STUDIES IN HOMOGENEOUS GAS REACTIOXS I BY LOUIS STEVENSOPI’ KASSEL
Introduction During the last few years the theory of activation by collision, rather than by radiation, has made rapid strides. Subsequent to the suggestion of activation by radiation by Perrin,’ Lindemannz showed that on the basis of the simple radiation hypothesis the inversion of sucrose must be enormously accelerated by sunlight, in disagreement with experiment. Many other cases are now known in which the frequency which is calculated from the simple radiation hypothesis is quite without effect on a system; frequently it falls in a region in the infra-red which is not absorbed at all. At about the same time, Langmuir? pointed out that there is not enough radiant energy in a system to account for the observed rates of reaction. Also, Lindemannj called attention to the fact that, if the rates of activation and de-activation are large compared to the rate of reaction, the reaction may be kinetically unimolecular regardless of the order of the activation process. Calculations have been made by Christiansen and K r a m e r ~by , ~ G. ?;. LewisJ6and by Tolman’ which deal with the maximum possible rates of activation by radiation and by collision. I n all of these calculations it was found necessary to make rather undesirably liberal assumptions to account for the observed rates of reaction; it cannot be said that they were very favorable to either method of activation. Since these last-mentioned calculations were published, two distinct argiiments in favor of activation by collision have appeared. The first of these is of an experimental nature; Hinshelwood and his co-workers* have found that the decompositions of propaldehyde, of diethyl ether, and of dimethyl ether, all of which are strictly unimolecular at high pressures, deviate from the unimolecular course at low pressures, the specific reaction rate becoming smaller with decreasing pressure; RamspergerQhas observed the same type of decrease in the specific rate in the decomposition of azomethane, and to a greater extent than in any of the cases studied by Hinshelwood. This decrease in the specific rate with decreasing pressure removes the very feature of unimolecular reactions which the radiation hypothesis was invented t o explain, 1“LesAtomes” 11913);Ann Phys., 11, j (19191, *Phil. Mag., 40, 871 (1920). 3 J. Am. Chem. SOC., 42, 2190 (1920). 4 Trans. Faraday SOC., 17, j98 (1922). 5 Z . physik. Chem., 104,451 (1923). 6 Lewis and Smith: J. Am. Chem. SOC.,47, I jO8 (192j). 7 J. Am. Chem. SOC., 47, 1524 (192j). 8 Hinshelwood and Thompson: Proc. Roy SOC.,113A. 221 (19261;Hinshelwood: 114A. 84, (1927); Hinshelwoodand Askey: 115A,2 1 5 (1927). J. Am. Chem. SOC.,49, 912, 1495 (1927).
2
26
LOUIS STEVEKSON KASSEL
and may be iegarded as experimental evidence that radiation alone is not sufficient to cause the thermal decompositions of the above-mentioned substances. The second of these arguments in favor of activation by collision is based upon the suggestion made by Christiansenlo that the number of degrees of freedom of the molecule must be considered. This idea has been the basis for calculations by Hinshelwoodll and by Fowler and Rideall* which show that the maximum rate of activation is very much increased by this assumption. I n the paper by Fowler and Rideal there are some other assumptions made as to the method of activation which seem rather questionable, and which have been criticized by Tolman, Yost, and Di~kinson.'~ The treatment given by Hinshelwood seems to be preferable, and will be discussed in a later section. I t may be said that as a result of this treatment it proves possible to account fairly well for the experimental results of Hinshelwood himself, but that the much faster decompositions of azomethane and of nitrogen pentoride would require the assumption of an zhsurd number of degrees of freedom. Hinshelwood assumed that the specific reaction rate of all activated molecules was a constant, independent of their energy content. I n this paper the consequences of replacing this assumption by the more reasonable one that the specific reaction rate for active molecules increases with their energy content will be considered. Just before the paper was submitted, an article appeared by Rice and Rarn~perger'~ in which this same assumption was made. The development and the final result are in many respects similar to that presented here, but there are enough differences to make publication of this article seem desirable.
