E. J. McAldutf St. F. X. Unwerslty Antigonish, Nova Scoiia B2G 1CO
An Introduction to Collision Theory Rate Constants Via Distribution Functions
In many physical chemistry textbooks (1-5), the simple collision theory is presented by counting the number of collisions made between rigid spheres in the gas phase and modifying this collision number by the fraction of collisons which possess the required energy for reaction. This fraction is often indicated to he exp(-EalRT) without further reference to its source and without reference t o the more general activation in many degrees of freedom rate constant. In a senior level chemical kinetics course the author develoos the topic as presented in this paper 11yarriving at the many rleerees of freedom rnte nmstant and showine that the ( - E a..t RT) is a special case of the former and corresponds to activation in 2 squared terms or the line of centers rate constant. The approach used to develop these expressions is through the use of distribution functions. A variety of distribution functions e.g. velocity, speed, momentum, and energy in one and three dimensions are considered and the process of obtaining average values from them is reviewed. Many of these topics are presented in physical chemistry textbooks (1,5), or in kinetic theory of gas texts (6).These distribution functions take the form of eqns. (1)and (2) for the one dimensional velocity distrihution F(u) and the three dimensional speed distribution F(c), respectively.
Resolution of velocities of wlliding molecules into wmponents parallel (11) perpendicular (1to)line-of-centersof collision pair. The speed c is defined as the positive square root of the sum of the velocity components squared and the coordinate system has been changed in eqn. (2) to spherical polar coordinates. Equipartition of Energy
The equipartition of energy principle states that if the energy of a collection of molecules may be written as a sum of squared terms in coordinates and momenta then each squared term contributes %kT to the total average energy of each molecule. The distribution function approach is used to show how 'hkT is the average energy contribution from one squared term. Recalling that eqn. (1)represents the fraction of molecules having velocity in the interval between u and u du, then the one dimensional speed distribution function will be twice the velocity distrihution function since the fraction of molecules having speed in one direction between c, and c, + dc, contains both molecules with velocity in the range u to u du and those with velocity in the range -u to -u - du. The one dimensional speed distribution function can be converted to a one-dimensional energy distrihution function by using the fact that E l = %mc2, and hence d E l = mc,dc, or dc, = dE1/(2mE1)1/2. Substitution of these relationships into twice eqn. (1)gives the one dimensional energy distribution function (3)
distrihution function by the energy, El, and integrating over all energy values El 112 g =J ~ E I F ( E I ) ~ E=I J m (m) exp(-EdkT)dE~ (4) This integral is a standard integral of the form given by eqn. (5)
where
+
+
-
W E= F(E),dE1= 2 -' I 2 exp -EllkTdE~ N (2rrmkT) (2mE)1'/~
(3) whirh reprewnts the fraction of molecules with enerw in the ranae hetween El and El + dEl distributed in unr squared term. Using the previous definition of average values, the average value of this energy is obtained by multiplying the
and
I-(LZIZ) = YZ fi Thus eqn. (4) integrates to -
E =
(
1
112
1 2
(kT)3'2- (a)112= %kT
(6)
Equation (6)shows that the average energy in one squared term is KkT. If the energy were expressed as the sum of 3 squared terms the average contribution would be 3/2kT. Since each translational and rotational degree of freedom have enerev which deoend uoon the sauare of a linear or rotational v e k t y then the averke contrilhtion to the total energy from each translational or rotational depree of freedom is %kT. On the other hand, the vibrational energy depends upon both a kinetic and ~otentialenerpv term (harmonic oscillator model). Thus the vibrational energy is written as the sum of two squared terms, one velocity and the other position, and each vibrational degree of freedom makes an average contribution to the total energy of kT. In general if the total energy of a molecule is written as a sum of n squared terms as in eqn. (7), then Volume 57. Number 9, September 1980 1 627
+ +
E = %rn(u12 uz2 us2+ . . . un2) (8) the prohahility distrihution function for distributing the energy among the n squared terms is given by
+
w=-
uz2 + . . . exp - ( u ~ ~ (2?rmkT)n/2
du~duz. . . du, (9)
In the three dimensional case the probability function is multiplied by the volume of a spherical shell to ohtain the fraction of molecules in a given range. Analogously in the ndimensional case the probability function must he multipljed by an n-dimensional shell. If the variahle r n 2is suhstituted for u12 uz2 . . . un2the volume of an n-dimensional hypersphere becomes (7)
+
+
or the volume of a shell in n-dimensional space becomes
. The ahove equation is based on the general relationships concerning the gamma function (8). Combination of the probahility function W with the volume element dV, gives the probability distribution function (12). (rn/2skT)"/2e - r n 2 / a 2 2sn&,,-L dr,, (12) r(nA) Since the energy is a sum of n squared terms then En = 11zmr,2, the substitution of energy expressions for r, can he made using the fact that r, = (2E,)1/2/m W,.
