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Comments on ‘studies on the mechanism and kinetics of the oxidation of copper sulfide’
Chemical Engineering Science, 1968, Vol. 23, pp 95 1.
Comments
Pergamon Press.
on ‘studies on the mechanism
Printed in Great Britain
and kinetics of the oxidation of copper sulfide’
(Received 6 December 1968) Dear Sir, In a recent article[l], Ganguly and Mukherjee have shown that their data for the oxidation of copper sulfide with gaseous oxygen follow a rate law developed by Gamer[2] for the decomposition of solids. Garner’s development of this rate law is based on nucleation theory. In an effort to justify the use of an expression based on a nucleation model for their case, the authors have presented a derivation starting with a chemical reaction model. However, this attempt contains a number of mathematical errors which vitiate their derivation. The authors have started with a rate model for a gas-solid reaction which they present as Eq. (I),
justified since it presents a relation between t and ll different from that assumed in Eq. (1). Since the object of the original derivation is to develop the functionality between t and 0, it is not proper to use two different relationships. Actually, the second equation is based on the contracting sphere model for gas-sold reactions[4]. (3) The approximation, k’t k’t (l-e)~/~=l--;r__
Rod, Rod,’ presented in Eq. (4) is not valid since the left side of the approximation is always positive while the right side is always negative. (4) The integration of Eq. (5) to get Eq. (6) is invalid since the integration takes place with respect to 0 assuming t is constant. Actually, t is a function of 0. (As pointed out previously, the object of the derivation is to find an expression relating t and 0.) (5) The solution for the constant C in Eq. (6) is invalid for the reasons just stated above. Actually, when 0= 1, t = t,, the time required for total reaction. Although, as the authors have pointed out, their final result, Eq. (9), can be achieved starting with Gamer’s model, it cannot be arrived at by their procedure. Incidentally, the formulation of the kinetic equation for random nucleation with overlapping of nuclei has been given incorrectly in Table I as,
$ = k,tIP,,“‘A. In this expression, 0 is fraction of copper sulfide converted to copper oxide at time r, P,” is partial pressure of oxygen taken to the mth power, A is reactive surface area per unit volume of particle, and k, is the rate constant. The authors also have presented an expression for RI, the radius of the interface, in terms of 6 and Ro, the initial particle radius, RI2 = RO’(1 -,9)2’s
(2)
from which it follows immediately that A
=
-~RI’ = +( 4?TRo3 o
1 -@Z/3.
This result can be substituted into Eq. (1) and, assuming P, is constant, the resulting expression integrated to give 0 as a function oft. When this is done, one gets, V3( tan-l
-log( 1 -e)
l+2(l-e)“3_tan-‘l_ ~3
-In ~i++i1-_)~~3+(i-e)2~~=
= k,t3
which rearranges to
V/31 1_(l_e)w
3k,Phm(t,--t) R0
where t, is the time required for total reaction. It should be pointed out that the assumed model needs an initiating step (since it gives a zero rate when t = 0 and 8= 0) as recently discussed by Walles and Platt[3]. Unfortunately, the authors have not chosen to adopt this direct approach, resorting instead to some erroneous mathematics. The inconsistencies in their derivation may be summarized as follows: (1) 4rr is omitted from Eq. (3). (2) The use of the equation 1 - (1 - @II3 = k’t/R,,d, is not [l] [2] [3] [4] [5]
-log(l-e)“3= kt. The proper form as given by Gamer is
Finally, we remark in passing that the nucleation equations which have been used by the authors (derived from Garner) apply only for solid decomposition reactions. For gas-solid reactions of the type studied here, Delmon[S] has treated the theory of nucleation and interfacial reaction in considerable detail. H. G. MclLVRlED F. E. MASSOTH GulfResearch & Development Company P. 0. Drawer 2038 Pittsburgh, Pennsylvania 15230
REFERENCES GANGULY N. D. and MUKHERJEE S. K., Chem. Engng Sci. 1967 22 1091. GARNER W. E., Chemistry ofthe SolidState, p. 209. Butterworths 1955. WALLES W. E. and PLAIT A. E., Ind. Engng Chem. 1967 59 41. LEVENSPIEL O., Chemical Reaction Engineering, p. 350. Wiley 1962. DELMON B., Revue Inst. fr. P&role 1963 18 491; 1965 20 938: Bull. Sot. chim. Fr. 1966 1966 2677.
