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Kinetics of Endothermic Decomposition Reactions (13) N. Bloembergenand L. 0. Morgan, J. Chem. Phys., 34, 842 (1961). (14) E. V. Goldammer and M. D. Zeldler, Ber. Bunsenges. Phys. Chem., 73, 4 (1969). (15) M. Gruner, Thesis, Karlsruhe, 1969. (16) E. v. Goldammer and H. G. Hertz, J. Phys. Chem., 74, 3734 (1970). (17) H. Versmold, Ber. Bunsenges. Phys. Chem., 78, 1318 (1974). (18) R . 0. W. Howarth, Chem. Commun., 286 (1974). (19) A. Abragam, "The Principles of Nuclear Magnetism", Oxford University Press, London, 1961. (20) H. G. Hertz, Prog. Nucl. Magn. Reson. Spectrosc., 3, 159 (1967). (21) H. G. Hertz, Wafer: Compr. Treatise 1973, 3,301 (1973). (22) H. G. Hertz, Ber. Bunsenges. Phys. Chem., 71, 1008 (1967). (23) D. E. Woessner, J. Chem. Phys., 36, l(1962). (24) D. E. Woessner, J. Chem. Phys., 37, 647 (1962). (25) D. E. Woessner, 8 . $. Snowden, and G. H. Meyer. J. Chem. Phys., 58, 719 (1989). (26) D. Wallach, J. Chem. Phys., 47, 8258 (1967). (27) H. Versmold, 2.Naturforsch. A, 25, 367 (1970). (28) A. R. Edmonds, "Angular Momentum In Quantum Mechanics", German translation, Blbliographlsches Instltut, Mannheim, 1964. (29) M. E. Rose, "Elementary Theory of Angular Momentum", Wiley, New York. N.Y., 1957. (30) H. Versmold, J. Chem. Phys., 58, 5649 (1973). (31) M. Gruner and H. G. Hertz, Adv. Mol. Relaxation Processes, 3, 75 (1972). (32) H. G. Hertz and M. D. Zeidler, "The Hydrogen Bond. Recent Development in Theory and Experiments", North-Holland Publishing Co., Amsterdam, 1975.
(33) J. R. Lyerla. Jr., and D. M. Grant, Phys. Chem., Ser. One, 1972-1973, 4, 155 (1972). (34) R. Goller, H. G. Hertz, and R. Tutsch, Pure Appl. Chem., 32, 149 (1972). (35) D. R. Herschbach, J. Chem. Phys.. 25,358 (1956). (36) H. G. Hertz and R. Tutsch, to be submitted for publication. (37) R. Sperling and H:,Pfeifer, Z.Nafurforsch. A, 19, 1342 (1964). (38) E. A. C. Lucken, Nuclear Quadrupole Coupling Constants", Academic Press, London, 1989. (39) R. E. Verral, Wafer: Compr. Treatise 1973, 3, 21 1 (1973). (40) D. E. Irish in "Structure of Water and Aqueous Solutions", Verlag Chemle, Weinhelm, 1974. (41) H. Hartmann and E. Muller, Z.Naturforsch. A, 18, 1024 (1963). (42) T. Chiba, J. Chem. Phys., 41, 1352 (1964). (43) R. M. Hammaker and R. M. Clegg, J. Mol. Specfrosc, 22, 109 (1967). (44) J. A. Pople, W. G. Schneider, and H. J. Bernstein, High-Resolution Nuclear Magnetic Resonance", New York, N.Y., 1959. (45) J. R. Homes, D. Klvelson, and W. C. Drinkard, J. Am. Chem. SOC.,84, 4677 (1962). (46) C. Franconl, C. Dejak, and F. Conti, "Nuclear Magnetic Resonance In Chemistry", Academic Press, New York, N.Y., 1965, p 363. (47) 2. Kecki, Rocz. Chem., Ann. SOC.Chim. Polonium, 44, 213 (1970). (48) Here and In the following second coordination sphere corrections are omitted, because such a neglect never effects the qualitative conclusions to be drawn. (49) H. Langer, private communication. (50) This result was obtained from the proton-Na+ intermolecular relaxation rate. A corresponding experiment can also be performed in MeOH. (D. S.Gill, H. G. Hertz, and R. Tutsch, to be submitted for publication.)
