T H E THEORY O F THIRD-ORDER GAS REACTIONS* BY LOUIS
s. WSSEL**
Introduction During the last few years considerable progress has been made toward putting the theory of homogeneous gas reactions upon a firm foundation, but in the midst of this development the subject of third-order reactions has been entirely neglected. There are two distinct kinds of trimolecular reactions which would follow a third order equation; in one kind the third molecule plays the role of a catalyst and permits the occurrence of a reaction which would be impossible at an ordinary collision of the two reactants; the most important action of the third body in this case is to remove some of the heat of reaction and thus make possible an association which otherwise could not occur; such a process is known to take place in the formation of hydrogen and of bromine molecules from their atoms; the theory of this type of thirdorder reaction has been considered from the standpoint of the quantum mechanics by Kallman and London' but no attempt a t a complete treatment has been made. The other type of trimolecular process, and the one which is of more chemical interest, is that in which all three molecules actually enter into the reaction. It is this latter type with which the present paper will be chiefly concerned. There are four third-order reactions known which are presumably trimolecular reactions of the kind in question. These are the action of nitric oxide upon oxygen, upon chlorine, upon bromine, and upon hydrogen. The first of these has a negative temperature coefficient, the second and third have quite small positive coefficients, while the last has one which is of the usual order of magnitude. I t is not universally admitted that these reactions actually have a trimolecular mechanism; the alternative view is that they occur in two bimolecular steps, the first being reversible and so rapid that the equilibrium is always maintained, and the second being slow and therefore governing the rate. A mechanism of this kind would of course lead to a third order expression for the rate of the reaction. Furthermore, it would be easy to understand the occurrence of a negative temperature coefficient, since the temperature coefficient for the measured rate would be the sum of two coefficients, one for the equilibrium, the other for the second of the two bimolecular steps. The first of these would be negative, the second positive, and the sign of the sum would thus be left unrestricted. It is not possible to make any sharp distinction between triple collisions and intermediate compounds. It is necessary in any case to consider the at* Contribution from Gates Chemical Laboratory, California Institute of Technology, NO. 249. * * National Research Fellow in Chemistry. Kallman and London: Z. physik. Chem., ZB,240 (1929).
1778
LOUIS S . KASSEL
tractive forces between the molecules, and these forces will give rise t o pairs, and perhaps larger clusters, which do not have enough energy to dissociate; it does not make any real difference whether we call these aggregates molecules or not; whatever we call them, we shall find it necessary to know their concentration, since at low enough temperatures they will be responsible for the major part of the reaction. There are two ways in which me might calculate the concentration of these clusters; if we want the concentration of stable pairs, it will be most natural t,o consider that they are molecules, estimate their entropy and free energy, and calculate the ordinary chemical equilibrium constant; if, on the other hand, we are not interested in the internal kinetic energy of the cluster, but only in its momentary configuration, we can estimate the intermolecular force and calculate by statistical mechanics the number of clusters which have a separation less than any specified distance. For the purposes of reaction rate calculations the latter procedure seems more suitable; it will be used exclusively in this paper, and the term pair from now on will be defined as two molecules whose separation is less than a specified distance; we shall speak also of transient and of stable pairs, according as they have sufficient energy to dissociate or not. The chief advantage of this method over the chemical one is that the critical separation in the definition of a pair may be varied t o suit the particular reaction in question. Still, by introducing an energy of activation when necessary, and by including a trrm arising from pure triple collisions of three previously-free molecules, the essential features of the theory to be presented could all be reproduced by a theory based on intermediate compound formation. Intermolecular Forces in an Imperfect Gas
'
Before we attempt to say anything further about trimolecular reactions we must examine in considerable detail the process of triple collision in an imperfect gas. Our knowledge of the intermolecular forces in an actual gas is based mainly upon the equation of state and the phenomenon of viscosity. The method used is to design a molecular model which is physically plausible and yet simple enough so that the mathematical difficulties are not insuperable, and then t o work out the kinetic theory expressions for the pressure and viscosity of a gas whose molecules are like the model; by comparison of the theoretical results with experimental ones it is found whether or not the model is suitable, and if, it is the parameters which occur in it may be evaluated. The problems involved in working out the properties of these model gases have received much attention, but as yet relatively little progress has been made. The most, general model which it has been possible to treat possesses a symmetrical central field of force f(r) = hr-" - pr-m
(1)
where m > 4, n > m. The latter condition must be satisfied in order for the force to be repulsive at small distances, and attractive at larger ones. The reason for the former condition will be seen shortly. The present state of the
THEORY O F THIRD-ORDER GAS REACTIOKS
IT79
problem has been admirably summarized by Lennard- Jones’ and the treatment given here is based largely on his. The equation of state is somewhat better suited to the problem of determining the attractive part of the field than is viscosity, and it IS about as good for the repulsive portion. We shall find it necessary to consider its use n some detail. The equation of state, at moderate dilution, may be represented by pl* = -4” Bv,V Cv/T? (2)
+
+
+
\\here A,, By,etc are the first, second, etc w i a l coefficients. The theoretical expression for spherically s j mmetric molecules is known to bez =
KkT
[
I
- __ 2;s
I@= r?
