. Sol. Chem., 4, 841 (1975). Dixon, W. J., Massey. F. J., Jr., "Introduction to Statistical Analysis", p 294, McGraw-Hill, New York. N.Y., 1957. Ezheleva. A. E., Zorin, A. D., Tr. po Khim. i.Khim. Tekhnol., 1, 37 (1962). Field, L. R., Wilhelm, E., Battino, R., J. Chem. Thermodyn.. 6, 237 (1974). Fleury, D., Hayduk, W., Can. J. Chem. Eng., 53, 195 (1975). Glew, D. N., Moelwyn-Hughes, E. A., Discuss. Faraday SOC., 15, 150 (1973). Harman. H. H., "Modern Factor Analysis", University of Chicago Press, 1970. Hayduk, W., Buckley, W. D., Can. J. Chem. Eng., 49, 667 (1971). Hayduk, W., Castaneda, R., Can. J. Chem. Eng., 51,353 (1973). Hayduk, W., Walter, E. B., Simpson. Ph., J. Chem. Eng. Data, 17, 59 (1972). Hildebrand, J. H., Lamoreaux, R. H., lnd. Eng. Chem., Fundam., 13, 111 (1974). Hildebrand, J. H., Prausnitz. J. M., Scott, R. L., "Regular and Related Solutions", p 201, Van Nostrand-Reinhold, New York, N.Y., 1970. Huber, J. F. K., Alderlieste, E. T., Harren, H., Poppe, H., Anal. Chem., 45, 1337 (1973). Jadot, R., J. lnf. Chim. Phys., Phys. Chim. Biol., 69, 1036 (1972). Jolley, J. E., Hildebrand, J. H., J. Am. Chem. Soc., 80, 1050 (1958). Jons, O.,Gjaldbaek, J. Chr., Acta Chem Scand. A, 28, 528 (1974). Kobatake, Y., Hildebrand, J. H., J. Phys. Chem., 65, 331 (1961). Kuntz. R. R., Mains, G. J., J. Phys. Chem.. 68, 408 (1964). Lawrence, J. H., Loomis, W. F., Tobias, C. A., Turpin, F. H., J. Physiol., 105, 197 ( 1946). Lenoir, J. Y.. Renault, Ph., Renon, H., J. Chem. Eng. Data, 16, 340 (1971). Michaels, A. S., Bixler, H. J., J. Polym. Sci., 50, 393 (1961). Nelson, W. T.. Hildebrand, J. H., J. Am. Chem. SOC., 45, 682 (1923). Pierotti, R. A,, J. Phys. Chem., 67, 1840 (1963). Pierotti, R. A,, J. Phys. Chem., 69, 281 (1965).
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Receiued for reuiew April 12,1976 Accepted June 8,1976
EXP E R IMENTA1 TECHNIQUES
Measurements of Concentration Fluctuations in Gaseous Mixtures William M. Edwards, Jorge E. Zunlga-Chaves, Frank L. Worley, Jr., and Dan Luss+ Department of Chemical Engineering, University of Houston, Houston, Texas 77004
A new experimental technique is presented for measuring concentration fluctuations in gaseous mixtures using a catalytic wire on which a mass-transfer-limited exothermic reaction occurs. This method utilizes a catalytic sensor in conjunction with a constant-temperature anemometer unit. It should be a useful tool for studies to improve the design of gas mixing equipment and of chemical reactors in which the yield and/or conversion depend on the degree of mixing.
Information about instantaneous concentration fluctuations in gaseous mixtures is very useful for the design of industrial gas mixing equipment and of reactors in which the yield and/or conversion are sensitive to the mixing of the reactants. The techniques now available for measuring these fluctuations (Corrsin, 1949; Blackshear and Fingerson, 1962; McQuaid and Wright, 1973) are either suitable only for mixtures in which the heat transfer properties (e.g., k , C,) of the two gases are significantly different, or require the simultaneous use of several sensors. Recently, optical techniques were developed which use interferometer or crossed Schlieren optical systems to detect selective index gradients for mixtures of gases with large density differences (Wilson arid Prosser, 1971). These can be related to density fluctuations and hence concentration fluctuations for isothermal gas flow. We deskribe here a novel technique for measuring rootmean-square concentration fluctuations. The method utilizes a catalytic sensor in conjunction with a constant-temperature anemometer unit. It is applicable to gaseous mixtures containing species which can react rapidly and exothermally on a platinum wire, such as hydrocarbons and oxygen. A major advantage of this technique is that it can be applied to mixtures of gases with similar physical properties.
Theoretical Background Consider a catalytic wire whose capacity parameter, defined as the ratio of the characteristic time for changes in wire temperature to the characteristic time for surface concentration changes due to changes in the rate of mass transfer, is large. When a mass-transfer-limited exothermic reaction occurs on such a wire its temperature may be described by the equation (Edwards et al., 1974).
For wires with negligible end effects, the first term in the right-hand side of (1)can be discarded (this term is normally ignored when dealing with hot wires with length-to-diameter ratios larger than 200 (Hinze, 1959)). If in addition the wire is heated by means of an electric current, eq 1 takes the form dT ApC, - = h P ( T , - T) P(-AH)k,C J12R (2) dt The temperature of the wire can be maintained a t a constant level by using a constant-temperature anemometer to control the heating current. Therefore, the time derivative in (2) vanishes. Moreover, the transport coefficients, the limiting reactant concentration, and the electric current can be expressed as the sum of a stationary time average and a fluctuating component thus enabling the reduction of (2) into
+
+
In this equation all second-order terms have been neglected and ( ) denotes a stationary time average. Time averaging of (3) yields
( h ) ( T- T g )= (-AH)( k , ) ( C ) + J ( I ) 2 R / P
(4)
When no electrical heating is used
T* - T , A (-AH)( k c ) ( C ) / ( h ) (5) and T* - T, is defined as the adiabatic temperature rise. Subtracting (4) from (3) and division by (5) yields
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