ABSOLUTE RATES OF SOLID REACTIONS: DIFFUSION ALLEN E. STEARN
Department o j Chemistry, University of Missouri, Columbia, Missouri AND
HENRY EYRING
Frick Chemical Laboratory, Princeton University, Princeton, New Jersey Received January 88, 19iu INTRODUCTION
It is ordinarily assumed (44) that diffusion through solids may take place by three different mechanisms, which may be termed solid solution, riit, and grain boundary diffusion, respectively. From another point of view all solid diffusion may be spoken of as ( a ) interface or ( b ) volume diffusion. Interface diffusion may occur (1) along the rifts ordinarily arising from mechanical working, (2) at the interface betn-een the grain and the surrounding cementing material, and (S) a t the surface of the crystal. Volume diffusion may be either through the grains or through the cementing material a t the grain boundary. Since ordinarily only one molecule in 10' is located on a surface, the corresponding diffusion coefficients will appear anomalous by such a factor if one makes the mistake of calculating them as for volume diffusion, Le., there will be a large negative apparent entropy of activation. On the other hand, if one calculates this interface diffusion on the basis of surface atoms only, then the only term which will in general differ from the D for volume diffusion will be the heat of activation, which will be smaller. Thus an inspection of an observed diffusion coefficient will sometimes suffice to distinguish between volume and surface diffusion. It'hatever the specific mechanism of solid diffusion, the rate can be expressed by the formulation
where D is the diffusion coefficient, A is a term which varies only slightly with temperature, and Q is the heat of activation of the diffusion process. A general equation has been developed on statistical grounds (17), 955
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ALLEN E. STEARN AND HENRY EYRING
which applies t o any rate process in which the slow step involves surmounting a potential barrier. It has the form
where k’ is the specific rate constant, T is the temperature, IC and h are the Boltzmann and the Planck constants, respectively, and EOis the activation energy a t absolute zero. 3:and Yn are the partition functions for the activated state and the initial state, respectively. In equation 2 as written we have taken the transmission coefficient as unity. The activated complex may always be considered essentially as a molecular species, and thus the rate constant can be pictured as an equilibrium constant between activated complex and reactants a t unit activity multiplied by the universal frequency kT/h. This permits formulation of the reaction rate in thermodynamic terms (71) in the form
k ‘ = - ekT h
-- = Ee AS% e -A-RHT% h AF: RT
(3)
where AF:, A S : , and A H t are, respectively, the standard free energy’ entropy, and heat increases when the activated complex is formed from the reactants. A H f can be obtained from the temperature coefficient of k’ by the relation AH: =
RT
d In k’ __
dT
- RT
(4)
Applied specifically to the rate process diffusion, it has been shown (18) that the diffusion constant is related to an activated rate process by the equation
D
= h2k’
(5)
where X is the distance travelled in a single jump. Substituting equations 2 and 3 into 5 we obtain
D
2kTF h 3s
= X --e
-2
In the derivation (17) of equation 2 the partition function for that degree of freedom along the reaction coordinate was included in the frefor the normal quency term, k T / h . Thus the partition function, 3”, state contains a term for the reaction coordinate not appearing in 3%.
ABSOLUTE RATES OF SOLID REACTIONS
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If we neglect the changes in all the terms of the partition function on activation except those in the reaction coordinate, we may write
where (1 - e-;)-' is the partition function, and Y the frequency of the normal state for that degree of freedom along the reaction coordinate. In the limit, when Y is extremely large, equation 8 reduces to
In the range where Y is small enough so that hv