kT
-
I)].
+ (I(+ (3) )
0
nhere Ekr) is the energy of t\so molecules the distance r apart, and N is the number of molecules in the volume V. JTe see by comparison of ( 2 ) and (3) that Av = S k T Bv/-Av = v,B’ (4) where vo is Loschmidt’s number, z i o X 1019, and
This integral cannot usually be evaluated, but in favorable oases it may be. X partial integration gives
sow -dE,dr = f(r) and for the particular field mentioned previously this becomes
(7)
-dE/dr = Xr-n - ,m-m (;a) We see now why m must be greater than 4 : othermtse B’ would be infinite. The integral (6) with the value (7a) has been treated by Lennard-Jones3 who gives the solution
2
Lennard-Jones, in Fowler: “Statistical Mechanics,” p. Fowler: “Statistical hlechanics,” pp. 172, 213. Lennard-Jones: Proc. Roy. SOL,106A,463 (1924).
217.
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LOUIS S. KASSEL
where
and
This result is in a form suitable for comparison with experiment. A plot is made of log IB,/A,J against log T, and another, on transparent paper, of log I F(y)/ against log y, for some chosen values of n and m. The scale of log IF(y)l is the same as that of log ~B,./A,~,and that of log y is ( n - ~ ) / (n-m) times that of log T, and increases in the opposite direction. Now, if a relative position for the two plots can be found, such that their axes are parallel and the curves are in satisfactory agreement, the model may be considered suitable; it is then possible to evaluate A and p in terms of the parallel displacement (X, Y) of the two curves by the equations logy
+ (n - m) /(n
log /B,/A,I
-
I)
log T = X
- log IF(y)I = Y
(94 (9b)
into which are to be substituted the values of y and F(y) as given by (Sa) and (8b). It is possible to obtain several still more special cases from this result by loss of some of the constants. In particular, if n is allowed to increase beyond bound, and X decreases in an appropriate way, we obtain for the model a rigid sphere of diameter u given by
surrounded by an attractive field. Upon comparison with experiment, it is found that the rigid sphere model is not satisfactory; the values of B,/A, for helium and neon possess a maximum at high temperatures which this model does not reproduce, and presumably other gases would show the same maximum at still higher temperatures. The more general model accounts satisfactorily for the observed values; it does not prove possible to determine m and n uniquely, since usually several sets of values can be found, all of which are satisfactory. By the use of viscosity data, and a study of the forces between ions in crystals it is possible to derive some additional information. Thus for helium viscosity requires n = 14.6, and the two best sets of values (n, m) for the equation of state data were (11, 6) and (14 1/3, 5). The second of these is thus to be considered preferable. Likewise for hydrogen we find that ( I I , 5) is almost uniquely singled out, even though the spherical model cannot be correct in this case. But when the attractive forces are not quite weak, the theoretical formula used in interpreting the viscosity data becomes unsound, and, just as we should expect, the agreement becomes worse. We may then have recourse to crystal data, from which we find values of n = 9, I O or 1 1 for repul-
THEORY O F THIRD-ORDER GAS REACTIOXS
1781