951
Comments
Pergamon Press.
on ‘studies on the mechanism
Printed in Great Britain
and kinetics of the oxidation of copper sulfide’
(Received 6 December 1968) Dear Sir, In a recent article[l], Ganguly and Mukherjee have shown that their data for the oxidation of copper sulfide with gaseous oxygen follow a rate law developed by Gamer[2] for the decomposition of solids. Garner’s development of this rate law is based on nucleation theory. In an effort to justify the use of an expression based on a nucleation model for their case, the authors have presented a derivation starting with a chemical reaction model. However, this attempt contains a number of mathematical errors which vitiate their derivation. The authors have started with a rate model for a gas-solid reaction which they present as Eq. (I),
justified since it presents a relation between t and ll different from that assumed in Eq. (1). Since the object of the original derivation is to develop the functionality between t and 0, it is not proper to use two different relationships. Actually, the second equation is based on the contracting sphere model for gas-sold reactions[4]. (3) The approximation, k’t k’t (l-e)~/~=l--;r__
Rod, Rod,’ presented in Eq. (4) is not valid since the left side of the approximation is always positive while the right side is always negative. (4) The integration of Eq. (5) to get Eq. (6) is invalid since the integration takes place with respect to 0 assuming t is constant. Actually, t is a function of 0. (As pointed out previously, the object of the derivation is to find an expression relating t and 0.) (5) The solution for the constant C in Eq. (6) is invalid for the reasons just stated above. Actually, when 0= 1, t = t,, the time required for total reaction. Although, as the authors have pointed out, their final result, Eq. (9), can be achieved starting with Gamer’s model, it cannot be arrived at by their procedure. Incidentally, the formulation of the kinetic equation for random nucleation with overlapping of nuclei has been given incorrectly in Table I as,
$ = k,tIP,,“‘A. In this expression, 0 is fraction of copper sulfide converted to copper oxide at time r, P,” is partial pressure of oxygen taken to the mth power, A is reactive surface area per unit volume of particle, and k, is the rate constant. The authors also have presented an expression for RI, the radius of the interface, in terms of 6 and Ro, the initial particle radius, RI2 = RO’(1 -,9)2’s
(2)
from which it follows immediately that A
=
-~RI’ = +( 4?TRo3 o
1 -@Z/3.
This result can be substituted into Eq. (1) and, assuming P, is constant, the resulting expression integrated to give 0 as a function oft. When this is done, one gets, V3( tan-l
-log( 1 -e)
l+2(l-e)“3_tan-‘l_ ~3
-In ~i++i1-_)~~3+(i-e)2~~=
= k,t3
which rearranges to
V/31 1_(l_e)w
3k,Phm(t,--t) R0
where t, is the time required for total reaction. It should be pointed out that the assumed model needs an initiating step (since it gives a zero rate when t = 0 and 8= 0) as recently discussed by Walles and Platt[3]. Unfortunately, the authors have not chosen to adopt this direct approach, resorting instead to some erroneous mathematics. The inconsistencies in their derivation may be summarized as follows: (1) 4rr is omitted from Eq. (3). (2) The use of the equation 1 - (1 - @II3 = k’t/R,,d, is not [l] [2] [3] [4] [5]
-log(l-e)“3= kt. The proper form as given by Gamer is
Finally, we remark in passing that the nucleation equations which have been used by the authors (derived from Garner) apply only for solid decomposition reactions. For gas-solid reactions of the type studied here, Delmon[S] has treated the theory of nucleation and interfacial reaction in considerable detail. H. G. MclLVRlED F. E. MASSOTH GulfResearch & Development Company P. 0. Drawer 2038 Pittsburgh, Pennsylvania 15230
REFERENCES GANGULY N. D. and MUKHERJEE S. K., Chem. Engng Sci. 1967 22 1091. GARNER W. E., Chemistry ofthe SolidState, p. 209. Butterworths 1955. WALLES W. E. and PLAIT A. E., Ind. Engng Chem. 1967 59 41. LEVENSPIEL O., Chemical Reaction Engineering, p. 350. Wiley 1962. DELMON B., Revue Inst. fr. P&role 1963 18 491; 1965 20 938: Bull. Sot. chim. Fr. 1966 1966 2677.
951