The Energy of Activation There seems to be some disagreement in the literature about the meaning to be given to the energy of activation. I n the original Arrhenius equation,
q is the energy per mol necessary for activation, that is, the difference between the average energy of the activated molecules and the average energy of all the molecules. T01man'~as the result of a rigorous and very general deriration, has obtained the expression
1OProc. Csmb. Phil. SOC.,23, 438 (1926). 11Proc. Roy. SOC., 113A, 230 (1926). l* Proc. Roy. soc., 113A, 570 (1926). 13 Proc. Nat. Acad. Sci., 13, 188 (1927). I4 J. Am. Chem. SOC., 49, 1617 (1927). 16 J. Am. Chem. Soc., 47,2652 (1925);"Statistical Mechanics," 26j (1927).
STUDIES I N HOMOGENEOUS GAS REACTIONS
227
where E is the average energy of the molecules that react and E is the average energy of all the molecules; this equation applies to unimolecular reactions in which the equilibrium quota of activated molecules is essentially maintained; for other cases, Tolman's paper must be consulted. Tolman, then, proposes to call this quantity E - E the energy of activation. Lewis,16 on the other hand, defines the energy of activation as the minimum internal energy that a molecule must have in order to react; this definition does not involve any equation. This definition seems preferable to that of Tolman, because many simple hypotheses as to the nature of the activated states give values of E - E which are not constant with respect to the temperature; it should certainly be a fundamental requirement for any definition of the energy of activation that it be independent of the temperature. Thus, for the simple case characterized by two internal degrees of freedom for the molecule, and maintenance of the equilibrium quota of activated molecules, Lewis'' has calculated the temperature coefficient of the reaction rate, assuming, (a), that the specific reaction rate of activated molecules is independent of their energy content, apd, (b), that the specific reaction rate is proportional to the excess energy over some critical amount, E,. Assumption (a) leads to the equation d_ In _ _K - - E, dT kT2' while (b) gives d- In -K dT
- E,
+
kT kT2
*
These same results might have been obtained directly from Tolman's equation. Thus for the first case
[=N k T
EdE
and
l m Nk T
Ed E
-
=k T
E = ,hi k T e -E'kT
dE
"LewisandSmith: J . . h . C h e m . S O C . , ~Ijr2 ~ , (1925). I' Lewisand Smith: J. Am. Chem. SOC.,47,15r3 (1925).
LOUIS STEYESSON KASSEL
228
Whence
_d l_n h_ ;_-_ Z- -_E- - E, dT
kT2
,
kT2
just as Lewis found. And in the second case
(E - E,) E d E
e -E,
and
kT
(E - Eo) EdE
is the same as before. Therefore d~lnh; dT
-
Eo
+
k T kT2
’
Thus we see that assumption (a) corresponds to a type of reaction in which Tolman’s definition of the energy of activation is the same as Lewis’s, but that assumption (b) corresponds to a type in which the energy of activation as defined by Tolman is not only larger than that given by Lewis, but is variable, increasing with temperature. Since many other examples could be given in which this same sort of variation would occur, it seems best to adopt the definition given by Lewis. Accordingly, in this paper, the energy of activation is defined as the minimum internal energy per mol which molecules must attain in order to react. This same definition has been adopted by Rice and Ramsperger alsb. The Specific Reaction Rate of Activated Molecules The paper of Rice and Ramsperger dealt almost entirely with two theories; that which they have called Theory I, which is simply the theory considered by Hinshelwood, and Theory 11, in which it is assumed that an activated inolecule does not react until some particular degree of freedom acquires energy of E, or greater. On the basis of this assumption it is possible to calculate, from statistical mechanics, the way in which the specific reaction rate of activated molecules changes with their energy contents. They have applied these two theories to Hinshelwood and Thompson’s results on propaldehyde, and found that the form of the curve obtained by plotting log K against log p (at a fixed temperature) is given fairly well by either theory. The author has also developed Hinshelwood’s theory, in much the same way as have Rice and Ramsperger, and obtained the same result. It is found that.