sive forces between atoms or ions with neon, argon, krypton, or xenon structures. This course is not open to us when we are dealing with molecules, or even with atoms that do not have a noble gas structure. I n our application we shall be especially concerned in finding the position of minimum potential energy and the value of the energy in this position. It is easy to see that, quite aside from any errors arising from the existence of non-central fields, we cannot calculate these quantities with any precision from the virial coefficients alone. Thus the pressure data for argon may be represented by n = 9, m = j, A = 101.0 X IO-?(, p = 1 6 2 . 0 X IO-^^ or by n = =, m = 5 , u = 3.13 x IO-^ cm., p = 70.4 X IO-(^. The first set of values gives for the distance of minimum energy ro = 4.94 X IO-^ cm., and for the energy u = -3.534 x 1 0 - l ~ergs. The second set gives ro = 3.13 X IO@ cm., u = - 18.34 X 1 0 - l ~ ergs. For nitrogen the uncertainty is even larger, and here it cannot be removed by the use of crystal data. Thus, in our applications to molecules we shall expect to use the virial coefficients only as a guide to the order of magnitude of u. The molecule in which we shall be most interested is nitric oxide, and we shall therefore proceed to apply the foregoing methods to it. The pressurevolume-temperature relations for this gas have been measured by Briner, Biedermann and Rothen,' for pressures between 30 and 160 atmospheres and temperatures between 9 and -78.6"C. These workers were interested in the possibility of the occurrence of (NO)? in the gas, but they concluded that there was no polymerization, since the nitric oxide did not depart as much from the perfect gas laws as did carbon dioxide or ethylene. They did not compute virial coefficients from their data, and indeed these are not well suited for that purpose; but since there is no other source of information about the intermolecular forces of nitric oxide, the attempt will be made to use these measurements. The pressures used are so high that we should expect that it would be necessary to go beyond the second virial coefficient, but it does not prove useful to do this, since the values which are obtained for the third and higher coefficients fluctuate violently with the temperature, and do not even show any regularity in sign. The procedure adopted is therefore to compute the ratio Bv/Av from the relative densities a t 30 atmospheres and at I atmosphere. The method by which this is done is evident. The results are given in Table I. We will consider as an example the model (14 1/3, 5 ) . For this particular model the expansion of F(y) has been given by Lennard-Jones2 in a convenient form. It is F(y) = y9/*' (1.1932 - 2.8857 y - 0.2667 y2 - 0.0017 y" - 0.0003 y6 -
-
0.0j02
')
~
Briner, Biedermann and Rothen: Helv. C h m . Acta, 8, 923 (1925). 2Lennard-Jones: Proc. Roy. Soc., lMA, 470 (1924). 1
y3 - 0.0095 y'
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LOUIS S. KASSEL
By the use of this equation the values of F(y) given in Table I1 are easily calculated. I n Fig. I the values given in Table I are plotted as points and those of Table I1 are shown by the curve, using appropriate scales. It may be seen that the
TABLE I Virial Coefficients for K O
T
&/A"
T
282
- 0 . 0 0 I 242 - 0 . O O I io9 -o.002042
2 I3
253
233
194.4
-0.002482 -. o ,003272
TABLE I1 log F(y) -0.587 -0.399 -0.261
F(Y) 587 -0,3993 -0.5485 -0.7061
Y
0.50
-0.2
0.jj
0.60 0.65
-0,151
points lie sufficiently well on the curve. From the relative position of the origins of the two coordinate systems we find
x
= 1,414
y = - 2.34 and the corresponding values of X and p are
X = 1.241x IO-^^' p = 2.008 X IO-^^
The position of minimum potential energy is given by ro = 4.919 x IO-^ cm and the energy in this position is u
=
-
5.97 X
ergs.
10-l~
I n a similar way we may work out the results for other models. The pressure data at our disposal extend over such a limited range of temperature that in no case is there any question of a model failing to fit the data; we can only be governed in the choice of model by analogy with other substances and by general considerations. For a rigid sphere with m = 5 we find u = - 1.83 X
10-l~
- 5.86 X
10-l~
and with m = 7 we find u =
ergs
ergs.
