O'Connell, J. P., Prausnitz. J. M., Ind. Eng. Chem., Process Des. Dev., 6, 245 (1967). Onnes, H. K.. Pros. Akad. Sci. (Amsterdam), 4, 125 (1902). Orantlicher, M., Prausnitz, J. M., Can. J. Chem., 45, 373 (1967). Peikins, A. J., J. Chem. Phys., 5, 180 (1937). P'tzer, K. S.,Curl, R. F., J. Am. Chem. SOC.,79, 2369 (1957). Ramalho, R . S..Frizelle, W. G., J. Chem. f n g . Data, I O , 366 (1965). Kwong. J. N. S..Chem. Rev., 44, 233 (1949). Redlich, 0..
Townsend, P. W., doctoral dissertation, Columbia University, New York, N.Y., 1956. Woolley, H. W., Scott, R. B., Brickwedde, F. G.. J. Res. Nat. Bur. Stand., 41, 379 (1948).
Received for review October 21, 1974 Accepted April 21,1975
Solubility of Nitrous Oxide in the Mixtures of Alcohols and Water: Comparison with Pierotti's Gas Solubility Theory Elzo Sada,' Shlgeharu Klto, and Yoshitaka It0 Department of Chemical Engineering, Nagoya University, Furo-cho, Chikusa-ku. Nagoya, 464 Japan
Solubilities of nitrous oxide in methanol-, ethanol-, 1-propanol-, and 2-propanol-water mixtures at 25OC are presented. By using these data, various features of Pierotti's gas solubility theory are discussed.
Introduction Solubilities of gases in liquids are of considerable industrial and theoretical importance. A vast amount of experimental work has been done and various behaviors found. Not a few theoretical or semitheoretical analyses have been presented also. A majority of these were, however, devoted to gas solubilities in pure liquids or in aqueous solutions such as aqueous electrolyte solutions. On the other hand, experimental as well as theoretical investigations of gas solubilities in mixed solvents are rare. O'Connell and Prausnitz (1964) and Boublik and Hila (1966) have analyzed the phenomena from purely thermodynamic viewpoints and derived semitheoretical correlations. The concepts of these thermodynamic treatments were briefly reviewed by O'Connell (1971). He pointed out the limitation of the thermodynamic treatments and proposed an approximate correlation on the basis of the solution theory of Kirkwood and Buff (1951). However, the applicable range and accuracy of correlations are still not satisfactory. All of these correlations require the thermodynamic data for the mixed solvent. Tiepel and Gubbins (1972) have recently proposed an analysis by using a statistical mechanical perturbation theory. The theory is in relatively good agreement with experimental results in predicting solubilities of gases even in a polar substance containing mixed solvents. The review of previous investigations indicates that it is necessary to make more experimental data available and to understand the phenomena better in order to advance the analysis of gas solubility in mixed solvents. In the present research, solubilities of nitrous oxide in the mixtures of alcohols and water were measured a t 25OC and 1 atm total pressure. Some qualitative features of gas solubility in mixed solvent are discussed by using the results of this experimental work and those of analyses by Pierotti's gas solubility theory (1963; 1965) extended to such a case. Experimental Method Solubility measurements were carried out by using the same apparatus as that used in the previous investigations 232
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
(Onda et al., 1970a,b; Sada et al., 1972, 1974). The full description of the apparatus and mode of operation have been given elsewhere (Onda et al., 1970a). Here, a mixed solvent prepared to a desired composition was boiled in a degassing vessel with a reflux condenser under a reduced pressure to remove dissolved gases. In this work gas solubilities were represented by Henry's constants. When Henry's constant was calculated from the directly measured gas volume dissolved in mixed solvent, vapor pressure data of the mixed solvent were needed. However, for 1-propanol- and 2-propanol-water mixtures the data in the literature where available were very inaccurate. Therefore, the vapor pressures were observed in this work by using an apparatus similar to that described by Taha et al. (1966). The observed vapor pressures of methanol, ethanol, and water a t 25OC agreed well with the published data (Riddick and Bunger, 1970). Moreover, those of the mixture of methanol and water were also in good agreement with the data by Butler et al. (1933). Nitrous oxide used was supplied from a cylinder specially prepared by Showa Denko Co., Ltd. (Tokyo) and the purity was confirmed to be more than 99.8% by a gas chromatographic technique. It has already been confirmed by the authors (Sada et al., 1974) that the purities of the alcohols are satisfactory. Water was carefully distilled. The composition of mixed solvent was determined by measuring its density at 25OC by using an Ostwald-type pycnometer. Experimental Results The observed results of vapor pressures are shown in Figure 1 for a 1-propanol-water mixture and in Figure 2 for a 2-propanol-water mixture. These results were correlated by the following five-parameter Redlich-Kister equation after the computation technique of Barker (1953).
