The Journal of Physical Chemistry, Vol. 82, No. 19, 1978
Communications to the Editor
2141
COMMUNICATIONS TO THE EDITOR 100 + X(C - 1) 100
Comment: Parabolic Klnetics of a Spherical Particle Publication costs assisted by CSIRO
Sir: Christie e t a1.l purport to have modified the Ginstling-Brounshtein equation (GBE) for the parabolic reaction kinetics of a spherical particle in order to account for a product coating which occupies a greater volume than the solid reactant from which it was formed. If F is fraction reacted, v the ratio of excess product volume to / ~ p, = (1- F)ll3then reactant volume, a = [(l F ) v ] ~and their eq 9 becomes (a- p ) 2 - 2(a - /3)3/3 = k’t (1)
+
However this equation does not reduce to the GBE and t reaches a maximum for a - p = 1. For their example, u = 1.357, this maximum occurs at F = 0.944 and thereafter the rate is negative. The correct equation has been derived by Valensi2 and Taplin’s v e r ~ i o nsupposes ~?~ that product formed a t the inner surface of the coating might differ from product formed at the outer surface and allows different diffusion coefficients in the inner and outer parts of the coating [ l - (1- F)2/3- 2F/3]/Di + [l- (1 + vF)’l3 + 2vF/3]/DOv = h’t
If, as the reader might expect, the experimental errors lie in F rather than t , then the comparison of correlation coefficients for calculated time vs. experimental time for different equations, as used by Christie et al., does not necessarily find the equation of best fit. For equations as nonlinear as these it is usually necessary to use nonlinear regression or a t least weighted linear regression. (The weighting factor for GBE varies by a factor of 1000 between F = 0.1 and F = 0.9.) References and Notes (1) J. R. Christie, A. J. Darnell, and D. F. Dustin, J. Phys. Chem., 82, 33 (1978). (2) M. G. Valensi, C. R. Acad. Sci. (Paris), 202, 309 (1936). (3) R. E. Carter, J . Chem. Phys., 34, 2010 (1981). (4) J. H. Taplin, Natl. Bur. Stand. (U.S.),Monogr., No. 43,263 (1982). (5) J. H. Taplin, J. Chem. Phys., 59, 194 (1973). Division of Mineralogy, CSIRO Private Bag, P.O. Wembley Western Australla, 60 14
John H. Taplln
Received Aprll 79, 1978
Response to “Comment: Parabolic Kinetics of a Spherical Partlcle”, by J. H. Taplln Publicatlon costs asslsted by Rockwell International
Sir: Equations 6 and 9 in our paper (Christie et a1.l) are in error; the incorrect fraction (100 + X)(C - 1) 100
should be
The correct eq 9 should be
2/3[
(
100 + 100 X(C - 1 ) ) 1 / 3 - (100100 - X)’/3])
This equation properly gives a value of zero to the left side at t = 0. Also, for C = 1, Le., where solid product volume equals solid reactant volume, this correct form of the equation reduces to the Ginstling-Brounshtein equation, eq 8 in the Christie et al. paper. As the experimental kinetic data were treated using the erroneous eq 9, the derived values, including Figures 3 and 5, are in error. The correct values of the rate constants, k (Table I, column 5), differ from those listed but are of the same order of magnitude. Also, it should be noted that the units for the rate constant should be pm2 h-l. The derived correlation coefficients (Table 11) are likewise in error. Taplin suggests use of his equation which takes into account variations in the diffusion coefficients in the coating. As the solid product found by Christie et al. in all cases was P-NaA102,with no evidence of solid solution formation, the assumption was made that diffusion coefficients were constant. The formation of P-NaA102 from A1203and Na2C03requires the net diffusion of some or AP+ along with neutral species (e.g., Na+ along with 02-, 02-,or both) for the reaction to proceed. In the absence of additional evidence, a detailed mechanism for the diffusion processes cannot be put forth. One may still conclude that the reaction rate is controlled by diffusion of some species with its diffusion coefficient being the pertinent one. Elucidation of the actual processes occurring through detailed study utilizing techniques not employed in this work would be of value. A paper by us on the reaction of molten sodium carbonate with ferric oxide is in preparation. It will be submitted for publication shortly. It is the authors’ intention to publish corrected values for the sodium carbonate-aluminum oxide reaction as part of that paper.
References and Notes (1) J. R. Christie, A. J. Darnell, and D. F. Dustin, J . Phys. Chem., 82, 33 (1978). Environmental Chemistry Atomics International Division Rock well International Canoga Park, California 9 1304 Recelved June 26, 1978
0022-3654/78/2082-2141$01 .OO/O
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0 1978 American Chemical Society
J. R. Christie A. J. Darnell D. F. Dustin H. L. Recht‘