These figures are sufficient to show the general dependence of u on the values chosen for m and n; it is apparent that increase of either m or n results
THEORY O F THIRD-ORDER GAS REACTIONS
I783
in decrease in u. A rigid sphere has the largest possible value for n, and for m values greater than 7 are not generally used, so that the value of -5.86 X 10-l~ergs is in some sense a lower bound for u. But it is not certain that the actual value of u nil1 not exceed this limit; in other cases for which the data are more extensive the models have been tested by their ability to reproduce viscosity and compressibility data, and they have proved fairly satisfactory for these purposes; but there is no assurance that they are sufficiently correct so that reliable values of u can be found; in particular, there is some reason to believe that the attractive forces between molecules fall off exponentially
FIG.I
with the distance, at least at large distances; what effect this would have upon the values found for u is uncertain. We shall discuss this subject further when we come to make use of the value of u in a later section. The Number of Collisions in an Imperfect Gas Our next problem is to examine the effect of the intermolecular forces upon the collision frequency. We shall consider explicitly only the case of spherically symmetric molecules. I t is always important to specify a collision in the way that is suitable for the purposes of the problem being studied; we shall therefore mean by a collision an encounter in which the two molecules approach t o within some specified distance I~ such that the mutual potential energy is - q a:we will consider a collision between unlike molecules, since this case may easily be reduced to that of like molecules whenever this is necessary. We assume that the gas is so dilute and the temperature so high that
1784
LOUIS S. KASSEL
clustering has not yet become important. Now consider some collision for which the distance of closest approach is I , and the relative velocity at that distance is V. This velocity will necessarily be perpendicular to the line of centers. We can write down expressions for the total energy
E = +pV2- P(I)
(11)
and the angular momentum p = pV1 (12) where p is the reduced mass; both of these quantities are conserved, and hence if we calculate back to the configuration some time before the collision, when the molecules were separated by a distance x for which q ( x ) is negligible, the velocityvand its angle Owith the line of centers must satisfy the equations apv2 = ipV2 - P (1) (13) p vx sin 0 = p VI
(14)
These equations determine v and 0 in terms of V and I , so that if we can calculate how often the constellation (v, e) will appear we will obtain also the frequency of the collisions (V, I ) . But since the intermolecular forces are negligible at the distance x, we can use the perfect gas theory; the easiest way t o do this is to imagine the colliding molecules a t the centres of spheres such that uI2,the sum of the radii, is equal to x. Then the frequency of the collisions with relative velocity between v and v dv, and the angle of the d0 is relative velocity and the line of centers between 0 and 0
+
+
dZ = 8N1 Nz x2 6 (p/2kT)”/’ e -wz/zkTv3 sin 0 cos 0 dB (IS) Now there is surely but one pair (v, e) corresponding to any (V, I , ) , and this pair has the property that any other pair (v, e’) such that 10’1 < 101 will cor. for every v from o to 00 there respond to a (V’, 1’) such that I ’ < I ~ Thus is some 0 = Bo such that for all tl from o to eo, the distance of closest approach is 5 I - . This 0, is in fact given by
+
sin e, = (I,/vx) d v 2 2p0jp Hence the frequency of the collisions in which we are interested is
(16)
Z = 8N1 N2x2 .\/y ( p / ~ k T )~~ m ’ ~~ o o e - ~ v zvs/ sin z k 0Tcos 8 de dv
+
= 8N1 N2x2dF ( ~ / 2 k T ) (~~’ ~ 0 2 / x0 z ) ~ m e - ~(1v 2 ~2 z~ ko /Tp v ~ ) v 3 d v = 4X1NZ
dy ( ~ ( / 2 k T ) ~I,*”
[2(kT/p)*
+
+ 2kT (00/lr2]
= 2N1Nz d / a r k T / p [I cp,/kT] (17) The number of collisions is thus increased over the ideal number by a factor I
+ %/kT
(18)
which is identical in form with the Sutherland assumption’
u* = 1
am2‘(I
+ C/T)
Jeans: “Dynamics1 Theory of Gases,” p. 284.
