THEPREEXPONENTIAL FACTORS FOR SOLID-STATE THERMAL DECOMPOSITION line configuration to move toward that of the stable form. If the affected regions average about 15 A in radius, each would contain about 200 molecules; and if two such regions form for each 100 eV, a dose of 20 Mrads should produce alteration of about one-third of the total material. If the altered material on further irradiation produces CHaCN with a yield characteristic of the stable form, the observed increase of G(CH3CN) with dose is not unreasonable. The altered material must then, however, be supposed to exhibit a G(H2)much smaller than that of either pure crystalline form. We have no specific suggestion as to how this might be possible. Many different mechanisms have been proposed for radiolytic Hz formation from organic compounds: (1) reaction of a positive ion or excited molecule with a neighboring molecule, with or without cross linking; (2) separation of H2from an excited state of a single molecule; (3) splitting out
2185
H:, on ion neutralization; (4)separation of atomic H, followed by abstraction of a second H from another molecule; (5) trapping or “solvation” of a free electron, followed by protonation to form atomic H, then by the abstraction reaction. The data in Table I1 on the ratio H2:HD show that different mechanisms predominate in Hz formation from the two crystalline forms. The peculiar effect of dose on G(H2) from the labile form points up how little we understand about the formation of this most common and prominent product of radiolysis. The apparent negative temperature coefficient of G(H2) for the labile form at 20 Mrads total dose presumably indicates that the alteration in crystalline form occurs to a much lesser extent a t lower temperatures, as would be expected since it is essentially a thermal effect. The low value of G(CH3CN) for the labile form a t -78” tends to confirm this explanation.
The Preexponential Factors for Solid-state Thermal Decomposition by Herman F. Cordes Michelson Laboratories, Naval Weapons Center, China Lake, California 03666 (Received December 11, 1067)
The preexponential factors for both unimolecular and bimolecular solid-state thermal decompositions are analyzed from the point of view of activated complex theory. The effect of molecular rotation in both the reactants and the activated complex is considered. If the activated complex has freer rotation than the reactant, the first-order preexponential factor is high. In the bimolecular case, the activated complex is likely to have restricted rotations leading to very low pseudo-first-order preexponential factors. This approach is compared with other treatments.
Introduction It is the general nature of solid-state decompositions that the observed macroscopic rates are extremely difficult to interpret in terms of elementary steps. One reason is that in contrast to gas-phase and liquid-phase reactions the time behavior is not controlled by the molecularity or chemistry alone, but also is controlled by the geometry of the system, its topochemistry.l Nearly all solid decompositions have rate “constants” that are given in units of reciprocal time regardless of the behavior of the rate with time. There is some confusion as to the interpretation of such constants.2*a Consequently the question of the molecularity of solid-phase rate constants has been neglected. The topochemistry will not be considered here. The constants to be considered here are the ones to go into the topochemical solutions. Electron-transfer reactions are excluded from this discussion.
Several applications have been made of the theory of the activated complex to solid de~ompositions.~-~~ Most of these treatments have assumed a monomolecular reaction, although Schultz and Dekkar considered a bimolecular reaction to form a surface (1) W.E.Garner, P. W. M. Jacobs, and F. C. Tompkins, Ed., “The Chemistry of the Solid State,” Butterworth and Co. Ltd., London, 1955,Chapter 7. (2) W.Gomes, Nature, 192, 865 (1961). (3) J. H.Taplin, ibid., 194, 471 (1962). (4) 8. Glasstone, K.J. Laidler, and H. Eyring, “The Theory of Rate Processes,” McGraw-Hill Book Co., Inc., New York, N. Y., 1941. (5) S. S. Penner, J . Phys. Chem., 52, 949 (1948). (6) 8. 5. Penner, ibid., 52, 1262 (1948). (7) 8. S. Penner, ibid., 56, 475 (1952). (8)R. D. Schultz and A. 0. Dekkar, J . Chem. Phys., 23, 2133 (1955). (9) R. D. Schultz and A. 0. Dekkar, J . Phys. Chem., 60, 1095 (1956). (10) R. D.Shannon, Trans. Faraday Soc., 60, 1902 (1964).
Volume 72,Number 6 J u n e 1068
HERMAN F. CORDES
2186 complex which then decomposed. The purpose of this paper is to correlate the previous treatments and explicitly to extend the treatment to the bimolecular case. This will be done in such a fashion as to blend with the better understood theory as applied to gasphase reactions.