=
and
This results in the energy distrihution function F(E,) dE, which is given by eqn. (13)
(13)
This expression can he reduced to 1 E 1"/2)-1
dE" exp(-EJkT) (14) kT Equation (14) represents the fraction of molecules with energy dE, distrihuted among n squared between E n and E, terms. Integration of this expression over all values of the energy should give account of all the molecules. However, in terms of chemical kinetics the quantity frequently of interest is the fraction of molecules ahove some critical energy E who have energy distrihuted among n squared terms. Integration of eqn. (14) from E, to will give the fraction of molecules which lead to reaction F(E")dE" =
r i a (6) +
-
- . .
..
.
Evaluating eqn. (15) by successive integration by parts results in eqn. (31) (9)
. . In eqn. (16) the ratio of any term to the previous term is E,l rkT. Therefore, provided E, >> nI2 k T all terms of eqn. (16) may be rejected except the last to ohtain the expression
(17) The condition for this approximation is sometimes erroneously given as E, >> kT. This fraction when multiplied by
628 1 Journal of Chemical Education
the collision frequency gives the activation in many degrees of freedom rate constant. Application to Kinetics When two atoms form a diatomic collision complex, the two atoms initiallv Dossess a total of six translational degrees of freedom. 'l'h' &mplex had only three translarional.degrees of freedom, hut there are two mtalional degreesof freedom and one vihrational degree of freedom. Thus, three of the translational degrees of freedom of the colliding atoms become the two rotations and the vibration of the complex. The rotations may he considered to arise from the two mutually nernendicular comoonents of the relative velocitv" oeroeu.. dicular to the line of centers while the vibrational degree of freedom mav he associated with the comoonent of the relative velocity parallel to the line of centers. Figure 1 shows the resolution of velocity into perpendicular and parallel components for two dimensional motion. For motion in three dimensions there is one additional perpendicular components which is perpendicular to the plane of the paper. The average energy of the two translational degrees of freedom of relative velocity perpendicular to the line of centers is k T ('12 k T for each degree of freedom). Whereas one might expect the average energy of the one translational degree of freedom of relative velocity parallel to the line of centers to he S/z k T actually the average energy for this degree of freedom is kT. The reason for this degree of freedom is that the collision requirement selects out the most rapidly parallel moving atoms. These make a contribution to the total numher of collisions which exceeds their simole . .roo.or ti on in the eas. An analogous result is obtained it' one rounrs the numh& of collisions made with the walls of R container hv. .vartirles with a distribution of velociti~s,i.e.. the Caster moving partirles makc! a larwr numher of collisions with the walls. Thus, k?, the average energy for the translational degree of freedom of relative velocity parallel to the line of centers, is equivalent to having energy distrihuted in.2 squared terms. If n = 2 is suhstituted in eqn. (17) B(E) becomes exp (-E,l kT). Therefore, the more common exp (-EJkT) term in collision theory rate constant expression is one specific case of the more general activation in many degrees of freedom rate constant. I t corresponds to the case where the critical energy E, is distrihuted in the translational degree of freedom of relative velocity parallel to the line of centers of the colliding atoms and hence forms part of what is known as the line-ofcenters rate constant. For n >2 contributions of internal degrees of freedom act to increase k. This increase in k is by a factor of
. .
&($Y-'
relative to the line of centers rate constant. This change represents an increase in rate constant by a factor of 1.14 X lo4 for n = 12 and a critical energy of 10 kcal at room temperature and afactor of 3.66 X 105for n = 12 and a critical energy of 20 kcal. Literature Cited (1) Maroo, S. H., and Laodo, J. B.. "Fundamentals of Physical ChemiAzy: MaeMillan. New York. 1974. (2) Sheehsn. W. F.."Physieal Chemiatry"2nd Ed.. Allyn and Bseon Inc.,Boston, 1970. (3) Castellsn. G. W.. "Physical Chemistry" 2nd Ed.. Addiaon-Wesley, Reading. MA, 1811.
(4) Andre=,
D. H., "lnhoduetion to Physical Chemistry," McCraw-Hill, New York.
1970.
Chemiatzy: 5thEd..John Wiley 61 Sans. New York, 1979. (6) Golden, S., "Elementr of the Thwry of Ganes: Addiaon-Wesley, Reading, MA. 1961. p. 92. (7) Kaufman. E. D.,"AdvandConeeptr in Physical Chemism: M f f i r a ~ - ~NmYork, l, 1966. p . m . ( 8 ) For a tableoferpnsaionsfor thegsmmsfunetiona~Benaon. S. W.,"TheFovndation of Chemical Kineti-," McGraw-Hill. New York, 1960, Appendix B, p. 660. (9) Fwler,R.,andGwonheim, E. A.,"StatisticalThormdynami~"C~bR~eUnlvasity Press, 1960. p. 496. ( 5 ) A1berty.R A., Daniels F8mington;'Physical