(19)
THEORY O F THIRD-ORDER GAS REACTIOSS
I785
if we put Polk = C. We shall expect to use values of p0 of the order of 5 x 10-l* ergs, corresponding to a constant C of 350 which is about the usual value; the meaning of our equation, however, is quite different from that of Sutherland’s. This result applies only when -p0 is negative, so that the intermolecular force is attractive. When -cp0 is positive, the limits of integration for v in the equations leading to ( 1 7 ) may no longer be taken as o to 00 ; the lower limit must be d - - z c p 0 / p and With this limit we find that the number of collisions differs from the ideal number by a factor eVp,/kT
(188)
in which, of course, cpo is negative; the number of collisions is thus decreased. This result for the case of repulsive forces might have been more easily obtained by using equation ( 2 1 ) of the following section with r = I ~ together , with the theorem of kinetic theory that the distribution of velocities is independent of the potential field; (18a) could then have been written down a t once. But when the field is attractive this treatment would fail, since it would include in the collision number the vibrations of stable pairs. We expect that in applications to reaction rate problems we shall need the form (IS), but the possibility that (I8a) should be used must not be overlooked entirely. We must consider also to what extent our formula is applicable to an actual gas. The weakest point in the derivation is the expression for the number of collisions of the hypothetical spheres; this will surely hold if NI and NPare the number of molecules per cc. which are actually free from the fields of other molecules, and as long as these numbers are not too different from the total number of molecules per cc. the equation will be valid. When we have calculated the number of pairs actually present, which is our next step, we will be able to return to this point more profitably. If the molecules are not spherically symmetric, the problem becomes more complex; we cannot attack it successfully, and we can only hope that some equation like (IS) will remain true, where cpo has become an average potential.
The Number of Pairs in an Imperfect Gas We shall now consider the number of pairs in our gas; we will define a pair as a constellation of two molecules within the distance p o , and we will continue to assume spherical symmetry. According to Fowler’ when the concentration is not too high we may write for the chance of finding molecules a and 8 in the volume elements dw, and dwp, whose distance apart is r N, Np e-E(r)’kTdw, dwp
(20)
where N a and Ng are the numbers of molecules per cc of the kinds a! and 8, and E(r) is the potential energy of molecules of these kinds separated by the Fowler: “Statistical Mechanics,” p.
182.
I
786
LOUIS S. KASSEL
distance r. If we integrate this over a unit volume with respect t o dw, we will get h-o e-E(r; kT dwp and we may write now dwp = qnr2 dr
ra
Then the number of pairs per unit volume for which r 5 po is given by
p = ~TN:,NbSP;e-E'r)/kT r2 dr We are interested in the temperature coefficient of this quantity; it is
We shall expect that usually in our applications p o will be about equal to T o , the distance at which the potential energy is a minimum. Therefore, since E(r) increases very rapidly as r becomes less than ro, the essential contribution to t'he integrals in ( 2 4 ) will come from a narrow range near ro, for which E(r) is nearly constant, and, to a first approximation d In P/dT = E(r,)/kT*
(25)
Since E(ro) is negative, the number of pairs decrea\ses with an increase in temperature, as of course it should. We may also write an accurate equation resembling ( 2 s), namely d In P / d T = E/kT* (26) where E is the average potential energy of all the pairs which exist a t the on temperature T. We can see qualitatively what the dependence of temperature must be; a t low temperatures the molecules seek positions of low potential energy, but at relabively high temperatures they pay less attention t o the potential energy, so that will increase with T ; that is, it will be more negative at lower temperatures. This is a conclusion that we wish to emphasize; on the basis of the present assumptions it is inevitable. We can see further what the order of magnitude of d E l d T will be; clearly it must be 3 1 2 k since E represents the potential energy in three degrees of freedom. But since the corresponding Hamiltonian is not even approximately quadratic, this cannot be considered as imposing any definite requirements on dE/dT.
e
a
We shall now investigate the deviations from ( 2 3 ) and ( 2 6 ) Iyhich may be expected to occur when the concentration becomes too high. The number of ternary and higher clusters will then become appreciable, so that ( 2 1 ) is invalid, although a similar expression in which E is replaced by a suitably defined W will remain true. Without inquiring too closely into the transition region, we may be sure that in liquids, and in very highly compressed gases,
THEORY O F THIRD-ORDER GAS REACTIONS
I787
the deviation from ( 2 1 ) and therefore from (26) is in the direction of fewer pairs, since we feel sure that at high concentrations the distribution is sensibly uniform. We see that along with this decrease in the number of pairs there will go an increase in their average energy, which will thus become closer to zero; as a result, the temperature coefficient of the number of pairs will be less negative a t high concentrations, and the deviations from the predictions of (23) will be least a t high temperatures and greatest a t low. We can see from (23) that the chance for any selected molecule of the kind a to participate in a pair is
p = 4,1\’gJPoe-EW/kT 0
rz dr
(27)
The condition for the number of ternary complexes to be negligible may be taken as Pa
