DOYLEBRITTON
1464
Vol. 63
ON THE EQUILIBRIUM CONSTANT FOR LOOSELY BOUND MOLECULES BY DOYLEBRITTON S'chool of Chemistry, University of Minnesota, Minneapolis 14, M i n n . Received February 23,1969
Bunker and Davidson, and Stogryn and Hirschfelder have recently described calculations of the equilibrium constant for loosely bound molecules using the methods of classical statistical mechanics. Alternative approximate methods of calculating this equilibrium constant using quantum mechanical partition functions for the species involved are described here.
I n a recent paper,l Bunker and Davidson have proposed a theory to account for the observed rates of atom recombination reactions. The reaction in question is X + X +M+X2 =
dt
+M
(1)
If er is small compared to LT, as is true here, then the sum may be replaced by an integral.
k~(M)(x)z
To allow for the possibility of homonuclear moleEquation 2 defines the rate constant for recombina- cules, QPas given in equation 7 should be divided by tion. The theory describes the dependence of k~ the symmetry number of the molecule CT. Howupon the temperature and upon the third body, M. ever, it should be noted that the rotational quanThe essential features of the theory are that inter- tum number K cannot increase indefinitely since it mediate molecules, MX, are formed from the third is unreasonable to allow the rotational energy body molecules and the atoms in question, that to be greater than the binding energy for MX.3 these molecules are present in their equilibrium con- This means that there is a maximum value of K , centration, and that the equilibrium constant for given by the formation of these molecules can be calculated K m (K, 1 ) e r = 60 (8) by the methods of classical statistical mechanics. Using the value of K as an upper limit to the inteAlternative approximate methods of calculating this equilibrium constant using quantum mechani- gral in equation 7 yields the correct expression4 (in cal partition functions for the species involved are which the symmetry number has been included). described here.2 The equilibrium constant for the formation of MX The vibrational partition function for a harmonic M + X S M X K = - -(MX) (3) oscillator is
+
(M)(X)
m
is given by the expression
Qv =
e-rv(u+l/2)/kT
(10)
v=o
K
=
&MX e w / k T
QYQX
(4)
which can be summed directly to give
where Q is the partition function of the species in question and eo is the binding energy of the molecule MX. If it is assumed that M is a monatomic spe- Again, however, t,he quantum number v cannot incies, then the partition functions QM and QX in- crease indefinitely. There is a maximum value volve only translational states. For the partition given by function QMX, rotation and vibration must also be e" (v, 1 / 2 ) = EO (12) considered. It is in the evaluation of QMX that the If this upper limit is placed on the sum in 10, then three approximations differ. e-cv/Zkt - e-(ro+cv)/kT -~ First Approximation : Rigid Rotor, Harmonic (13) Qv = 1 - e-rv/kT Oscillator.-The translational partition function is If E" is small compared to €0 and IcT, then
+
Q~ =
(2q)8'z molecules/cc.
(5)
The rotational partition function for a rigid rotor is usually written (1) D. L. Bunker and N. Dayidson, J . .4m. Ciirm. Soe., 80, 5090 (1958). ( 2 ) This problem has also been considered by D. E. Stogryn and J. 0. Hirschfelder ("The Contribution of Bound, Metastable and Free Molecules to the Second Virial Coefficient," WIS-ONR-32a, N7onr28511, 4 November 1958) who have carried out a more complete calcrilation by the methods of classical statistical mechanlcs. The first two approximations given here more closely parallel the treatment of Bunker and Davidson; the third, that of Stogryn and Hirschfelder. I aould like to thunk Th. Himohfelder for making his work availahle to
me.
Qv g
!!.? (1 - e-ra/kT) EV
(14)
I n order to evaluate equation 4 for the equilibrium constant, we need to know €0, €,and e". The first of these can be obtained from virial coefficient data's5 which also give the distance of closest ap(3) This corresponds to the crucial assumption of Bunker and Davidson that the integral over phase space should include only those parts of momentum space which correspond to a bound molecule. (4) I n the cases usually encountered ro>>kT so t h a t the term in parentheses is equal to unity. This is not the case here, however. ( 5 ) J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, "Molecular Theory of Gases and Liquids," John Wiley and Nons, New York, 1454, pp. 1110-1112.