STUDIES I N HOMOGENEOUS GAS REACTIONS
229
where PI is a constant at any fixed temperature; this is equation ( 5 ) in Rice and Ramsperger's paper. The author has found it very convenient in testing this theory to plot I / K against ~ / p . According to the equation, this plot must be a straight line, from whose intercepts I< and pi may be found a t once. It is thus very easy to tell whether the data for any particular reaction
i/V
/
FIG.I I Propaldehyde 849" I11 Diethyl Ether 798" V Azomethane 603'
11 Propaldehyde 796" IV Dimethyl Ether 777' TI Azsmethane 563'
are in agreement with this theory or not. In this way it is found that the experimental data for diethyl ether, dimethyl, ether, and azomethane are in definite disagreement with the requirements of the simple theory; propaldehyde gives straight lines within the accuracy of the experimental results. The experimental results for all four of these substances are plotted in Fig. I . Thus it seemed that, although the assumption of a reasonable number of degrees of freedom was capable of accounting for the order of magnitude of the reaction rates of most monomolecular reactions, it was not able to account quantitatively for the variation of specific rate with pressure. Also, it was found that the theory was unable to account for even the order of magnitude
LOUIS STEVENSON KASSEL
230
in the case of some of the faster reactions, notably the decompositions of aaomethane and nitrogen pentoxide, without assuming an unreasonable number of degrees of freedom.
A second theory was then developed, based upon the following argument. I n a complex molecule it is not sufficient to have a large amount of energy stored to cause decomposition. I t is necessary for this energy, or a sufficient part of it, to be concentrated at some weak point in order to rupture the molecule. Now the simplest possible assumption is that there is some single bond in the molecule which is easier to break than any of the others, and that all but an inappreciable part of the reaction is initiated by a break of this bond. Then, if we assume that the bond breaks whenever it acquires energy in excess of some critical amount, E,, it should be possible to calculate from statistical mechanics a relation between the specific reaction rate of activated molecules and their energy content. This differs from Theory I1 of Rice and Ramsperger in one way: they assumed that it was necessary for the energy E, t o be concentrated in a single degree of freedom; the author has regarded a chemical bond as similar to a simple oscillator, and hence possessing two degrees of freedom. Save for the differences resulting directly from this one variation, the two developments are very similar. Following Rice and Ramsperger, let a molecule of Class A be one whose total internal energy is between E and E d E, and a molecule of Class C be one which is of Class A and in which some particular bond (that is, some particular two degress of freedom) has energy in excess of E,. We assume that the rate at which molecules of Class A enter Class C is proportional to the fraction of the molecules of Class A which, at equilzbrzum, would belong to Class C. The assumptions involved in this are very clearly outlined by Rice and Ramsperger on page 1 6 2 2 of their article, and are exactly the same in this theory as in theirs.
+
We wish, then, to determine the fraction of all molecules which have energy d E and which would have also, if the distribution law between E and E held, energy in excess of Eo in some definite two degrees of freedom. This calculation may be made in a way analogous to that used by Rice and Ramsperger. I t may also be done in a different way, somewhat more elegant mathematically; a similar method is not available for the case treated by Rice and Ramsperger, because of the non-integrability of one of the equations. It is this second method which will be given here. For a single oscillator it is well known that the probability of energy in the range from E to E d E is I -E/kT d E P l , E , d E = -e J kT
+
+
and that the probability of energy greater than E is P,,~ = e-E’kT.
23'
STUDIES IN HOMOGENEOUS GAS REACTIONS
Then it follows at once that the probability that a system of two oscillators will have energy greater than E is
In this expression the first term within the brackets is the probability that a dE, and chosen one of the oscillators will have energy in the range E, to E, the second term is the probability that the other will have energy greater than E - E,, so that the two together will have energy greater than E. The integral of this product from E, = o to E, = E plus the probability that the first oscillator will have energy greater than E, is evidently the total probability that the two together will have energy greater than E.
+
I t is then easy to find the probability that the two oscillators together will dE. This is have energy in the range from E to E
+
PZ,E,dE
'&E dE = -a -
aE
.
