Kinetics of Endothermic Decomposition Reactions
The Journal of Physical Chemistry, Vol. 82,No. 2, 1978
further elucidate the detailed effects of the irradiation.
163
( 8 ) E. G. Prout and F. C. Tompkins, Trans. Faraday Soc., 42, 468 (1946). (9) M. Avrami, J . Chem. Phys., 9, 177 (1941); B. V. Erofeyev, C. R . Acad. Sci., USSR, 52, 511 (1946). (10) E. G. Prout, J . Inorg. Nucl. Chem., 7, 368 (1958). (11) E. G. Prout and P. J. Herley, J . Phys. Chem., 66, 961 (1962). (12) P. J. Herley and E. G. Prout, J . Inorg. Nucl. Chem., 16, 16 (1960). (13) E. G. Prout and P. J. Herley, J . fhys. Chem., 65, 208 (1961). (14) E. G. Prout and M. J. Sole, J . Inorg. Nucl. Chem., 9, 232 (1959). (15) E. G. Prout, personal communication. (16) P. J. Herley and P. W. Levy, J . Phys. Chem., 65, 208 (1961). (17) P. W. Levy and P. J. Herley, J , Phys. Chem., 75, 192 (1971). (18) P. W. M. Jacobs and F. C. Tompkins, “Chemistry of the Solid State”, W. E. Garner, Ed., Butterworths, London, 1955, Chapter VII. (19) F. C. Tompkins, “Solid State Chemistry”, Vol. 4, N. B. Hannay, Ed., Plenum Press, New York, N.Y., 1976, p 193.
References and Notes (1) W. E. Garner and E. W. Haycock, Proc. R. Soc. London, Ser. A , 211, 335 (1952). (2) V. I. Mikheeva, M. S. Selivokina, and 0. N. Kryukova, Dokl. Acad. Nauk. SSSR, 109, 541 (1956). (3) V. I. Mikheeva and S. M. Arkipov, Russ. J. Inorg. Chem., 12, 1066 (1967). (4) J. Block and A. P. Gray, Inorg. Chem., 4, 304 (1965). (5) S. Aronson and T. J. Salzano. Inoro. Chem.. 8. 1541 11969). (6j M. McCarty, Jr., J. N. Maycock, a n i V . R. Pai Verneker; J. Phys. Chem., 72, 4009 (1968). (7) P. J. Herley and P. W. Levy, J. Phys. Chem., 49, 1493, 1500 (1968).
Kinetics of Endothermic Decomposition Reactions. 2. Effects of the Solid and Gaseous Products Alan W. Searcy” and Darlo Berutot Materials and Molecular Research Division, Lawrence Berkeley Laboratory and Depa~mentof Materlals Science and Mineral Engineering, College of Engineering, University of California, Berkeley, California 94720 (Received February 25, 1977) Publication costs assisted by Lawrence Berkeley Laboratory
Decomposition rates are predicted to decrease linearly with increased pressure of the product gas if a chemical step for the gaseous component of the reaction is rate limiting, but to be a constant function of the difference between the reciprocals of the equilibrium decomposition pressure and the product gas pressure if a step for the solid reaction component is rate limiting. An equation which was derived by Sandry and Stevenson to describe the effect of tubular capillaries on the rates of vaporization from surfaces of known vaporization coefficients is adapted to describe the effect of porous solid barriers on the rates of decomposition reactions. The equations for the effect of barrier thickness take two forms, one form when a chemical step for the solid reaction component is slowest and another form when a step for the gaseous component is slowest. Equations are also described for use when both the product gas pressure and barrier thickness influence the reaction rate. The various rate equations are functions only of the activities of the reaction components, of measured rates in vacuo with negligible barriers, and of measurable transmission properties of the barriers. The theory is applied to interpret data for calcite decomposition.