EQUILIBRIUM CONSTANT FOR LOOSELY BOUNDMOLECULES
Sept., 1959
proach, re. (Both eo and rs were obtained assuming a Lennard-Jones potential function describes the interaction between M and X.) The quantum of rotational energy
1465
tor of two. However, this is balanced, somewhat, by another approximation, that the vibrational energy levels are evenly spaced. If the Morse function is used, the correct distribution of vibrational states is8 e(v) =
E"(V
f
1/z)
- 4E.60" (v f
'/2)2
(21)
where p is the reduced mass of the molecule and ro This leads to just twice as many levels as the har(which is equal to 2'/6rs) is the equilibrium internu- monic oscillator does, so that Q, is underestimated clear distance. Rather than use the Lennard-Jones by a factor of two if the vibrational excitation is esfunction, it is more convenient to use a Morse po- sentially complete, as is the case in these weak comtential function plexes. At high temperatures the partition function for vibration and rotation depends only upon E(r) = a o [ l e41-~/~0)]2 (16) where CY is a parameter which determines the shape the total number of states available to the molecule, of the curve. The Morse curve can be adjusted to while at lower temperatures the distribution of these be roughly the same shape as the Lennard-Jones by states also matters. It is therefore reasonable that choosing a suitable value of CY. If the behavior of the derivation described here gives an answer which the molecule is described by a Morse function, then6 is close to correct a t high temperatures and which is poor at low temperatures. E" = 2CY(E0Er)'/Z (17) The above arguments as well as those that follow The results in equations 5 , 9 and 14 can now be also apply to complexes between molecules if one assumes that the complexes are loose enough so that substituted into equation 4 the two molecules are free to rotate within the complex. I n this case all the internal parts of the partition functions of the separated molecules will also This can be simplified further if the exponentials are appear in the partition function of the complex and expanded into series and the various series are com- will cancel out. bined. Second Approximation : Rigid Rotor, Morse Oscillator.-In this approximation it will be asK = 4as/2 UCY Tos ( $'/*[I o . sumed that the rotational energy levels are those of the rigid rotor, the vibrational levels are those (19) This may be compared with the result of Bunker for a molecule that is described by the Morse potential function, and that the total internal energy and Davidsonl cannot exceed the binding energy for the molecule. This last restriction means that the rotational and vibrational partition functions may not be summed Since r03 = 2'/2rs3, the terms outside the series are independently. As usual the sum over the rotathe same if CY = 37r/2/2 = 6.67. For comparison: tional states is approximated by an integral. if CY is evaluated by matching the curvature of the Morse curve at ro to that of the Lennard-Jones a t To, then CY = 6; if CY is evaluated by making the Morse potential have a value of EO a t r = rs, then CY = Zn2/(1 - 2 - 9 = 6.38. It can be seen that both expressions show the same temperature dependence a t high temperatures and have about the same abso- The upper limits indicated are given by lute value. At temperatures where eo kT,equaVm 2 0 -1 tion 19 gives a poor picture of the temperature de€" z pendence. 1 (V 1/2)2] The separation of the vibrational and rotational K d K m 1) = -er [eo - €42) l / 2 ) parts of the partition function involves two approxi(23b) mations.' The first of these is the familiar one that the energy can be expressed as a sum of terms The integral in equation 22 can be evaluated exinvolving the rotational quant,um number only and actly. The resulting summation can be performed terms involving the vibrational quantum number by expanding the exponentials in power series and only, and that there are no cross terms. This is summing the individual terms. The result is discussed more fully in the third part of this paper. [l - 0.700 2'- + 0.271 The second, which is considered in both of the other &," = kT parts, is that the limits to be taken in summing the 0.074 . ..I (24) separated partition functions are independent,. This is clearly not true and is a t first glance a poor assumption since a t high vibrational levels we are Each of the coefficients in this power series should The constant terms clearly counting too many rotational states. A be a finite power series in E J E O . rough estimate would be that Qr is too large by a fac- are those given, the terms in + / E O are all exactly zero and the terms in ( e v / ~ o ) z and higher only contribute (6) A. G. Gaydon, "Dissociation Energies," Dover Publications, a per cent. or so and are omitted.9 Inc., New York, N. Y., 1950, p. 29. (7) I would like to thank Professor N. Davidson for a clarifying (8) G. Herzberg, "Spectra of Diatomic Molecules," 2nd Ed., D. discussion on this point and also for some good advice. Van Nostrand Co., Inc., New York, N. Y., 1950, p. 101.
-
+ ($ + (g)'..]