Continuing in this way, it is found that the probability that S oscillators have energy greater than E is
and the probability that they have energy in the range from E to E PS,E,dE
-
=
+ dE is
'8,E de - aaE
ES- I -E (S - I ) ! (kT)' e
kT
dE
These are of course all well-known results. Evidently the integrand of ( I ) , namely,
LOCIS STEVEXSOX KASSEL
232
I E_ - E_ , I!
kT
+
E-E, 11
e - F
is the probability that qome chosen one of the S oscillators will have energy in the range E, to E, dE, and that the whole group will have energy greater than E. Hence the probability that the whole group will have energy in the range E to E d E and that the chosen one will have energy in the range E, to E, dE, is given by
+
+
+
-~ -
I
(S - z ) !
(E - E O ) ~ - ' ~ - Ld ~E TdE, (kT)'
But it has already been found that the probability that the whole group will have energy in the range from E to E d E is
+
Hence the probability that when the Soscillators have energy in the range E to E d E some chosen one of them has energy in the range from E, to E, dE, is
+
+
Then, finally, the probability that when the whole group has energy in the range E to E d E some chosen one of them has energy greater than E, is
+
Hence the specific reaction rate of molecules with e n e r a E is
where X is the proportionality constant
It only remains to evaluate this constant in terms of the rate at high pressures. When the equilibrium quota of activated molecules is maintained, the d E that decompose per number of molecules with energy between E and E second is
+
STUDIES IN HOMOGENEOUS GAS REACTIONS
233
where N is the totalhumber of molecules. Then the total number of niolecules that decompose per second is r m
Hence the observed specific reaction rate at high pressures will be
and hence the specific reaction rate of activated molecules of energy E is
This is the quantity that Rice and Hamsperger have called bE; thecorresponding equation in their theory is (18) ; it is
It is of interest to calculate % and E in Tolman's equation d-l-n K - -E - E dT kT2 Evidently
234
LOUIS STEVENSON RASSEL
and
Hence and d In K dT
-
E, kT2’
as could have been seen directly.
Thus if the assumption made by Rice and Ramsperger that reaction occurs whenever some single degree of freedom in the molecule acquires energy in excess of E, be replaced by the very similar assumption that reaction occurs whenever some bond (two degrees of freedom) in the molecule acquires energy in excess of E,, a slightly simpler, but very similar result is obtained for the specific reaction rate of activated molecules. If it is remembered that S in the present theory is equal to n/2 in the notation of Rice and Ramsperger, since the number of degrees of freedom is twice the number of oscillators, it is evident that the difference is in the direction of a relatively greater specific reaction rate for the molecules of high energy in the Rice and Ramsperger theory than in that of the author; this effect, however, is not a large one.
The Rate of Activation of Molecules The next problem is to find the actual reaction rate when the pressure is not so high that the equilibrium quotas of activated molecules are maintained. If, following Hinshelwood and Rice and Ramsperger, we assume that at all temperatures and pressures the rate of activation is equal to the rate at which activated molecules would participate in collisions if the equilibrium quota of them were present, we may again use part of Rice and Ramsperger’s derivation. They have shown that
where WEdE is the equilibrium quota of molecules with energy between E and E dE, bE is the specific reaction rate of these molecules, and
+
STUDIES I N HOMOGENEOUS GAS REACTIONS
235
Here we have
and
When these substitutions are made, the resulting equation reduces to
K
C" -
a form which is suitable for calculation. It is not possible to perform the integration algebraically, and it is thus necessary t o obtain numerical values by graphical means or by quadratures.
Test of the Theory The data of Ramsperger on the decomposition of aeomethane afford the best test of any theory which concerns itself with the decrease of reaction rate with pressure, since the reaction has been studied a t pressures low enough to give a very considerable decrease, at two temperatures. This last fact is of great importance, as d l be seen. The high-pressure measurements on azomethane may be fairly well represented by l o g K = 15.96512 - 11180.12/T, or K = 9.228 X 1 0 1 6 e--51130 RT. Comparison of this with the theoreticalequation for high pressures showsthat
A
X IO^^,
= 9.228
and E,
=
51130.
