By using the principle of microscopic reversibility, unimolecular decompositions are considered from the point of view of the reverse reaction, which is a bimolecular association. In cases where the immediate product of the decomposition is a pair of free radicals, it is assumed that the probability of these free radicals recombining upon collision is independent of their energy. An explicit expression for the activation energy in terms of the energy of reaction and the thermal properties of the molecules involved is thus obtained. From this one may calculate the energy of activation at absolute zero and the energy necessary to break the bond which falls apart in the reaction, quantities which may differ considerably from each other and from the energy of activation at the temperature of reaction. The results are applied to several cases of interest, and are critically compared with those of the earlier theories of Rice and Ramsperger, Kassel and Rice. A brief discussion is given of the calculation of the rate constant as a function of pressure, and its modification according to the point of view of the present paper.
REFERENCES
1.
Presented, in part, at the Boston meeting of the American Association for the Advancement of Science, December, 1933.
2.
Thayer Fellow.
3.
(c)
O. K.
Rice
, Proc. Nat. Acad. Sci.
14
, 114
, 118
(1928
). For a more detailed analysis of these theories than will be given in this paper and for a more complete list of references to original papers, see(d) Kassel, Kinetics of Homogeneous Gas Reactions, Chem. Cat. Co., 1932, especially Chapter V.
4.
More exact definition of the various energy quantities will be given below.
5.
O. K. Rice, L’Activation et la Structure des Molecules, Réunion Internationale de Chimie Physique, Paris, 1928, p. 309. Kassel, reference 3 (d), pp. 105–6.
6.
La
Mer
(J. Chem. Phys.
1
, 289
(1933
)) has also considered energy of activation as a function of temperature but has applied his results only to reactions in solution.7.
The possibility of bimolecular association has been discussed by
Kassel
, J. Am. Chem. Soc.
53
, 2143
(1931
).8.
See Tolman, Statistical Mechanics, pp. 261–2, Chem. Cat. Co., 1927. Our definitions of and are the same as Tolman’s definition of the corresponding quantities.* But see reference 11.
9.
The exact significance of the quantity E is perhaps best understood by considering the corresponding quantity for diatomic molecules, namely, the energy from the minimum of the lowest potential energy curve representing the bond to its asymptote (if potential energy curves cross, to the lowest asymptote). It is therefore identical with the quantity that spectroscopists call (Jevons, Report on Band Spectra, Chap. IX, Cambridge University Press, 1932). It must be distinguished from the quantity that is experimentally determined spectroscopically, and which differs from it by the amount of the zero point energy. In this paper we shall call E the energy of the bond, and distinguish it from the energy change of the reaction at absolute zero, which gives the difference between the energy of the associated molecule in its lowest energy state and the energy of the dissociated particles in their lowest energy states. The difference between E and will be given by the difference between the zero point energy of the associated molecule and the sum of the zero point energies of the dissociated particles. In the literature the concept of bonding energy for the case of polyatomic molecules has not been unequivocally defined.
10.
See for example
Rice
and Ramsperger
, J. Am. Chem. Soc.
49
, 1618
(1927
);and Kassel, reference 3 (d), p. 21.
11.
In the case of the bimolecular reaction our definition differs slightly from that of Tolman. He defines the energy of activation as the average energy of all pairs of reacting molecules minus the average energy of all pairs of colliding molecules. This is not very clearly brought out in his book, but is clearly stated by Kassel, reference 3 (d), p. 26. We define the activation energy as the average energy of all pairs of reacting molecules minus the average energy of all pairs of molecules. The latter quantity is less than the average energy of the colliding pairs (i.e. we have an “energy of activation of collision”), so that in this way we avoid the term which appears in Tolman’s Eq. (614). If we write and for the concentrations of the respective molecules and define by the equation where represents the rate of increase of due to reaction between A and B, we have simply . If we combine this with the equation for the unimolecular reaction, it is readily seen from the immediately following development of this paper that we get the correction thermodynamic relation for the temperature dependence of the equilibrium constant It must be noted, however, that if the bimolecular rate constants are defined in terms of partial pressures rather than concentrations a slight correction must be made. Such a correction has in fact been applied in the case of the data on the association of ethylene and hydrogen discussed below.
12.
For thermodynamic quantities we use the notation of Lewis and Randall, Thermodynamics (McGraw‐Hill, 1923).
13.
For example, they have been used by
Bodenstein
and Jung
(Zeits. f. Physik. Chemie
121
, 129
(1926
)),Trautz
, 3
, 81
(1926
), Berlin). The relation used here was not exact being where where Q is the heat of reaction.14.
15.
16.
Pease
, J. Am. Chem. Soc.
54
, 1876
(1932
). We have made our calculations for Pease’ temperature, about 800 °K. Marek and McCluer worked at slightly higher temperatures, but this difference can be neglected.17.
18.
Parks and Huffman, The Free Energies of Some Organic Compounds, p. 80, Chem. Cat. Co., 1932.
19.
A consideration of this assumption, and how our results would be affected if it does not hold, will be given later on in the paper. We should remark that there is some evidence that free methyl radicals do not combine in the gas phase (Ogg, reference 14;
Bates
and Spence
, J. Am. Chem. Soc.
53
, 1689
(1931
)). Since the recombination does not occur over a considerable temperature range it is unlikely that need of activation energy is the cause but rather some quantum‐mechanical selection rule concerning rotation. Work is now being done on this problem.20.
To verify this point the spectrum of the vibrational energy levels of the cyanogen molecule was plotted as far as the level representing 27.43 kg cal. per mole. Between 26.01 kg cal. and 27.43 kg cal. there are ten separate energy levels. As the total energy of the molecule increases the energy levels become closer together. Since the energy of dissociation for this molecule is about 77 kg cal. the energy levels in this region will be exceedingly close together. See also, Kassel, reference 7.
21.
22.
Reference 21, p. 436.
23.
24.
25.
We refer to calculations by the variation method which manipulate the electronic wave functions holding the distance between nuclei fixed.
26.
27.
28.
29.
(c)
Ramsperger
, 50
, 714
(1928
). The energies of activation used are those calculated by Kassel, reference 3 (d)., J. Am. Chem. Soc.
30.
F. O.
Rice
and Evering
, J. Am. Chem. Soc.
55
, 3898
(1933
)., J. Am. Chem. Soc.
31.
Kohlrausch, Der Smekal‐Raman Effekt, pp. 304 ff. Springer, 1931.
32.
Such a rearrangement is postulated in theories of chemical binding advanced by Heitler and Schuchowitzki, reference 26.
33.
From the data of
Kharasch
, Bur. Standards J. Research
2
, 359
(1929
), the heats of hydrogenation of both propylene and butylene can be calculated to be 32 kg cal. as compared with the 31 kg cal. experimentally determined for ethylene.34.
35.
36.
Reference 3a, p. 1622, Eq. (8).
37.
For the treatment of a similar problem see reference 3a, p. 1622.
This content is only available via PDF.
© 1934 American Institute of Physics.
1934
American Institute of Physics
You do not currently have access to this content.