-
+
sv
+
+$ +
(gj3
DOYLE BRITTON
1466
If the value of Qrv just obtained is used to evaluate the equilibrium constant, then
yt
K=-r3
n
($,)*" [l + 0.300 i ,+ + 0.014 (g)' . . .] (25)
This may be compared with equations 19 and 20, and with the result of Stogryn and Hirschfelder2 IC
450
($)"' [l + 0.254 kT2 + 0.057 (i;)'+ 0.011 (5)'. . .] (26)
r.3
The leading term in equation 25, which corresponds to the high temperature limit, is about 25% smaller than the result of Bunker and Davidson, and about 30% smaller than the result of Stogryn and Hirschfelder. The power series which gives the change in the temperature dependence a t lower temperatures is about the same as both of their results.
Vol. 63
Pion of the centrifugal potential gives rise to the possibility of metastable molecules; that is, molecules with total internal energy greater than eo but which can only dissociate by leakage through the barrier caused by the centrifugal potential. Stogryn and Hirschfelder have obtained equilibrium constants for both the bound and metastable molecules by integrals over classical phase space. Similar calcrilations will be made here in terms of thc quantum inechanical partition functions. The wnt.e equation has been solved for a molecule with the potential energy given by equation 27 by llorsc,llwho assumed that r remained constant a t 1-0 in the second term in the potentid energy expression, and by Pekeris12who obtained a more exact solutiou in the neighborhood of the minimum of the potential. Both of these results give the energy of the molecule as having the following relationship t o the quantum numbers K and v.
K(K
+ I) + 2 K ( K + 1) - s 9 K ~ ( K+ i y EO
(28) 2
Morse found y = l/z and Pekeris found y = 3(a 1)/2a2 = 5/24 for a = 6 . Both found 6 = ~ / Q Z= 1/36. In the calculation given here, y = 2/5 and 6 = 1/30 will be used, for reasons which will be developed below. I J .Fig. ~ 2 the values of the maxima and minima in the potential energy curves of Fig. 1 have been plotted as a function of erR(K l)/eo. The value of the energy a t the minimum should be given by equation 28 with (v i/z) set equal to zero. The fit here at high energies is much improved if 6 = 1/30 is used rather than 6 = 1/36. For the ground rotational level, the maximum value of the vibrational quantum number v occurs at the point where de/bv = 0. If this same criterion is chosen to find the maximum value of v for excited rotational levels, and this is a convenient but purely arbitrary choice to make, then y = 2/5 gives
+
c
2 1
+
+
0
2 ' 3 4 rho. Fig. 1.-The Morse potential function plus the centrifugal potential for various rotational states. From the bottom the values of s K ( K l ) / are ~ ~ 0, 1/2, 1, 3/2, 2 and 2.215. The last is the highest rotational state for which the potential function shows a minimum.
0
1
+
Third Approximation : The Potential Function of the Molecule Given by the Morse Potential Plus a Centrifugal Potential.-As Stogryn and Hirschfelder have pointed out12a calculation of an equilibrium constant of this nature should be based on a potential function which includes the centrifugal potential for all rotational levels above the ground level. If this were included with the Morse potential given in equation 16, the proper potential function for the K t h level'o would be
(vmsx
+
'/z)~v
(2K
= v
(9) If the sum over rotational states is replaced by an integral, as in equation 22, then the order of the sumination and integration must be considered. The order given is correct. If the summation over v is performed first (with the appropriate changes in the limits) then errors of the order of €"/eo are introduced in each of the coefficients in the series. (10) The Kth level is that level where the square of the total angular 1)&2. momentum due to the rotation of the molecule equals R(K The energy must be determined for this state by solving the radial wave enuation.
+
4
5 &(K
+ 1)
(29)
which, when substituted back into equation 28, gives a good approximation to the position of the maxima, as Fig. 2 shows. It should be noted that this choice for the upper limit of v gives the maximum possible number of vibrational levels. The partition function that we wish to evaluate is QrV
This potential function is pictured in Fig. 1 for a Morsepotential with a = 6, and for various values of K . As inspection of this figure will show, the inclu-
= 2 ~ 0-
+ l)e-+J)/k*
(30)
K
where E(U,K) is given by equation 28. This is evaluated by completely espanding the exponential, replacing the sum over K by an integral, integrating between limits which depend upon v, and finally summing the individual terms of the expansion over v W 9 These sums are tedious to obtain, the more so as the exponent of eo/lcT becomes larger, but there are no unusual features involved in the summations so the details will be omitted here and only the final answers given. (11) P. M. Morse, Phya. Rev., 34, 57 (1929). (12) C. L. Pelteris, ibid., 46, 98 (1934).
