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Application of this effect toward explanation of some catalytic phenomena is readily apparent. Consider, as an example, an adsorbed gas with a total moment of 1 D = 3.33 X C cm at 27” and in a surface field of 9 X lo7 V/cm. (pD/KT) = 7.24, and Ql(D) = Ql(0)(sinh 7.24/7.24) = 96.4Q1(0), corresponding to a change in entropy of -3.32 cal/(mol deg) for the nonlocalized model, - 1.38 cal/(mol deg) for the localized model, and a change of two orders of magnitude in the partition function. This leads to an increase of from lo2 to lo4in the value calculated for the rate constant, provided that the dipole moment of the activated complex is also 1 D and depending on the assumptions made concerning the rotational mobility of the activated complex. Because of residual, though weakened, attractive forces between the surface and the original reacting dipoles, the activated complex will be restricted in its rotational freedom to a substantial degree, and the values expected for the rate constant should be toward the higher end of this range. It may be noted that attempts to apply absolute rate theory to surface react i o n ~have ~ been fairly successful in several cases, but that the rate constants calculated for some, primarily
polar, molecules have proved to be lower than the experimental rate constants by several orders of magnitude.l0I1l It is possible that this discrepancy could be accounted for by making a correction for the effect postulated here. It should be emphasized that the model used for this calculation is grossly oversimplified with respect to the actual situation in heterogeneous catalysis. Further, the magnitude of the surface field D can only be determined indirectly and approximately. This or some equivalent effect should exist to a significant degree, however, in the case of surface reactions of polar molecules. Acknowledgments. I am indebted to Dr. V. I(.Wong of the Department of Physics and to many of my colleagues in the Department of Chemistry at The University of Michigan for their assistance in the formulation of this work.
DIFFUSION-LIMITED LIFETIMEOF
A
(9) K. J. Laidler in “Catalysis,” P. H. Emmett, Ed., Reinhold, New York, N. Y., 1954, Chapter 5. (10) A. B. J. Robertson, J . Colloid Sci., 11, 308 (1956). (11) K. J. Laidler, “Chemical Kinetics,” McGraw-Hill, New York, N. Y., 1965, p 292.
Effect of Rapid Homogeneous Reaction on the Diffusion-Limited
Lifetime of a Soluble Sphere of Arbitrary Density’ by Daniel E. Rosner2 Department of Engineering and Applied Science, Yale University, NewHuven, Connecticut (Received November 16, 1970)
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Publication coats assisted by the U.S . Air Force Ofice of Scientific Research
The time required to completely dissolve a stationary sphere can be shortened by deliberate addition of a reactive solute to the solvent. It is shown that if the homogeneous reaction is rapid and the reactants have nearly equal diffusion coefficients with respect to the solvent, then the shortened sphere lifetime is readily obtained analytically from the corresponding sphere lifetime in the absence of homogeneous chemical reaction. This interrelation, valid for spheres of arbitrary density (dense particles, droplets, or bubbles), follows from a fully transient analysis of the resulting nonlinear boundary value problem for constant physical properties. Results are shown to be related to, but more general than, previously obtained lifetime ratios based on quasisteady and/or sparing solubility approximations. Quantitativecriteria ale derived to define when the “rapid” reaction condition is satisfied for any particular finite second-order reaction rate constant, and illustrative calculations are included to demonstrate the computational algorithm and its expected domain of validity.
I. Introduction I(nowledge of the total time required to dissolve a motionless sphere (gas bubble, droplet, or particle) in a liquid solvent is necessary in the design of mass transfer equipment and in understanding geophysical
and even pharmaceutical phenomena. I n ref 3 and 4 the author showed that useful correlation formulas
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The Journal of Physical Chemistry, Vol. 76, No. 19, 2971
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DANIEL E. ROSNER
2970 and lifetime predictions can be obtained for systems in which interfacial solute detachment kinetics and/or long-range solute diffusion limit the rate of dis~olution,~ regardless of the magnitude of the mutual solubility or solute/solvent density ratio. While these and previous6-8 authors assumed that no further chemical reaction occurs in the solvent enveloping the sphere, it is well known from work in related fields’-“ that homogeneous chemical reactions (acting as a local solute sink) can accelerate diffusion-limited mass transfer rates by steepening the concentration gradient of the dissolving substance in the vicinity of the solute/solvent interface. This raises the important but hitherto unanswered question of how much the total dissolution lifetime tlif. of a sphere a can be shortened by taking advantage of homogeneous chemical reaction with a reactant od deliberately added to the surrounding solvent.15 The purposes of this paper are to (i) show, mathematically, that for the commonly encountered case of nearly equal Fick diffusion coefficients ( D A DB = D)the effect on h i r e of rapid irreversible homogeneous reaction between a and od can be predicted exactly in terms of results for the dimensionless nonreactive (“physical” dissolution) lifetime i m recently given in ref 4 and 5 , (ii) present the graphical correlation and construction illustrating the resulting computational algorithm, (iii) briefly discuss the relation between this dissolution problem and analogous problems involving the reactive augmentation of mass transfer rates in the fields of metallurgy,” chemical engineeringg.l0 and omb bust ion,^^-'^ and (iv) clarify the simultaneous conditions required to adequately satisfy the “rapid” homogeneous reaction assumption.