By assuming that vapor phase is an ideal gas mixture, the parameters in eq 1 were determined from the observed
30
0.2
0
04
06
mole fraction of n-propanol,
1.0
0.8 x2
Figure 1. Vapor pressures of 1-propanol-water mixtures at 25OC.
Table I. Values of Redlich-Kister Parameters at 25°C 20
Alcohol
A
B
C
tz-PrOH i-PrOH
1.711 1.578
-0.373 -0.390
0.370 0.324
D
0
E
-0.544 -0.270
-0.188 -0.352
02 0 4 0 6 mole traction of +propanol,
0 8
xp
Figure 2. Vapor pressures of 2-propanol-water mixtures at 25'C.
Table IV. Solubilities of Nitrous Oxide in n-PrOH-H20 Table 11. Solubilities of Nitrous Oxide in MeOH-H20
Mixtures a t 25°C and 1Atm
Mixtures a t 25°C and 1Atm x2
Hi m
x2
HIm
0 .o 0.022 0.048 0.054 0.060 0.077 0.095 0.140 0.155 0.175 0.197 0.204 0.208
2320.1 2208.7 2106.7 2090.7 2062.1 2025.7 1980.2 1814.0 1670.8 1724.3 1631.6 1609.0 1951.1
0.398 0.408 0.435 0.595 0.670 0.690 0.770 0.828 0.870 0.928 0.932 1.o
921.6 892.8 827.1 515.7 420.8 401.1 327.7 295.6 255.2 222.1 219.6 190.6
Table 111. Solubilities of Nitrous Oxide in EtOH-H20 Mixtures at 25°C and 1 Atm x:
HIm
s
Hl m
0.442 0.537 0.651 0.731 0.783 0.797 0.853 0.868 0.978 1.o
539.1 437.8 303.6 256.6 216.5 218.5 193.3 189.6 151.1 145.8
.V?
Hl rn
V?
HIrn
0 .o 0.040 0.085 0.167 0.267
2320.1 2131.6 1809.6 1053.5 636.5
0.517 0.655 0.707 0.8 52
302.1 224.6 204.5 160.4 125.8
1.o
Table V. Solubilities of Nitrous Oxide in i-PrOH-HzO Mixtures a t 25°C and 1 Atm s2
Hi m
,V 1
HIrn
0 .o 0.008 0.028 0.051 0.070 0.076 0.140 0.267
2320.1 2273.3 2230.0 2237.7 2164.7 2096.6 1471.3 718.1
0.338 0.433 0.572 0.708 0.760 0.865 1.o
524.4 382.0 266.3 201 .o 182.9 148.5 125.8
~~
0 .o 0.023 0.036 0.058 0.077 0.130 0.192 0.197 0.251 0.34 9
2320.1 2201.8 2155.6 2102.3 2083.1 1889.7 1500.8 1458.6 1127.7 761.1
vapor pressure data by a nonlinear least-square method and tabulated in Table I. The correlated results of vapor pressures are shown in Figures 1 and 2 by solid lines. In the evaluation of Henry's constants, nonideality in the gas phase was corrected by using the second virial coefficient of nitrous oxide a t 25OC, i.e., -133.2 cm3/mol, which was interpolated from the data compiled by Dymond and
Smith (1969). The numerical values of Henry's constants are tabulated in Tables I1 through V and the variations of the Henry's constants with solvent composition are shown in Figures 3 through 6 in semilogarithmic plots.
Discussion Both alcohol and water are polar associated liquids and therefore they may form highly nonideal liquid mixtures when they are brought together. Moreover, when gases are dissolved in them, the thermodynamic behaviors may hecome more complex. This can be seen from the gas soluhility curves in Figures 3 through 6. In these figures the abnormal points in the solubility curves can be observed at low alcohol concentrations with the exception of the system consisting of the 1-propanol-water mixture. This feature is very significant in the mixture of 2-propanol-water as can be seen in Figure 6. Ind. Eng. Chem.. Fundam.. Vol. 14, No. 3, 1975
233
8
5
8 ,
0.2
0.4
,
os
mole fraction of methanol,
a8
I
X2
F i g u r e 3. Solubilities of nitrous oxide in methanol-water mixtures a t 25OC and 1 atm total pressure.
a
02
04
06
08
male fraction of n-propanol, X 2
F i g u r e 5 . Solubilities of nitrous oxide in 1-propanol-water mixtures a t 25" and 1 atm total pressure.