Sept. , 1959
EQUILIBRIUM CONSTANT FOR LOOSELY BOUND MOLECULES
1467
Equation 30 has been evaluated first over all energy levels where e ,< eo. This corresponds to the bound molecule and gives 2
+ 0.292 (k!l')' 0 + 0.082 (si)3 . . .] (31) This in turn gives 3.378 K = -ro3
(5) [l + 0.265 5 + 0.057 (5)' + 0.009 (2)' ...] (32) 812
These are both assuming a = 6, y = 2 / 5 , 6 = 1/30, and with each coefficient evaluated to about 1% accuracy. The individual coefficients should be finite power series in E ~ / E O but the first power terms are all zero and the higher terms have been omitted as contributing 1%or less to the total. Equations 24 and 25, which were special cases of this solution with y = 6 = 0, gave about the same result. The leading term in equation 32 agrees to within about 1% with that of Bunker and Davidson (equation 20), and is about 6% less than that of Stogryn and Hirschfelder (equation 26). The series are almost identical.l8 Equation 30 also has been evaluated over all energy levels with energy greater than €0 but less than the height of the centrifugal barrier, that is, over the region in Fig. 2 indicated as the metastable region. Qrvm
590e02 [l
-
1.359
U€,€V
mrtsrta blc
1
x bound
0
ho/ccukS
0
molecules
1 2 sK(K l)/eo. region of summation for the vibrational-
+
Fig. 2.-The rotational partition function: heavy line, limits as given by the potential curves in Fi 1; dashed line, limits given by e uation 28 wit,h y = 275 and 6 = 1/30. Li h t line, bounlary between the region of bound molecules an8 that of metastable molecules.
.
2 -+ - 0.475 ($)*. . .] (33)
0.965
0.106
B
7;.
2
(ki)a
+ 0.023 ($)a.. .] (34)
This may be compared with the result of Stogryn and Hirschfelder
0.032 (j$)3
- 0.005
(i;)'.. .] (35)
In calculating Km the assumption was made here that the energy levels for the metastable molecules were given by the same equation that gave the energy levels for the bound molecules, and that there was no leakage through the barrier to dissociation. If these assumptions are made, it appears intuitively correct that the partition function for vibrat,ion and rotation should have the same temperature dependence a t high temperatures as the partition function for the bound molecules, since in both cases the partition function approaches the total number of vibrational-rotational states. This is the result that mas obtained, and a t the high temperature limit the equilibrium constant for the formation of metastable molecules is slightly greater than that for stable molecules. The relative importance of the metastable molecules decreases at lower temperatures. This is essentially the same result as that of (13) It should be mentioned t h a t Stogryn and Hirschfelder give the coefficients for the first eight terms in this series and also for the first eight terms in the series for the metastable molecules.
-8
I
-4
I
1
-2 0 2 log ( ~ o / k T ) . Fig. 3.-The equilibrium constant for bound molecules as a function of temperature: solid line, value a t high temperatures given by equation 32; dashed line, value at low temperatures given by equation 38.
Stogryn and Hirschfelder except that the numerical agreement is much worse than was the case with the equilibrium constant for the bound molecules. Further Discussion.-The calculations made above for the bound molecules all agree quite well with the results of the classical statistical mechanical derivations at temperatures where Ea < kT, and have the advantage that they use functions which are in general more familiar. There is one additional advantage in this approach which appears when an attempt is made to carry out similar calculations which will be good at lower temperatures.
1468 P. D. MERCER AND H. 0. PRITCHARD Vol. 63 If eo < kT, then the high rotational and vibrational where this has been put in a form to resemble equastates become unimportant, and the upper limits for tions 19,25 and 32. Figure 3 shows K as calculated the sums can be taken as infinite with very little by equation 32 displayed with K as calculated by error. I n this case, the expressions for the partition equation 36. It can be Seen that they join quite functions are the usual ones, given in equations 7 smoothly in the neighborhood Of = k T * Acknowledgments.-I wish to thank Professors and 11, and the equilibrium constant is 2 2. Hugus and Rufus Lumry for helpful discuse(ro-rv/2)/kT sions. This work was sponsored in part by the k ' 2 TS/p (1 - e-w/KT) (36) Office of Ordnance Research, U. S. Army, 3u
(%)"*2
THE GAS PHASE FLUORINATION OF HYDROGEN-METHANE MIXTURES BY P. D. MERCER AND H. 0. PRITCHARD Chemistry Department, University of Manchester, Manchester I S Received February
1969
A study of the reaction between fluorine gas and hydrogen-methane mixtures has been made over a temperature range from 25-150" in both inconel and quartz reaction vessels. The kinetics are complicated by surface effects but may reasonably be inter reted in terms of a chain mechanism in which the rate-determining steps are F HZ --c HF H (1) and F CHI -+ %F CH3 (2) with E, - Ez = 0.5 f 0.2 kcal./mole.