Here E, and R are in calories per gram molecule. If it is assumed that u2 = 5 X
- l6 cm2
IO
then
B
= j.214
x
10-11.
Then it only remains to select values for T and S;for, when this h m been done, D may be calculated, and K is thus expressed as a function of N only. The resulting integrals have been found by quadratures for a number of values of S and N. The results are summarized in the following tables.
LOUIS STEVENSOX KASSEL
S = 6,T I 023
6.2 X 105 .9162
=
603
1922
I021
I020
6.2 X 104 .6528
6.2 X 1 0 3 .3055
6.2 X IO? so93 7
S = 9,T
=
603
I021
1019
1018
I 017
1016
6200 7889
62 ,2091
6.2 .06383
.62 .01495
,062 ,002905
S
I 019
6.2 X I O .0193
I 015
,0062 ' 0004537
= 12, T = 603
'
1018
I 017
1016
1015
6.2
.62 ,1008
.062
,0062 ,006242
3203
S=
12,
T
=
,03005
563
I 020
1019
1018
I oil
1016
I 015
580
58
5.8
.58
,058
,9601
,6847
,3708
,1425
,04158
,0058 ,009975
I n considering these values several points are to be kept in mind. If the value of u2 is altered, it is not necessary to recalculate the integrds; new tables may be prepared from the ones given by reducing the values of N and P in the same ratio in which u2 has been increased; an inspection of the equation which governs K shows the correctness of this procedure. The values of h and E,, used in this calculation are those obtained from the measurements in Ramsperger's first paper. It is likely that both X and E, are a little too small; the values of K, determined by extrapolation at 603' and 563' are 3.09 X IO-^ and 1.38 X IO-^ as compared with 2.66 X IO-^ and 1.28X I O - as given by the equation. For this reason, in comparing observed and calculated values of the reaction rates, the comparison is made between values of K/K, and not between values of Ii; when this is done the error caused by inaccurate values of A and E, is small. I n Fig. 2 the calculated values of K/K, are compared with the experimental values. Here u? has been taken as 1.5 X I O - 14, and the values of P given in the tables have been corrected accordingly. I t is evident that t'he experimental results are fitted very well by taking s = 12, u? = 1.5 X I O - l i ; they could be fitted equally well by a smaller value of s, in conjunction with a larger value for u2. Thus, so far, the theory has proved satisfactory; but the real test is yet to come. It is now possible to calculate the specific reaction rate at any temperature and pressure, all of the arbitrary constants having been chosen. I n Fig. 3 the theoretical curve for T = 563 is compared with the experimental one; the agreement is little short of remarkable. Thus Ramsperger's results on the decrease in specific reaction rate with decrease in pressure may be quantitatively explained by taking u? = 1.5 X
STUDIES I N HOMOGENEOUS CAS REACTIONS
237
I O - 14, S = 12. They could be explained about as well, probably, by larger or smaller values of S, with correspondingly changed values of uz. It may be advanced in criticism of this theory that the values chosen for S and az are both larger than is probable for such a molecule as azomethane. But it must be remembered that an activated molecule has a very high “internal temperature” and that many of the degrees of freedom that are normally frozen
FIG.2 The lines are the theoretical ctirves for s = 1 2 , s = 9 , s = 6 respectively (T = 603”, uZ = I .s X 10-14). A horizontal displacement of these curves corresponds to a change in the value of u*. The points represent some of Ramsperger’s experiments on azomethane at 603OK. 4 vertical dmplacement of these points corresponds to a change in the assumed value of lim.