8
7
E
r-
C
6
5
a2 0.4 06 mole traction of ethanol, xg
08
Figure 4. Solubilities of nitrous oxide in ethanol-water mixtures a t 25OC and 1 atm total pressure.
1
0
From very careful experimental work, Ben-Naim and Baer (1964) indicated that the solubility of argon in ethanol-water mixture exibits an abnormal variation with the solvent composition and that the abnormal point is encountered a t an ethanol mole fraction of 0.1. They suggested that it is due to the abnormality of the solvent structure a t the corresponding solvent composition. I t is natural to consider that the abnormality originates mainly from the hydrogen bonds between like or unlike molecules in the solvent since in the mixture of alcohol and water, hydrogen bonds probably play an important role in the thermodynamic behavior. I t can also be found in Figure 4 that the solubility curve of nitrous oxide in ethanol-water mixtures shows such an abnormal point a t the ethanol mole fraction of around 0.1. This solvert composition coincides with that of argon in 234
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
I
0.4 0.6 08 mole t r a c t i o n of I-propanol, X 2
0.2
F i g u r e 6. Solubilities of nitrous oxide in 2-propanol-water mixtures a t 25OC and 1 atm total pressure.
the same solvent as reported by Ben-Naim and Baer (1964). From this fact it is suggested that the solvent composition a t which the abnormality is observed is almost independent of the gas species and therefore the solubility of a gas seems to be considerably influenced by the structure of mixed solvent itself. With these considerations, a statistical mechanical approach can be thought to be a powerful means for analyzing gas solubility phenomena in mixed solvents. However, a t the present time, there is no exact statistical mechanical solution theory. Instead, various kinds of approximate theories or model approaches have been presented and advanced.
In the present research a somewhat intuitive gas solubility theory presented for the cases concerned with pure liquids including water by Pierotti (1963, 1965) was extended to the system of a mixed solvent treated here. According to Pierotti (1963), the Henry's constant of a gas is given by
Table VI. Solubilities of Noble Gases in 1-Propanol and 2-Propanol at 25°C and 1 Atm
-_
Gas
Alcohol
H,! x IO-:, atm
He
Ii-PrOH i-PrOH u-PrOH i-PrOH
1.119 0.981 0.126
Ar
Here, gc is the reversible work associated with the creation of a cavity in which a solute gas molecule can be accomodated and g i is the partial molar internal energy for the introduction of a gas molecule into the cavity against various intermolecular interactions. When the solubility of a gas in a mixed solvent is considered, g , must be obtained from the scaled particle theory for a rigid sphere mixture. This was given by Lebowitz et al. (1965) as
0.107
Table VII. Physical Parameters Lsed in This Work lo?:, cm'
N x
Substance
a, A
NZO H?O MeOH EtOH rr-PrOH i-PrOH
4.59b 2.75d 3.679 4.321 5.100 5.088
~ , , / h ' ,K 1.Debye
...
...
176.87 242.45 280.07 407.05 383.47
1.84d 1.70" 1.69' 1.66"
1.63''
2.921'
...
... ... ...
...