+
+
The fluorination of a hydrocarbon such as methane may be expected to take place by one of two mechanisms. The first, which is analogous to the chlorination and bromination reactions, may be represented by the scheme F2+F+F
+ CHI+ H F + CH3 CH3 + Fz +CHIF + F
F
( 1)
(2) (3), etc.
Steps 2 and 3 are both highly exothermic and should have small or zero activation energies. Because of the extreme reactivity of fluorine and because of the uncertainty over chain ending mechanisms a t the walls and by impurities, such a reaction is most conveniently studied, not in isolation, but in competition with another similar fluorination. The second possibility, which was considered unlikely, arises because the process CH4
+ Fz +CHz + ZHF
(4)
is exothermic; our evidence will show that it does not seem to be important. Experimental
+
+
when VS and Va were closed; the fluorine was compressed into CZusing the piston, and VZagain closed. After coming to the thermostat temperature, VI was momentarily o ened to allow the COZ-F2mixture to expand into CIand t f e reaction was initiated by illumination with a medium pressure mercury lamp. After a period of 0.5 to 2 hours, the vessel was reattached to the vacuum system and the contents of CZand CSwere pumped away; the contents of C1 were then pumped through absorption tubes containing NaF pellets and CaCl2-KI mixture (to remove, respectively, HF and any unused Fz), through tt li uid nitrogen trap (to remove COS, fluorocarbons and anypz), the remaining Hz-CHI mixture being pumped into a second sample tube (taking care to avoid differential pumping).' The relative ?ate of reaction of fluorine with hydrogen and methane was found by a direct comparison of the initial and final mixtures on a mass-spectrometer. Provision was also made for fractional distillation of the contents of the liquid nitrogen trap in order to find the nature of the fluorinated products. In later experiments, a Pyrex system with a quartz reaction volume was constructed on the same principles (except that the transference of Fzfrom C3 to CZwas effected by expansion only) with the three sections separable from each other; the stopcocks and joints were lubricated with Kel-F vacuum grease without any undesirable effects.
Results Experiments in the Metal Reaction Vesse1.I n planning the experiment, it was thought desirable to These may be divided into two phases, the earlier guard against any possibility of ignition (perhaps wall- one where the reaction occurred spontaneously in catalyzed) upon mixing the fluorine with the hydrocarbon the dark, and the later one where photo-initiation mixture. The apparatus was therefore designed so t,hat the final reactant mixture should consist of about 1 mm. of Fz, was necessary. Phase 1.-Initially, qualitative experiments were 3 mm. of Hz, 3 mm. of CHI and 350 mm. of inert gas (Nzor COz). The arrangement of the metal reaction vessel is performed to find out the nature of the reaction shown diagrammatically in Fig. 1 . C1 and CI represent two cylindric31 inconel chambers of 150-ml. capacity each ; products between Fz and CH4. The first run gave one end of C1 was fitted with a transparent polychlorotri- mainly CH3F with some CHzF2, but after about six fluoroethylene (Kel-F) window and a light-tight shutter. runs, the product of fluorination of methane was alV1, Vp, V3 and V4 represent needle valves, V3 carrying a most completely CsF6; the products from the insuitable oonnection for attachment to the vacuum system tervening runs revealed a steady transition from the and V4 carrying one suitable for connection to the fluorine generator. The volume C3 consisted of a steel cylinder of expected to the unexpected behavior (among the about 1-ml. capacity, fitted with a piston P which could products being CF3H, CFI, GF,, CzF4Hz and sweep the contents of C3 through Vz. CzF6H in varying proportions). All subsequent An individual run was performed in the following manner. runs gave the same reaction product, Le., C2FB deA mixture of Hz and CH4 was made up and led into the reaction vessel via Va, giving a total pressure in C1 of about spite a threefold variation in the pressure of each 0.5-1.0 cm.; VI was closed and the contents of CZ only component of the reaction mixture. The addition were pumped into a sample tube to provide a standard of oxygen to the reaction mixture had no effect and against which to compare the reaction products. Volume Cz in some quantitative experiments, the relative rates was filled to approximately 1 atm. with COZ and V2 closed. The system was then attached to the fluorine cell and C3 was flushed out with FZ(freo of HF1) for about 30 minutes,
(1) It should be noted that electrolytic fluorine always contains some oxygen, and there is no convenient method of separation.