Introduction Decomposition reactions, that is reactions which can be described by the general equation AB(so1id) = A(so1id)
+ B(gas)
(1)
where symbols A and B represent not only the two reaction products but also the two chemical components of a binary or pseudobinary system, have probably been studied as much as any class of heterogeneous reactions. Nonethe-less, in a 1974 monograph on solid state reactions Schmalzried commented, “There is no general theory of decomposition reactions”.l Recently, we provided the central elements of a general theory by identifying the four essential steps of a decomposition reaction, by deriving rate equations for six possible rate-limiting processes in vacuo, and by evaluating the effect of a metastable solid product on the rate equatiom2 Here we complete the theory for the postnucleation period of reaction by deriving expressions for the dependence of decomposition rates on pressures of product gases and on thickness of layers of porous solid products or alternately on the time periods during which +Permanent address: Istituto Tecnica, Facolt5 di Ingegnerik, Universita di Genova, Via Opera Pia 11, 16145 Genova, Italy. 0022-3654/78/2082-0163$01 .OO/Q
the product layers have grown. The resulting equations predict two distinct classes of dependence of rates on pressure and on product layer thickness. In the discussion section we will briefly illustrate for calcium carbonate some applications and tests of the theory.
Effect of Pressure of the Gaseous Product Several points about the conceptual basis of the model which is used in ref 2 and is extended in this paper should be emphasized. 1. The model assumes, in agreement with experimental observation for most endothermic decomposition reactions, that the solid product forms a porous layer on the reactant. This physical arrangement requires that, as the reaction front moves into a particle of the reactant AB, some volume elements of AB along the reaction front are replaced by the solid product phase A and other volume elements are replaced by pores. This process can only be accomplished if in addition to transfer of A across the reactant-solid product interface and of B into the vapor, there is also movement parallel to the reaction front of that fraction of A in volume elements of AB that are replaced by pores and of B in that fraction of AB that are replaced by the solid product phase. These two diffusional pro0 1978 American Chemical Society
164
A. W. Searcy and D. Beruto
The Journal of Physical Chemistry, Vol. 82, No. 2, 1978
The rate equations that correspond to the chemical processes 2-5 are respectively
Figure 1.
cesses as well as the interfacial and surface transfer steps are all necessary steps of steady state decompositions which yield a porous solid product. Previous treatments have neglected the diffusion steps. 2. The model is appropriate not only for reactions in which A and B of eq 1 represent elements, but also for reactions in pseudobinary systems. For example, the model can be applied to decomposition of Mg(OH)2with A interpreted as MgO and B as HzO. 3. The model need not, and does not, specify how the diffusion parallel to the reaction front occurs. The form of the rate equations does not depend on whether the principal path of diffusion of the gaseous reaction component is along the interface between the reactant and solid product phase or is inside one of those phases near the interface. When the reactant is ionic the process which is formally described by the model as diffusion of the gas probably actually consists of ionic movements which have the net effect of transport of the gaseous product from the interface between reactant and solid product to a pore. For example, the diffusion step for COz, with C 0 2 viewed as the gaseous component of the CaO-COZ binary system, may occur by countercurrent diffusion of C032- and 02ions. The fluxes would be coupled by the requirement of electroneutrality and the driving force for the diffusion of the coupled flux of ions would be the gradient in thermodynamic activity of GOzbetween the interface and the surface of a pore, as assumed below. Figure 1 is a schematic representation of the steps of decomposition. (a) A flux jBformed from the portion of chemical component B which is at an interface between the solid reactant phase AB and the solid product phase diffuses to the surface a t a pore. (b) A flux JBof component B transfers from the AB surface to the gas phase. (c) A flux jAformed from that portion of chemical component A which is at the AB surface fronted by a pore diffuses on or in the AB phase to a particle of the solid product phase A. (d) A flux JAtransfers from the AB phase across the interface to the solid product phase. Using the symbols i for interface, s for surface, p for solid product, and g for gas, these four reaction steps can be written in the order described as Bi 2 B, (2) B, t B, (3) A, 2 Ai A, 2 A,
(4)
(5)