should be read as that of interaction with NzO. Hirschfelder et al. (1967). L' Moelwyn-Hughes (1961). Pierotti (1965). e Smyth (1955). 0 C1,,R
where 1
' i=l
Equations 2 and 3 are also used by Shoor and Gubbins (1969) in the analysis of gas solubility in aqueous electrolyte solutions. Although Pierotti's theory was developed for a nonpolar gas molecule, nitrous oxide, which is studied in this work, has a small dipole moment. Here, it was regarded as a nonpolar molecule for simplicity in discussion and this approximation will be checked later in this paper. As a result, on modifying the calculation method of Pierotti, gi in this case is given as
In carrying out the calculation along the line mentioned above, some mixing rules for the molecular parameters ulj and e l l are necessary. Pierotti (1965) carried out the calculation on the basis of the Lorentz-Berthelot mixing rules. In this respect, while the arithmetic mean rule for the molecular diameter holds exactly for the rigid sphere mixture, it is not suitable to use the geometric mean rule for the energy parameter of highly polar nonspherical molecules such as water or alcohol. De Ligny and van der Veen (1972) checked Pierotti's theory for various systems and pointed out that it was necessary to give attention to the evaluation of ell. In this research, therefore, tl,/k themselves were considered as the parameters determined from gas solubility data in each pure constituent solvent a t the same temperature. This estimation method contrasts with that of Pierotti (1965). However, by doing so, the ambiguities due to the mixing rule can be avoided and the features concerned with the mixed solvent seem to be extracted more significantly. In carrying out the computations, various kinds of other physical parameters are needed. Among these, the rigid sphere diameters of the alcohols and water are the most important. For a water molecule, Pierotti (1965) has given the value a t 25OC. However, for alcohols, those which are suitable to Pierotti's theory have not been given in previous investigations. Therefore, these values must be obtained according to Pierotti's estimating procedure. The rigid
sphere diameters of methanol and ethanol molecules were obtained by using the solubility data of helium and argon in each alcohol given by Lannung (1930). Thus obtained values of diameters were 3.679 8, for methanol and 4.321 8, for ethanol. These values are close to those of the force constants in the Stockmayer potential law concerned with polar substances. As can be seen from this fact, the values obtained in the present work seem to be reasonable. On the other hand, solubilities of the noble gases in 1propanol and 2-propanol could not be found in previous investigations. So in the present investigation, the solubilities of helium and argon in each of the solvents were measured a t 25OC by the method described above. Here, the high purity gases supplied by Matheson Gas Products were used. The experimental results are tabulated in Table VI. By using these solubility data the rigid sphere diameters of 1propanol and 2-propanol were evalulated as 5.100 and 5.088 8,, respectively. These values were used in the theoretical calculations. For the rigid sphere diameter of nitrous oxide molecule, the value of 4.59 8, was taken from the force constant in the Lennard-Jones potential function of nitrous oxide (Hirschfelder et al., 1967). (This is the value calculated from the second virial coefficient data of nitrous oxide gas.) The values of other molecular parameters used here are listed in Table VI1 as well as their sources. As mentioned above, it must be confirmed that nitrous oxide can be treated as a nonpolar molecule in the theory. Here, the authors checked Pierotti's theory as well as its applicability to the case of nitrous oxide as follows. If Pierotti's theory is plausible, the values of f ~ , / kcalculated from observed solubilities of one gas in one pure solvent must be shown to be constant over not too wide a temperature range. In this work values of c l z / k for an aqueous system a t various temperatures of 5°C through 50°C for eight gases. The solubilities used here were taken from the Latidolt-Bornstein Tabellen (1962) and the physical parameters in Table VI1 were used. The temperature dependences are shown graphically i t 1 Figure 7. It is found from the figure that f ~ . J for k a nitrous oxide containing system shows a good constancy and hence, nitrous oxide can be approximately treated as a nonpolar molecule in the theory despite its small dipole moment. Ind. Eng. Chem., Fundam.. Vol. 14, No. 3, 1975
235
ax
N20 I5(
Y
1
\ ’4-
C Ha IOC
02 Ar N2
x
I
10
20 30 temperature, “C
40
50
Figure 7. Variations of q3/k with temperature (calculated from the solubilities in water appearing in Landolt-Bornstein Tabellen, 1962.