General Approach The rate R of a chemical reaction can be defined in terms of the transition-state activated complex b y 4 3
where iis the contribution due to the products. For the condition of constant products and N = N , N*, the free energy is a minimum for
+
N* = Na-&* &a
The bulk rate constant then is
R = KV*N* where R is in molecules per second, N* is the number of activated complexes, v* is the frequency with which the complexes pass to the product side of the defining barrier, and 1 - K is the probability of reflection at the barrier. There are various methods of interpreting this equation, but for the purpose of this paper it will be assumed that the activated complexes are in equilibrium with the reactants. The problem then reduces to the finding of N*. To find N* the theory of perfect solutions will be e m p l ~ y e d . ‘ ~ ?That ’ ~ is, it will be assumed that the energy of a molecule is not altered by the substitution of an activated complex for a normal nearest neighbor and that the products do not affect the course of the reaction, except perhaps to define a boundary line for purely surface reactions. The Helmholtz free energy will be written for the solid phase and minimized to find the equilibrium expression for N*. As long as the volume change for the production of activated complexes is negligible, the Helmholtz free energy is a suitable choice even though the system is open as far as the over-all reaction to produce products is concerned. The partition functions, Qi per molecule, will all have the same origin to avoid carrying around extra Boltzmann terms. &* will be the total partition function (including the reaction coordinate) for the complex. After obtaining the expressions for N* in terms of number of molecules and partition functions, then the partition functions can be interpreted in terms of separable coordinates. Since most of the reactions considered are expected to have finite activation energies, N* will be much less than the number of reactant molecules.
Formulas for N* and K A. Bulk Decomposition. I n this case any molecule is as likely to react as any other, and no preference is shown toward corners, edges, surfaces, defects, or sites of previous decomposition. A I , Monomolecular Reaction. A 7-)A*
4products
The Helmholtz free energy, F per molecule, for this system is12113 The Journal of Physical Chemistry
lCbulk
= R =
KV*&* sec-’
N,
&a
This is the result obtained by Shannon.10 A$. Bimolecular Reaction.
A
+ A e Az*
-t
products
I n this case two neighboring A molecules react to form a bimolecular complex which occupies two sites. 2 will be the number of nearest neigh6ors and N p will be the number of product molecules of all types, one product molecule per site. Q will be the symmetry factor for a diatomic molecule. The Helmholtz free energy for this system is
F = -kTln
where N* ’7should be used, although the extension to three-dimensional external rotations is by no means obvious. The partition function for the molecule in the solid state can then be written in terms of the partition function for the gaseous state of the same energy (Le., neglect the sublimation energy) &solid = YtYrYinQgas
(nonlinear)
where yt = ( ~ T ~ T ) ~ / ’ / V I F yr~= / ’(uJ8n2) /~; [(2nICT)*/’/ l+rl”Z], nonlinear; +yr = (ar/4n)2nk~/lcj+l1/’,linear; and yi = (q/27r) [(Z~kT)”’/$i‘/*], for each internal rotor which is frozen. Numerical Values. There are four special cases which will indicate the range in values for the preexponential factors for the rate constants. For these purposes the preexponential factor A is defined as A keTIWnk)/bTl I
Case I . There is no change in degree of rotational excitation between the reactants and the complex. There will be two cases-completely free rotation and completely restricted rotation. The first case probably can only refer to unimolecular reactions. Case II. The complex has a “freer” condition than the reactants. This may be most likely to occur on a surface where the complex might extend itself from the surface and perhaps rotate parallel to the surface, This is essentially the case treated by Shannon.lO The reactant is assumed to be completely restricted. (14) D. R. Herschbach, H. S. Johnston, and D. Rapp, J . Chem. Phys., 31, 1652 (1959). (15) G. Herzberg, “Infra Red and Raman Spectra of Polyatomio Molecules,” D. Van Nostrand Co., Inc., New York, N. Y.,1945, p 534. (16) K.S. Pitzer and W. D. Gwinn, J . Chem. Phys., 10, 428 (1942). (17) K.S.Pitzer, “Quantum Chemistry,” Prentice-Hall Inc., Englewood Cliffs, N. J., 1953, pp 239-243. Volume 78, Number 6 June 1968
HERMAN F. CORDES
2188
Case 111. The complex is highly restricted in rotation. For the unimolecular case the complex would be expanded in size and hence interact more strongly with its neighbors. For the bimolecular case the two reactants are tied down at separate sites and over-all rotation of the complex is not possible. For the bimolecular complex the bond length between the two parts of the complex is shorter than the van der Waals distance and the two halves may be jammed against their mutual neighbors in such a way as to freeze some internal rotations. For this case the reactants are assumed to have completely free rotation. This means near-spherical tops and elevated temperatures.ls Case I V . This is the case considered by Schultz and Dekkar.slg I n this case the reactants are in equilibrium with a surface adsorbed layer. The adsorbed species on the surface then react via the activated complex to give products. The activation energy will be the activation energy for the last step, providing that the adsorbed species completely cover the surface (and hence their number is temperature independent). These adsorbed species are considered as the reactants, and for the specific model of Schultz and Dekkar this amounts to a tight complex. Their model is that the complex and reactant(s) have identical forces and geometries except in two modes. Two very weak vibrations in the adsorbed species become the reaction coordinate and a strong vibration in the complex. They consider the stronog vibration to be a translation over a distance of 0.2 A, but as far as the configuration integrals are concerned, this is equivalent (at the temperature of interest of 596°K) to a vibration with a force constant of 1.3 mdyn/& I n this case, v* is approximated by