where each k is the rate constant (or for diffusion, the composite rate constant) for the foreward direction of the reaction step indicated by the subscript and each k’ is a rate constant (or composite rate constant) for the reverse of one of the four steps. U B ~ ,for example, is the thermodynamic activity for component B at the interface between the reactant and the solid product, and uBsis the activity of B on the surface of the reactant a t the bottom of a pore. The diffusion flux j B is defined as the flux of B per unit area of that part of the reactant which forms an interface with the solid product and j A is defined as the diffusion flux of A per unit area of that part of the reactant which is fronted by pores, while J A and JBare fluxes per unit of total reactant surface plus interfacial area. It should be noted that these definitions are contrived to make all four fluxes numerically equal during steady state decomposition even though the numbers of A and B particles that must diffuse are not equal to each other or to the number of A and B particles that undergo reaction. The rate constant for the forward direction of an elementary reaction divided by the rate constant for the reverse direction equals the equilibrium constant for any fixed set of experimental conditions3 even if the reaction is far from equilibrium and reactants and/or products have activity coefficients that differ from those found at equilibrium: For reactions 2-5, the equilibrium constants are all unity, and 3 and 5 are elementary reactions so k3 equals ki and k5 equals hi. Reactions 2 and 4 are diffusion reactions which require a sequence of steps which may not have the same activation free energy barriers at all points along the diffusion path. However, for decomposition reactions the diffusion paths are short, and we will use the reasonable assumptions kz = ki and k4 = kq/. Suppose first that there is no significant pressure gradient of B along the pores through the product and that diffusion of component B in or on the reactant is slower than any of the other three necessary steps of the reaction. Then all steps of the overall reaction except step 6 have nearly equal forward and reverse fluxes, so that (1) component A can be expected to be essentially the same activity on both sides of the reactant-solid product interface, that is uAp = u A ~ ,and (2) when the activity of B in the gas phase is made as high or higher than the activity of B in the surface in vacuo, the surface activity becomes almost the same as that of the gas, that is a B s c~B~. Integral free energies of formation of the binary phases which undergo decomposition reactions usually change only negligible amounts with changes in composition of the phase, but the activities of the components can change by many orders of m a g n i t ~ d e . It ~ ,is ~ convenient to assign unit activities to the solid reaction product when it is formed in its thermodynamically stable form and to the gaseous product, not when its pressure is 1atm, but instead when its pressure has the equilibrium value for the decomposition of AB(so1id) to the stable form of A(so1id). With these definitions of activities, the product of the activities in a local region of the AB phase, for example, at the interface, is close to unity. Thus, C ~ A ~ U B ~1, and U B ~= l / a A i = l/aA,. Substitution of a B i = l / a A , and C ~ B ~ U B into ~ eq 6 gives for the net flux of B or A
Kinetics of Endothermic Decomposition Reactions
The Journal of Physical Chemistry, Vol. 82, No. 2, 1978
C
J
-=
If AB is a t equilibrium with the stable form of solid A during decomposition, uAp = 1 and eq 10 reduces to =
(k2/PBeq)
(PBeq - P B g
)
(10’)
where PBeq is the equilibrium decomposition pressure for reaction 1. If AB is at equilibrium with the metastable form of solid A during decomposition, uAp > 1 and a ~ = i l/aAp < 1. Then J=
(k2/PBeq)(PBm
- PBg)
(1or,)
where P B m , the equilibrium decomposition pressure for formation of metastable solid A, is less than PBeq, Equation 10”predicts that if diffusion of B is rate limiting and a metastable form of solid A is produced, the rate of decomposition will be reduced to zero by a pressure PB,,, of the product gas which is lower than the equilibrium decomposition pressure. In practice, a decomposition reaction which is governed by eq 10” a t low pressures of background gas is likely at higher pressures to yield the stable solid product and follow eq 10’ because the net flux for eq 10’’ reaches zero when PBg= P B m while the net flux for eq 10’ is finite until PBg reaches P B>~P B ~ ~. If a surface or desorption step of B is the slowest of the four steps necessary to the overall reaction, a similar line of argument leads to uBs u B ~N l/aAi = l/aA, and the net flux of B is given by J = h3[ ( I / a A p ) - aBg1
(11)
Equation 11 can be expressed in alternate forms that differ from eq 10’ and 10” only in that k3 replaces kz. If the rate constant k5 for the interfacial transfer step for component A is smaller than the rate constants for both of the steps for component B, a very different dependence on the pressure of B is predicted. Let k B equal kzk3/(kz + k3),7which reduces to k2 if k3 >> kz or h3 if h3