Also it can be seen from Figure 7 that, as might be expected from their character of nonchemical interaction, t12/k of nitrogen, oxygen, argon, and methane remain constant with variation of temperature. On the other hand, those for carbon Gioxide, hydrogen sulfide, and acetylene vary with temperature. The latter results seem to be natural since these gases are considered to partly dissociate into ions or form charge-transfer complexes with water molecules in aqueous solution. From these facts it may be concluded that Pierotti’s gas solubility theory works well in a system in which molecules interact only physically. The correlated results of Henry’s constants of nitrous oxide in methanol-water, ethanol-water, l-propanolwater, and 2-propanol-water mixtures are shown in Figures 3 through 6 by solid lines. It is found in the figures that Pierotti’s theory can represent the solubility curves qualitatively. Pierotti’s gas solubility theory is clearly a kind of oneparameter first-order perturbation solution theory in which the Percus-Yevick compressibility equation of state for rigid sphere mixture is used as the reference state. At the present time such a perturbation theory is considered to be the most successful solution theory. However, the theory assumes pair-wise additivity for system energy. This hypothesis is strictly correct for the rigid sphere mixtures. However, for real solutions, interactions higher than threebody must be considered. Therefore, use of the hypothesis does not take into account that the introduction of gas molecules into the mixed solvent would have some influence on the intermolecular interactions between solvent molecules. This shortcoming, however, does not seem to be serious as can be seen in the comparison of our results with those of Ben-Naim and Baer (1964), which was mentioned above. In addition, this kind of statistical mechanical theory cannot give the entropy term associated with the perturbed part of the intermolecular potential, Le., the attraction potential in this case. This can be seen from the fact that the radial distribution functions considered are those of rigid sphere mixtures. As a result, the contribution from the liq236
Ind. Eng. Chem.. Fundam., Vol. 14, No. 3, 1975
uid structure is taken into account only by intermolecular repulsion forces. Although this was found to apply to simple liquids such as argon, the entropy associated with hydrogen bonds which are very important in alcohol-water mixtures as well as other attraction forces cannot be enumerated at all. Besides, neither Pierotti’s theory nor any other statistical mechanical theory can take into account even the internal energy attributed to hydrogen bonds since they have not yet been analytically formulated. With the considerations mentioned above, Pierotti’s theory seems mainly to represent the contribution of the repulsive part of the intermolecular potential to gas solubility phenom en a. It is only natural to think as follows: since the abnormalities of the solubility curves would be caused mainly by hydrogen bonds in the mixed solvents, Pierotti’s theory cannot reproduce them. From the above discussion it is interesting to conclude that even in a complicated mixture of alcohol and water, the principal nature of gas solubility phenomena would be determined to a large extent by the repulsive part of the intermolecular interaction. The calculated results from Pierotti’s theory are found to be very sensitive to the values of the rigid sphere diameters used. This seems to be due to the infinite steepness of the repulsive potential function which was used as the reference state. The quantitative disagreement between experimental and theoretical solubilities may be considered to be partly due to this problem.
Concluding Remarks Solubilities of nitrous oxide in methanol-, ethanol-, 1propanol-, and 2-propanol-water mixtures were measured at 25OC and 1 atm total pressure by using a gas volumetric method. Vapor pressures of 1-propanol- and 2-propanol-water mixtures were also observed a t 25OC and correlated by the Redlich-Kister five-parameter equation for use in the evaluations of Henry’s constants. Pierotti’s gas solubility theory was extended so as to apply to gas solubility in a mixed solvent. Experimental and calculated solubilities were compared and showed qualitative agreement. A check of Pierotti’s theory was also carried out by using published data. From the comparison it is deduced that the general behavior of gas solubility phenomena is determined by the repulsive part of the intermolecular potential but that the abnormality in the solubility curves which is thought to be due to hydrogen bonds cannot be reproduced as expected. Nomenclature A = parameter defined by eq 1 B = parameter defined by eq 1 C = parameter defined by eq 1 D = parameter defined by eq 1 E = parameter defined by eq 1 GE = excess Gibbs free energy H = Henry’s constant k = Boltzmann constant P = total pressure P, = saturation vapor pressure R = gasconstant T = absolute temperature V = molar volume of solvent z = mole fraction in liquid phase Greek Letters a = polarizability t = energy parameter in Lennard-Jones-type potential
function number density
p =
= dipole moment rigid sphere diameter of molecule
u =
Subscripts j = solvent species; m = mixed solvent 1 = gas 2 = alcohol 3 = water L i t e r a t u r e Cited Barker, J. A,, Aust. J. Chem.. 6, 207 (1953). Ben-Naim, Baer. S..Trans. Faraday SOC., 60, 1736 (1964). Boublik, T., Hala. E., Collect. Czech. Chem. Commun., 31, 1628 (1966). Butler, J. A. V., Thomson, D. W., Maclennan. W. H., J. Chem. SOC.. 674 (1933). de Ligny, van der Veen. Chem. Eng. Sci,, 27, 391 (1972). Dymond, J. H., Smith, E. E.. "The Virial Coefficients of Gases," Oxford University Press, London, 1969. Hirschfelder, J. 0.. Curtis, C. F.. Bird, R. E., "Molecular Theory of Gases and Liquids," Wiley, New York, N.Y., 1967. Kirkwood. J. G.. Buff, F. P.. J. Chem. Phys.. 19, 774(1951). "Landolt-Bornstein Tabellen," II Band, 2 Teil. Springer, Berlin, 1962. Lannung. A.. J. Am. Chem. SOC.,52,68 (1930).