Kinetics of Endothermic Decomposition Reactions. 1. Steady-State Chemical Steps Alan W. Searcy* and Dario Beruto Inorganic Materials Research Dlvlsion, Lawrence BerkeleyLaboratory and Department of Materials Science and Engineering, College of Engineering, Univers/fy of Callfornia, Berkeley, California 94720 and lstltuto di Technologle, FacoltA di lngegneria. Universlta di Genova, Italy (Received October 3 1, 1974; RevlsedManuscript ReceivedNovember 14, 1975) Publication costs asslstsdby the US.Energy Research and Development Administratlon
When the solid product of an endothermic decomposition reaction is porous, the rate-limiting chemical step is usually assumed to be a surface step of the gaseous product or of a precursor of that product. It is shown here that the rate of such a reaction may also depend upon (a) rates of diffusion in the reactant phase, (b) the rate of transfer of the solid reaction product at the reactant-product interface, and/or (c) the thermodynamic stability of the solid product. Rate equations are derived for the six possible limiting cases when either a single step or a coupled pair of steps of a decomposition reaction significantly influence its rate. Data for calcite (CaC03) decomposition are shown to be most simply explained as reflecting formation of a known metastable modification of calcium oxide with near equilibrium conditions maintained for each reaction step except desorption of carbon dioxide. If this explanation is correct the free energy of formation of the metastable oxide from the stable oxide should be found to be about +7500 - 5T cal/mol.
I. Introduction Reactions in which a solid reactant yields a new solid phase plus a gaseous product, that is, reactions which can be described by the general equation AB(so1id) = A(so1id)
+ B(gas)
(1)
are usually called decomposition reactions.* When the solid decomposition product is porous, -the rate-limiting chemical step is usually assumed to be a surface step of the eventual gaseous reaction p r ~ d u c t . ~ - ~ This assumption may often be wrong. During steady-
* Address correspondence t o this author the University fornia.
of Cali-
state decomposition, such as characterizes calcium carbonate6 and barium sulfate6 single crystals heated under vacuum, the solid reactant is converted at a constant rate to the solid product plus pores through the solid product. Decomposition at a constant rate is only possible if four different steps occur a t the same rate (see Figure 1).(a) A flux jg formed from that portion of chemical component B that is at an interface between the solid reactant phase AB and the solid product phase must undergo solid state diffusion to the surface a t a pore. (b) A flux J B of component B must transfer from the AB surface to the gas phase. (c) A flux j A formed from that portion of chemical component A that is at the AB surface fronted by a pore must undergo diffusion on or in the AB phase to a particle of the solid product The Journal of PhysicalChemistry, Vol. 80, No. 4, 1976
426
Alan W. Searcy and Dario Beruto (1 + ~ ) A ( s )+ (1 - 6)B(g) = Al+6Bi-a(S) (6) The integral free energy change in reaction 6, AG,, can be written as a function of the partial molar free energy changes of the two components, and the partial free energy changes can be expressed in terms of activity or fugacity changed0
AGs = (1 4- 6)RT In a A
+ (1- 6)RT In fb
(7) where the activity aA and fugacity fB are those for components A and B in the AB phase of the composition described in reaction 6. For phases of narrow composition limits, the integral free energies of formation, unlike the partial free energies of formation, usually vary by negligible amounts with composition. Therefore, -AG, is for practical purposes equal to AGIO, the standard free energy change for reaction 1,independent of composition. With this substitution and neglecting the small 6, eq 7 yields
-
Figure 1. Schematic cross-sectional drawing of the spatial relatlons for the steady state reaction AB(solid) A(solid) B(gas).jB is a diffusionalflux of chemical component B from interfaces between solid AB phase and the solid product phase to the AB surface fronted by pores, and j A is the oppositely directed diffusional flux of component A. JA is the flux of A across the interface between AB and the solid product phase, and JB is the flux of B from the AB surface to the gas phase.
