3182
NOTES
+ o*
&Oq+-Lnz
+0 > At, i # j
(13)
then absorption data caused by mechanism 5 will fit eq 11. Table I shows that approximation 12 is not fulfilled. According t o 13, the inverse of the measured relaxation time would be proportional to that particular eigenvalue whose relaxation strength is dominant; the T h e Journal of Physical Chemistry, Vol. 76, No. 30,1971
proportionality factor is equal to IC-, (see eq 6) and is independent of alcohol concentration. The eigenvalue which fulfills this criterion best is X I , the numeri(10) J. Rassing and F. Garland, Acta Chem. Scand., 24, 2419 (1970). (11) H. C. Van Kess, J. Van Winkle, H. H. Richtol, and H. B. Hollinger, J. P h y s . Chem., 71, 1483 (1967). (12) L. J. Bellamy, R. J. Morgan, and R . J . Pace, 8Z;cctroehim. Acta, 22, 526 (1960).
NOTES
3185
cally largest eigenvalue. The resulting value of k-, is 1.2 X 108 sec-’. Figure 2 shows that the mechanism used here describes the ultrasonic results up to about 0.2 M . The common depolymerization rate constant, k-,, obtained from this mechanism is a factor of 2 larger than the value of k-1 which can be obtained from ultrasonic data alone by means of the extrapolation lim
( 1 / ~ )= ICdl
[At1+0
Considering the approximations involved in the mechanism and the uncertainty involved in the extrapolation, we consider this to be good agreement.
transition probability increases with increasing collision velocity v for b less than the zero potential distance u, while for b > u it decreases as v increases.2 Furthermore, it is thought that the probability normally decreases with increasing b a t a given v. However, the v and b dependences of vibrational transition probabilities are much more complicated than those predictedqa I n this paper we study the dependence of the rotationaveraged transition probability on b at a given v by use of the sudden appr~ximation.~We shall use the same terminologies and collision model presented in ref 3b; see Figure 1 of ref 3b for the collision model and definitions of the collision coordinates. For numerical discussion, we consider 02 Ar. According to the sudden approximation, the probability of the m -t n vibrational transition is4tG
+
Table I1 : Ultrasonic Relaxation Parameters Resulting from Fitting t h e Data to E q 11 by Means of a Least-Squares Fitting Procedure [ A 11, M
A x 1017, seca om-1
0.0467 0.0752 0.1144 0,1928 0.3128
108 165 195 175 163
P,,(v,b,e) = (nlexp 12idv,b,e,t)lIm)2 sec
B x 1017, secs cm-1
0.743 0.556 0.415 0.287 0,217
184 183 183 200 190
TO
x
108,
At alcohol concentrations above 0.2 M the predicted value of 1 / is~smaller than the actually measured value. Contributions from other relaxation times, polymerpolymer interactions, polymer-solvent interactions, or the breaking down of the equilibrium description itself might cause the observed deviation. Acknowledgments. The authors are deeply indebted to Professor Thor A. Bak for his constant interest and helpful advice in connection with this work. The authors acknowledge the support of ARO (Durham) under Grant DA-AROD-31-124-G818 for the part of the work carried out at Maryland.
Excitation of Molecular Vibration on Collision. Dependence of the Rotation-Averaged Transition
(1)
where the phase shift is 1
/.-
and where 6 = vibrational coordinate, 0 = orientation angle (see Figure l of ref 3b), V = interaction potential energy between BC and A, r = distance between the atom and the center of mass of BC. We use the linear trajectory approximation; then we haveab r2 = b2 z2 and z = ut. The assumed form of the interaction potential between the collision partners is the inverse-power law
+
2
U(r1,rz) = 2OCI(u/r,)’2 - (u/rJ61 i=l
(3)
Here the atom-atom distances are r1,2
=
+ t)X2,1r cos 0 + (d + mB,C/(mB+ mc) and d is the bond dis-
[r2 F 2(d
[)2S2,12]”3
where Xl,2 = tance of BC. After introducing these distances, eq 3 can be expanded in a power series to obtain approximately the &dependent perturbation energy p,s6 [v,b,@,t,r(t)1 =
Probability on the Impact Parameter’”
by Hyung Kyu Shin Department of Chemistry,Ib University of Neoada, Rem,Nerada 89607 (Received April 92, 1971) Publication costs assisted by the A i r Force Ofice of Scientific Research
One of the pressing problems of vibrational energy transfer in a three-dimensional collision is the dependence of vibrational transition probabilities on the impact parameter b. Simple analyses predict that the
(1) (a) This work was carried out under Grant AFOSR-68-1354 from the Air Force Office of Scientific Research; (b) Theoretical Chemistry Group Contribution KO.6-1032. (2) T. L. Cottrell and J. C. McCoubrey, “Molecular Energy Transfer in Gases,” Butterworths, London, 1961, Chapter 6. (3) (a) H. Shin. Chem. Phys. Lett., 7, 436 (1970); (b) H. Shin, J. Phy.3. Chem., 75, 923 (1971). The Journal of Physical Chemistry, Vol. 76, N o . 20, 1972