Lebowitz. J. L.. Helfand, E., Prestgeard, E.. J. Chem. Phys., 43, 774 (1965). Mayer. S. W.. J. Chem. Phys., 36, 1803 (1963). Moelwyn-Hughes. E. A.. "Physical Chemistry," 2nd ed, Pergamon, Oxford, 1961. O'Connell. J. P.. Prausnitz, J. M., lnd. Eng. Chem., fundam., 3, 347 (1964). O'Connell, J. P., A.l.Ch.E. J., 17, 658 (1971). Onda, K., Sada, E., Kobayashi. T., Kito. S.,Ito, K., J. Chem. Eng. Jpn., 3, 18 (1970a). Onda. K., Sada. E., Kobayashi. T.. Kito, S..Ito, K., J. Chem. Eng. Jpn., 3, 137 (1970b). Pierotti. R. A., J. Phys. Chem., 67, 1840 (1963). Pierotti. R. A,, J. Phys. Chem., 69. 281 (1965). Riddick, J. A., Bunger, W. B., "Organic Solvents," 3rd ed. Wiley. New York, N.Y., 1970. Sada. E., Kito, S..KagakuKogaku, 36, 218 (1972). Sada, E., Kito. S.,Ito. Y.. J. Chem. Eng. Jpn., 7, 57 (1974). Sada. E., Kito. S., Oda. T., Ito, Y.. Chem. Eng. J., to be submitted, 1975. Shoor. S.K., Gubbins, K. E.. J. Phys. Chem., 73, 498 (1969). Smyth. C. P., "Dielectric Behavior and Structure," McGraw-Hill. New York, N.Y.. 1955. Taha, A. A., Grisby, R. D., Johnson, J. R., Christian, S.D., Affsprung. H. E., J. Chem. Educ., 43, 432 (1966). Tiepel, E. W., Gubbins, K. E., Can. J. Chem. Eng., 50, 361 (1972).
Receiued f o r review October 25, 1974 Accepted April 9, 1975
Diffusion in Ternary Ideal Gas Mixtures. 1. On the Solution of the Stefan-Maxwell Equation for Steady Diffusion in Thin Films Lewis E. Johns* and Anthony E. DeGance Department of Chemicai Engineering University o f Florida Gameswile Florida 326 7 7
The use of the Stefan-Maxwell equation for the prediction of steady transport across thin films is investigated for systems wherein the material flux ratios satisfy conditions of chemical stoichiometric origin. In part I we identify conditions under which the matrix 8-the infinitesimal generator of x(z)-exhibits complex eigenvalues. We show that certain stoichiometric conditions give rise to complex eigenvalues, we investigate two such conditions, and we show that even for films bounding highly active surfaces -systems under a strong diffusive limitation-the constant multicomponent diffusion coefficient approximation is qualitatively, and often quantitatively, accurate.
1. Introduction
Strongly exothermic, catalytic chemical reactions often operate under high temperature, diffusion-controlled conditions in which the chemical state of the catalyst surface is either close to a state of chemical equilibrium, for reversible reactions, or close to a boundary of the chemical state space on which a t least one reactant-the limiting reactant-vanishes, for irreversible reactions. For an equiaccessible planar surface fed via diffusion across a thin film separating the surface from a spatially uniform reservoir of infinite extent, Frank-Kamenetskii (1969) gives the rule that the limiting reactant for a single, diffusion-controlled irreversible reaction vlAl U Z A ~. . . = 0 is that for which (Di/ui)xi(O)takes on its minimum value for i ranging over the chemical reactants, i.e., for ui > 0. This result, however, is predicated on the use of the diffusion law ni = -cDiVx, which generates straight line diffusion paths in state space for steady diffusion across a thin film and which simplifies the prediction of the boundary point through which a diffusion path, originating a t x(O), escapes the state space. The foregoing procedure is useful for systems in which the chemically active components are dilute in a chemically
+
+
inert carrier. For other systems a rational investigation including the determination of the limiting component must be based on a more suitable diffusive flux law. It is our purpose, therefore, to investigate the solution of the StefanMaxwell equation for steady diffusion across a source free, isobaric, thin film which separates a spatially uniform reservoir of potentially reactive constituents from a planar catalytic surface. This equation is descriptive of ideal gas systems. In the context of the diffusive-convective decomposition of material fluxes, it predicts a fully general diffusive coupling via composition dependent multicomponent diffusion coefficients in systems of three or more components. Thus, we investigate one of a small class of systems wherein the composition dependence of the diffusion coefficients can be stated with certainty-deriving here from the differing molecular masses of the differing constituents. Because of the likelihood of severe temperature gradients in systems of interest, we assume that the temperature dependence of c:Oi, is the same for all i, J pairs, e.g., that ci),,/dT is independent of temperature. We let c v ! , be evaluated a t a reference temperature in what follows and interpret the spatial parameter designating the position in Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
237