y $(&J 6* and m* are, respectively, the amplitude and the reduced mass for the reaction coordinate. Rather than actually calculating the complete partition function for highly hypothetical complexes, it is easier t o concentrate on the difference between the gas-phase rate constants and the solid-phase constants. The general orders of magnitude of the gas-phase preexponential factors are well understood and will be approximated as 1016 sec-l for the unimolecular case and 10l2 cm3/mol sec for the bimolecular case. The range is =k2 orders of magnitude from these values.lY The lattice force constants will be taken as 0.50 mdyn/S for the translational modes,20corresponding to a frequency of 90 cm-’ for a lOO-g/mol molecule. For the “monomolecular” complexes and molecules, this same force constant acting on each end of a 2-8 diameter yields a rotational force constant of lo-” erg/radian2, corresponding to a frequtncy of 130 cm-’ for a moment of inertia of 100 g A2/mol. For bimolecular complexes this diameter is taken as 4.5 8. The Journal of Physical Chemistry
The number, x , of nearest neighbors is taken as 8; N,) ci 1; and 6 Y 10 8. The ratio of volume to surface for the particles is taken as 10 p. For case I1 the computation is made for the freeing of one mode from vibration to free rotation. For case I11 the calculation is made for the restriction of three modes. Further changes due to internal rotation can easily be made. For case IV the weak vibrations in the adsorbed species y e assumed to have force constants of 3 X lom3mdyn/A ( v Y 3 X 10” see-l). The strong force constant in the complex is taken as 1 mdyn/A. The reduced mass along the reaction coordinate is taken as 11.6 g/mol. K is taken as unity. This is the set of constants used in ref 9. If IC,“ and kbi are, respectively, a high-pressure, gas-phase unimolecular rate constant and a bimolecular gas-phase rate constant then
N/(N
+
IC,”
=
KV*-
Q* ,gas
sec-I
&avgas
=
kbi
KV*
Q*m -
( N ~cm3/mo1 v ~ ) sec
Qa,wsQb,gas
where N o is Avagodro’s number. These formulas then enable the solid-state rate constants to be expressed in terms of hypothetical gas-phase rate constants having the same K , v*, and internal partition functions. The resulting equations for the preexponential factors are given in Table I. A numerical Table I : The Arrhenius Preexponential Factors A Eq for calcn of factor
Case
A. Monomolecular Reaction
Case I Case I1 Case I11
Asolid
=
AIolid = Aaolid =
Agas
e-1/2yr-1/8Agam e8/%Asaa
B. Bimolecular Reaction’ Case Ia, free rotation Not applicable Case Ib, restricted A s o l i d = ayt-lyr-1e-8Agaa rotation Case I1 Asoiid = ae-’/Zyt-~y,-‘~/BAgas Case I11 Aaolid = ayt-lyrAgas
C. Adsorbed Intermediate with Complete Surface Coverage
(18)R. E. Richards and T. Schaefer, Trans. Faraday SOC.,57, 210 (1961). (19) A. A. Frost and R. C. Pearson, “Kinetics and Mechanism,” John Wiley and Sons, Inc., New York, N. Y.,1961,Chapter 6. (20) Estimated from the Debye temperature for KCl: F. Seitz, “The Modern Theory of Solids,” McGraw-Hill Book Co., Inc., New York, N. Y.,1940, p 110.
THETHERMAL DECOMPOSITION OF SOLID ALKALIPERCHLORATES
summary is given in Table 11. (Ti is taken as 2 for each rotational mode (Le., urot = 8 for a three-dimensional rotor).
Table I1 : Numerical Summary of Preexponential Factors at 400°K Y P r e e x p o n e n t i a l factor---Mono-
Case
Bimoleoular
molecular
Bulk Decomposition throughout the Solid, A (sec-1) Case Ia Case I b Case I1 Case I11
1016 10'6 10'6 1012
Surface Decomposition, A Case I a Case Ib Case I1 Case I11 Case I V
(sec-1)
Not applicable 10'8 1018 10'0 for 10-p Particles
10" 10" 10'2 108
...
... . 1012 1014 106 108
A , (cm/sec) for Regression Rate
Case I a Case Ib Case I1 Case I11 Case IV
108 108 109 106
...
2189
... 109 10" 103 108
Conclusions It can be Been from Table I1 that, in contrast to the case of gas-phase reactions, the preexponential factors for solid-phase reactions are expected to have a wide range of values (six or seven orders of magnitude), even after the effect of surface area is corrected for. Empirical first-order preexponential factors may vary from lo5 to 10ls sec-'. The low factors will often indicate a surface reaction, but if the reactions are not dependent on surface area, the low factor may indicate a "tight" complex. The high factors will usually indicate a "loose" complex. Even higher factors (after correction for surface area) can be obtained if the complexes have free translation on the surface. Since in many cases the concentrations in the solid are not controllable, it would have been convenient if the magnitude of the preexponential gave an indication of the molecularity. This appears to be true only for nonsurface-controlled reactions having low (