+
phase A. (d) A flux JAmust transfer from the AB phase across the interface to the solid product phase. Using the symbols i for interface, s for surface, and g for gas, these four reaction steps can be written in the order described as Bi F= B,
(2)
B,
(3)
Ai
(4)
B*
A,
+
Ai e A,
(5)
The central purpose of this paper is to derive rate equations which describe the kinetics of steady-state decomposition reactions under vacuum when any of the four steps (2-5) is slow enough to influence the rate. Application of the analysis is illustrated in the Discussion section with data for calcite.6
11. Thermodynamic Considerations Darken7 showed that rate equations for highly nonideal systems are better expressed in terms of activity gradients than of concentration gradients. An important advantage will be gained by writing the rate equations of the four coupled steps of a decomposition reaction in terms of activity gradients. The activities of one of the two chemical components of a binary (or pseudobinary) phase can be calculated from the integral free energy of the phase and a measured activity of the other component even though the variations of composition that produce the changes in activity are too small to m e a ~ u r e .It ~ ?is,~ for example, thermodynamically meaningful to calculate the activity of calcium oxide, viewed as one component of a system, in a calcium carbonate phase from the activity of the other component, carbon dioxide, at pressures different from the dissociation pressure. Consider the reaction between a solid and a gas in their standard states to form a solid of a particular composition Ai+aBi-s Tho Journal of Physlcal Chemistry, Val. 80, No. 4, 1976
aAPB = exp(-AGlo/RT)
(8)
in the experimental range where the partial pressure of B can be substituted for its fugacity. For the purposes of this paper it is convenient to define the activity of B for any particular composition of the AB phase, not in the usual way, but as the ratio of the partial pressure of B for that composition, PB, to the equilibrium decomposition pressure PB(d) for each temperature. This definition combined with eq 8 yields aAaB = 1 for the AB phase at any composition when it is at internal equilibrium. A decomposition reaction can proceed at a finite rate only if the AB phase is supersaturated with respect to component A so that the activity of component A in the AB phase is greater than 1. For supersaturated solutions, provided that internal equilibrium is maintained, the relationship aAaB = 1 should be essentially as good an approximation as it is for the thermodynamically stable composition range because the phase boundary does not reflect any discontinuity in properties of the AB phase but only the coincidence in activities of ,the chemical components in phase AB and in solid phase A. However during decomposition, local equilibrium may not be maintained if one of the chemical components is much less mobile than the other. The less mobile component may not be able to rearrange locally under the steady-state reaction conditions to produce the atomic coordinations and distances characteristic of the equilibrium phase of the particular local composition. If local equilibrium is not maintained the local products aAaB at surfaces and at interfaces (which will be called K,and Ki) will be greater than unity. Decomposition reactions often yield as the direct solid product a metastable crystal modification or an amorphous form of the ~ o l i d In . ~ either ~ ~ event, the activity of the product, which can be called a A p , is not unity but aAp = exp(AG,/RT)
(9)
where AGP is the positive free energy of formation of the metastable form of solid A from the stable form. When the interphase transfer of component A (eq 5) is a near equilibrium process, the activity of component A on the AB side of the interface, which can be called aAi, approaches as a limit U A ~ Then, . if local equilibrium is attained in the AB phase at its interface with the solid product so that Ki = 1
427
Kinetics of Endothermic Decomposition Reactions where aBi is the activity of B on the reactant side of the interface. Equation 10 shows that, when the solid reaction product is metastable, the maximum activity that can be attained by component B, regardless of reaction mechanism, is not unity but exp(-AG,/RT), which has a value less than unity. 111. T h e Rate Equations for Steady-State