may be excited; thus the specific heat data would not be of value in fixing a value for s, since they apply to molecules with much lower internal energy. As for the value given for u2,it must not be forgotten that this is the molecular diameter for collisional deactivation, which need not be the same as the diameter as determined by measurements of viscosity or thermal conductivity; indeed, the experiments of Stuartt8 with mercury vapor have shown that in that case at least the diamete‘r for collisional deactivation is about three times the normal kinetic theory diameter. Three other unimolecular reactions are known in which the specific reaction rate decreases with decreasing pressure, the decompositions of propaldehyde. of diethyl ether, and of dimethyl ether. In Fig. 4 are given the 18
2. Physik, 32, 262 (1925).
238
LOUIS STEVENSON KASSEL
plots of log K against log P for these reactions. For propaldehyde the curve is similar in form to the theoretical curves, but in order to reconcile the others with the theory it is necessary to assume that in the case of dimethyl ether the highest pressures used have been somewhat insufficient to give K while for diethyl ether the insufficiency must have been still greater. In order t o decide this point, the reaction rate would need to be measured for these sub-
u
z
LOG P
I
.ZB-i
FIG.3 The curve is drawn for a = 12,u* = 1.5 X IO-*', T experimental results on azomethane at 563'K.
= 563'.
The points are some of the
stances a t pressures up to about five atmospheres. Unless it should be proved that the reaction rate does not increase further with increasing pressure, none of these results can be considered to be in disagreement with the theory. The only other unimolecular reaction which has been studied a t low pressures is the decomposition of nitrogen pentoxide. It seems wise to refrain from theories of this reaction until it has been further studied experimentally. If the rate actually does increase at very low pressures, in the way found by H i n t and Rideallg it is probably because some other mechanism of reaction is becoming the dominating factor. Until more is known about this phase of the reaction it is useless to guess what this low-pressure reaction is like. Even the maintenance of the rate down to pressures of about .I cm is hard l9
Proc. Roy. SOC., 109A,526 (1925).
239
STUDIES I N HOMOGENEOUS GAS REACTIONS
to account for. A rough calculation was made, taking S = 7 , T = 300. The results were
P K/K oa
I500 '
994
I5 .891
uz = I O
- 14 and
'15
,0015
I249
,0178
FIG.4 The points ive some of the experimental data for propaldehyde a t 84 (I)and a t 796" (111, for diethyfether a t 798" (111) and for dimethyl ether a t 777' (IV). g h e ordinates are log K, the distance between the hdriaontal lines being .,5 unit, but the relative vertical postions of the curves are not significant.
Thus a t a pressure of .IS cm the reaction rate has falIen to a fourth of its high pressure value. Hence it would be necessary either to take s greater than 7 , or to assume a molecular diameter of more than IO-' to account for the experimental results.
Discussion The results which have been presented here seem to show that the simple type of theory which assumes a single specific reaction rate for all molecules is not satisfactory; this is the theory that Rice and Ramsperger have called
2 40
LOUIS STEVENSON KASSEL
Theory I. Of the four reactions now known in which the specific reaction rate decreases with decreasing pressure, only the decomposition of propaldehyde was known to show this decrease a t the time Rice and Ramsperger submitted their paper, and the experiments on this substance have not been extensive enough to reveal the inadequacy of the simple theory. But when it is attempted to apply this theory to the decompositions of diethyl and dimethyl ethers, and of azomethane, it is found unsatisfactory; this is shown by the curves of Fig. I , which should be straight lines if Theory I is correct. There is every reason t o believe that, if the experiments on propaldehyde were extended to lower pressures, here too the I/K, I/P plot would curve.