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11. Physicochemical Model
With the exception that we now allow homogeneous chemical reaction in solution, our continuum model and nomenclature are identical with those of ref 3 and 4, viz. we consider an isolated sphere of pure solute a, having an initial radius Ro dissolving into an isothermal, constant property, otherwise quiescent solvent containing reactant od but (initially) no a (see Figure 1 and the Appendix). The only fluid motion considered is the spherically symmetric ra&iZ motion induced by the interfacial mass transfer process itself (when the solvent and solute densities are not equal). In the fluid, components (“L and CE react irreversibly in accord with the effective stoichiometry (“L
+ dod -+
soluble products
(1)
The reaction is assumed to proceed rapidly enough so that both CB and bes/br a t the sphere surface (T = R(t))are negligible (see section IV-D below), although, as discussed in section IV, it is not really necessary that the reaction be “instantaneous” ( i e . , confined to a sharp reaction “front” of thickness negligible compared to 8).Solution properties, including density and temThe Journal of Physkzl Chemistry, Vol. 76,No. 10, 1071
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I
i
Figure 1. Configuration and notation, diffusion-controlled dissolution of an isolated sphere in t h e presence of rapid homogeneous chemical reaction.
perature, are considered constant, and the presence of reaction products in solution is assumed to have a negligible effect on the equilibrium (saturation) mass fraction CA,**&,which therefore remains constant during the dissolution process. Both components are assumed to diffuse in accord with a Fick-type flux law, with the resulting effective diffusion coefficients DA, DB being approximately equaP to some common value, hereafter written as D. At time t = 0 the sphere of pure a is confronted with an unbounded constant density solution of (2) Associate Professor, Chemical Engineering broup. (3) D. E. Rosner. J. Phys. Chcm., 73, 382 (1969). (4)D. E. Rosner and M. Epatein, iW., 74,4001 (1970). (5) In the latter case, accurate numerical lifetime predictions (using finite differenoe methods) have just baen reported by D. L. Duda and J. S. Vrentas [Int. I.X e d Moas T~ansfei. 14, 395 (1971)l. To convert their notation into that used in the present work and ref 3 and 4. simply make the replacements N. ( p / p ~ .B. ) Nb B. (6) P. S. Epstein and M. S. Plesset, J. Chem. Phys.. 18, 1505 (1950). (7) M.Cable and D. J. Evans. J. Appl. Phys., 38, 2899 (1967). (8) See, e a , (a) E . A. Moelwyn-Hughes. “Physical Chemistry,” 2nd rev. ed. Pergamon Press, Oxford. 1964; (b) E. A. MoelwynHughes, “The Kinetics of Reactions in Solutions.” 2nd ed, Oxford University Press. London, 1947,p 374 ff. (9) (a) H. . I . Den Hartog and W . J. Beek, Appl. Sn’. Res., 19, 311 (1988); (b) S. K. Friedlander and M. Litt. > 1. Moreover, in the latter regime a logarithmic scale for T l i f a (Figure 2) would be preferable, or a more suitable lifetime variable should be adopted for the "bubble limit."
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SOLUBLE SPHERE
the principal results [here given by eq 11-13] outside of their expected domain of validity. It is interesting to observe that while we are dealing with an important effect of homogeneous chemical reaction on a rate (phase change) process, our results are independent of chemical kinetic parameters. However, in obtaining eq 11-13 we have assumed that the homogeneous reaction is sufficiently rapid to prevent @ from “penetrating” to the sphere surface over most of the sphere’s lifetime. More explicitly, the analysis of section I11 requires satisfaction of the following three inequalities over most of the time interval t l i f e
Table I: Estimated Physical Dissolution Lifetimes for a Stationary Benzoic Acid Sphere in 25’ Water Initial
radius, RQ,ern 100 10-1 10-2 10-3
10-4
Lifetime, tolife, SEC
1.9 1.9 1.9 1.9 1.9
x 107 x 106 x 108 x 10’ x IO-’
that for any particular solute-solvent system the required inequalities can be satisfied, provided (i) the relative concentration of @ (and, hence, FRXNand t1ire-l) is not too large, and (ii) the initial sphere size Ro (and, hence, hire)is not too small. C . Relation to Droplet Combustion Theory. While the problems of fuel droplet vaporization and “envelope flame” droplet combustion are usually discussed in terms of energy transport limitation^,'^-^^ there is a close similarity between these problems and the isothermal dissolution problem treated herein and in ref 4. Indeed, using a quasi-steady model applicable when the droplet density far exceeds that of the surrounding vapor, SpaldingI2 and others have shown that the ratio of the “combustion time” to the (physical) “vaporization time” is expressible in terms of a In (1 B ) ratio, where combustion merely modifies the effective mass transfer driving force parameter B. We now observe that, in the limiting conditions considered by Spalding, T l i f e = 1 (cf. Figure 2) since, as pointed ~ 0. out in ref 3 and 4, regardless of B, 7 l i f e 4 1 for p / p -t Our present work would therefore indicate that when the droplet and vapor densities became comparable (as they do when the prevailing pressure level becomes comparable to the thermodynamic critical pressure of the combustion should have a greater effect on the droplet lifetime than that expected from a simple In (1 B ) ratioz3since 7 1 i f e < 1 for systems with appreciable ( p / p ~ ) B . D. Instantaneous Location of the Reaction Zone and Validity of the Rapid Homogeneous Reaction Assumption. While similar mathematical and physicochemical approximations have been used in treating simpler mass transfer problems in the literature, unfortunately, i’ew authors have discussed the limitations of the rapid homogeneous reaction model in any detail, much less offer convenient quantitative criteria for the expected domain of validity. Yet, despite the complexity of this highly nonlinear boundary value problem, a combination of physical reasoning and relevant results derived from simpler configurations can be invoked, as done below, to derive sufficient conditions for the validity of the present mathematical and physicochemical approximations. As in all problems, it is prudent to keep such conditions in mind to avoid applying
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