Decomposition The average diffusion distances that must be traversed by particular atoms or molecules are functions of their initial positions relative to the advancing pores and particles of the solid reaction product. For example, an A atom or molecule originally located in a volume element of the AB phase that is swept through by a growing particle of the solid reaction product need not diffuse at all, and the minimum distance over which an A atom or molecule that is originally under a pore must diffuse is its distance to the boundary between the pore and the solid product phase. Suppose that the flux of component A that must diffuse per unit area of that part of the surface of AB which is fronted by pores is j A . That flux is jA
=
n
(dnaAsn - dn’aAin)
jB
= k z a ~i kz’a~,
(12)
=ksa~,
(13)
= k 4 a ~, k4‘a~i
(14)
JB
jA
JA = ksa.t,i - k5’aAp (15) There are important restrictions on the steady-state reaction: JA= J B jA
=j B
(16) (17)
(18)
aAsaBs = K , 1 1
(19)
Equation 16 expresses the restriction that the total fluxes of A and B must be equal during steady-state decomposition. Equation 17 is a similar restriction on the steadystate diffusion fluxes in or on the AB phase. Furthermore, since j B is the flux that diffuses per unit area of AB-solid product interface and j A is the flux per unit area of AB fronted by pores, j B = J B numerically and j A = J A numerically. The four restrictions on the steady-state system can be used to obtain a general solution in which all the activities other than the activity of the product phase have been e1iminated.l’ Here we derive rate equations for the six limiting cases possible for these four steps. Setting eq 12 equal to eq 13
k z a ~i k 2 ’ 0 ~= ~k 3 a ~ ~ so that
aBs = kza~i/(k2’+ k3) and from (13)
JB= k2k3asi/(kzr + k3)
(11)
where, for example, dn is the rate constant for movement in the forward direction over one of the characteristic steadystate paths, d,’ is the rate constant for the reverse direction over the same path, aAsn is the activity of component A at the particular point of the AB phase surface a t which the nth diffusion path is initiated, and aAin is the activity of A a t the point in the interface between the solid reactant and solid product at which the nth path is terminated. To simplify eq 11, it will be assumed that each activity of the kinds aAsn and aAin can be replaced by average activities at the surface, aA,, and interface, aAi, respectively. The summation E d , can be called k4 and the summation Zd,’ can be called k4’. Then the diffusion flux of component A is j A = k r a ~, k4‘a~i.Similarly, if j g is defined as the flux of component B that must diffuse per unit area of interface between the solid product phase A and the reactant then j g = kzasi - k z ’ a ~ where , k2 and k2’ are similar summations for diffusion reaction 2. The flux in the surface step for component B (reaction 3 ) can be written J B = k a a ~ where , J B is the total flux of B leaving each unit area of AB phase. The net flux of component A that leaves each unit area of the AB phase is J A = k5aai - k ~ ’ a where ~, k5 and k5‘ are the forward and reverse rate constants for step 5. The steady-state decomposition of AB is thus characterized by four interdependent rate equations, which are for steps 2-5 of the overall reaction respectively:
aAiaBi = Ki 2 1
(20)
For an elementary (single step) reaction the rate constant at any particular composition for the foreward direction divided by the rate constant at the same composition for the reverse direction equals the equilibrium constant, even if the rate constants are functions of composition.12 The equilibrium constants for steps 2 and 4 are both unity. However, as explained in the discussion that follows eq 11, the constants k2, k4, and k4’ are not rate constants for elementary reactions, so we do not know that k2 = k2‘ or that kd = k4‘. We expect each k‘ to be of the same magnitude as the corresponding k and assume the equalities to simplify eq 20. When component A in the reactant phase is a t equilibrium with both component B and with the solid product phase A, aBi = l/aA, and eq 20 yields two limiting solutions. For k2 = k2’ >> k3, that is when the rate constant for diffusion of component B is large relative to its rate constant for desorption
J B = k 3 / a ~ =, k3 exp(-AG,lRT)
(21)
When k2’ > k3, or JB = (k2k&i)1/2 (24) when k2’