It thus appears necessary to resort to a theory in which the specific reaction rateincreases with the energy; there are now twosuchtheoriesavailable, the one developed by Rice and Ramsperger, and called by them Theory 11, and the one presented here, which may be called Theory 111. These two are very similar: in the development of Theory I1 it is assumed that decomposition takes place whenever some single degree of freedom in the molecule acquires energy in excess of a critical amount; in Theory 111, it is assumed that it is some particular bond (two degrees of freedom) rather than some single degree of freedom which must acquire the critical energy. I n the development of the theories certain simplifying assumptions have been made. The discussion of these assumptions which Rice and Ramsperger have given is so good that it seems useless to consider them further here; the same assumptions occur in both theories. It seems to the author that the physical basis for Theory I11 is perhaps a little better than for Theory 11; rupture of the molecule might be expected to follow at a bond of the critical energy, and a bond represents two degrees of freedom. Rut such arguments are not conclusive and a decision between the two must rest upon experimental evidence. It is to be hoped that in the near future Rice and Ramsperger will attempt to apply their theory to azomethane. I t is possible that the two theories will prove sufficiently different to decide between them by a careful application to this reaction. As to the details of the theories, there is a considerable chance for change. It may be possible to avoid some of the assumptions that have been made, particularly in the calculation of bE, but this would not be expected to produce any considerable change; the rotational energy might be more explicitly considered, though this would probably be very difficult, or a quantum treatment might be given. None of these changes could modify the general character of the theory. summary I. Reasons have been adduced in support of the definition of energy of activation, due to G. N. Lewis, as the minimum internal energy which molecules must have in order to react.
STUDIES IX HOMOGENEOUS GAS REACTIONS
241
It has been shown that the theory t’reatedby Hinshelwood, and further 2. developed by Rice and Ramsperger, which these authors have called Theory I, in which it is assumed that all activated molecules have the same specific reaction rate, regardless of their energy content,, is in disagreement with the experimental results for the decompositions of diethyl ether, dimethyl ether and azomethane. 3 . On the basis of statistical mechanics an expression has been deduced for the variation of t,he specific reaction rate of activated molecules with their energy content. 4. By t,he use of this expression, an equation has been derived which gives the specific reaction rate for a unimolecular reaction at any pressure.
5 . This equation has been shown to be in quantitative agreement with the experimental results for the decomposition of azomethane.
6 . The differences between this theory and the similar one just published by Rice and Ramsperger (Theory 11) have been pointed out. Xote added in proof: Attention is specially called to the fact that the term degree of freedom is used in this paper in the same sense as it has been used by Rice and Ramsperger. This disagrees with the customary usage, according to which an oscillator has only a single degree of freedom, although it possesses both kinftic and potential energy. As the term is used here. an oscillator has two degrees of freedom.
I t is evidently necessary to discuss in more detail the theory of Fowler arid Rideal (ref. 12). I t was pointed out by Tolman, Tost and Dickinson (ref. 13) that this theory required very large diameters for collisional deactivation to account for the reaction rate of nitrogen pentoxide. I t has since been shown by Bernard Lewiszothat there is a numerical error in these calculations, and that the necessary diameters are not as large as Tolman, Yost and Dickinson had supposed. They are, however, very large; to make the rate of production of activated molecules equal to the rate of reaction at 300’ K. and a pressure of .os mm., it is neccssary to assume a diameter of 6 x IO-^ cm, which is certainly very large. h still larger diameter would be necessary to maintain the reaction rate a t its full value at this pressure, and this is necessary, since thc recent work of Hibben?’ shows that the rate is surely maintained at a pressure of .03 mm. and probably at one ten-fold lower. Lewis’s suggestion that this large diameter is “apparent and not real” and that if we take into account the fact that every collision involving an activated molecule does not result in its destruction we will find smaller deactivational diameters, seems t o the author to be entirely incorrect. lye can calculate the necessary rate of production of activated molecules from the rate of reaction. If me set this equal to a rate of collisional deactivation, we get a number of collisions from which Science, 66, 331 ( 1 9 2 7 ) . Proc. Nat. Acad. Sci., 1 3 , 626 (1927).
242
LOUIS STEVENSON KASSEL
we calculate a diameter; but if only some of the collisions are effective, since the number of effective ones is prescribed for us, we must have more collisions altogether, and hence larger diameters. The idea of a target area is one to which the author subscribes, but it cannot lead to decreased diameters. Lewis has made calculations for azomethane22 similar to those for nitrogen pentoxide and has, remarkably enough, succeeded in accounting for a rate of activation so large as to permit of the maintenance of specific rate in this reaction a t pressures much lower than those at which Ramsperger subsequently found the rate was not maintained. The University of Chicago, July 83,1987. **Proc.Kat. Acad., 13. 546 (19271.
