VIBRATIONAL TRAN~IITIONS IN BC
+ A COLLISIONS
923
Dependence of Vibrational Transition Probabilities on the Rotation Angles amid Im..pact Parameter in BC
+ A Collisionsla
y I-Iyung Kyu Shin i3epartment of Chemistry, University of Nevada,Ib Reno, Nevada 89607 ~pvb&alioncosts assisted by the
(Received October 6 , 1970)
U.S . Air Force Ofice of Scientij$c Research
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Vibrational transitions in BC ,4 collisions are investigated by use of the sudden approximation. The transition probability P,, is formulatedfor an arbitrary system of BC A collisions, but the 0 2 Ar system is chosen for specific consideration. Pol is calculated as a function of the collision velocity u, impact parameter Dependence of POIon these collision parameters at different values of t i , and molecular rotation angles 0, 6 . b 1s discussed. For b < u, the angle-averaged transition probability increases with increasing u, while for h > O- it decreases as u increases, where CT is the Lennard-Jones potential parameter.
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Introduction The problem OS vibrational energy transfer at high collision energies (in the electron-volt region) is a subject of great interest from both experimental and theoretical standpoints. The recent progress in highenergy beam experirnents is beginning to provide important information on absolute magnitudes of vibrational trans Ition probabilities and related quantiWith rhe advent of large memory highspeed computers, theoretical calculations of vibrational transition probabilitw at high collision energies are becoming a ~ a i l a b l e ~ and - - ~ are contributing to our understanding of vibrationally inelastic scattering. However, almoat all of such calculations are based on the model of collinear collisions, which does not adequately represent the physics obf molecular collisions, and a realistic model should explicitly consider collisions taking p l a x at nonzero impact parameters and a t different molecular orjentations. I n a recent papw,8 we considered low-energy collisions by use of the method of“ distorted waves and investigated collisions at different molecular orientations without specific reference to the role of impact parameters. I n a subsequent piaperig we treated collisions at noiizcro impact parameters and a t different molecular orien1,ations. The transition probability was formulated such that it can be used to investigate the problem of vibr:t.eior,aIenergy transfer at various impact parameters and orientation angles in high-energy collisions. Chsby and JIoran3 showed that the model and formulation reported in the latter paper gave 0’ vibrational transition probabilities of 0, in 0 2 collisions which are in good agreement with their measured data in th.: collkion energy range of 10-20 eV. I n studying T-ibratjonal energy transfer in the threedimensional col’ision, we must recognize that the orientation angle chmges during the course of energy transfer. Although this situation is intuitively clear, there
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has been no rigorous consideration of it’ in the calculation of vibrational transition probabilities. I n the present paper we consider t,his problem in diatomic A) collisions by extending the molecule-atom (BC collision model given in ref 9. The collision system is simple enough to allow an explicit consideration of this problem in calculating transition probabilities. After attempting several different methods for the derivation of vibrational transition probabilities in high-energy collisions, we found that the “sudden” approximationlo*”is most suitable and simple to apply here; we use it in this paper. For BC A collisions at energies in the electron-volt region, the use of this method should be appropriate. The orientation dependent interaction potential will be assumed to be of the Lennard-Jones form. In numerical illustrations we consider the 0 3 1 transition in 0, Ar collisions.
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Formulation of Vibrational Transition Probability I n Figure 1 we show the three-dimensional collision (1) (a) This work was carried out under Grant AFOSIt-68-1354 from the Air Force Office of Scientific Research. (b) Theoretical Chemistry Group Contribution No. 5-1029. (2) (a) P. F. Dittner and S. Datz, J. Chem. P h u ~ . 49, , 1969 (1968). (b) J. Schottler and J. P. Toennies, 2. Ph.ys., 214, 472 (1968). (3) P. C. Cosby and T. F. Moran, J . Chem. Phys., 5 2 , 6157 (1970). (4) D. Secrest and B. R . Johnson, ibid., 45, 4556 (1966). (5) J. D. Kelley arid M . Wolfsberg, ibid., 44, 324 (1966). (6) D. J. Wilson, ibid., 53, 2075 (1970); also see earlier papers by Wilson ,and his associates. (7) Also, see D. Rapp and T. Kassal, Chem. Rev., 69, 61 (1969); they present an excellent review of recent work. (8) H. Shin, J . Chem. Phys., 49, 3964 (1968). (9) H. Shin, J . Phys. Chem., 73, 4321 (1969). (10) K. Alder and A. Winther, “Coulomb Excitation,” Academic Press, New York, N. Y., 1966, pp 209-280; the original article was published in Kgl. Dan. Vidensk. Selsk., Mat.-Pus. Medd., 32, No. 8 (1960). (11) K. H. Kramer and R . B. Bernstein, J. Chem. Phys., 40, 200 (1964), yresents the application of the sudden approximation to rotational transitions. Recently, M . A . Warteili and R. J. Cross, Jr., applied this approximation to vibrational inelastic scattering, Chem. Phys. Lett., in press.
The Journal of Phgsical Chemistry, Val. 7b, N o . 7 , 1971
HYUNG KYU SHIN
924
z
+ +
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where r1,2 -- r2 F 2(d t ) S ~ ,cos a 8 (d t)2Sz,12, S1,Z= mB,C/(mB nzc), d is the equilibrium distance of the B-C bond, and D, u are the Lennard-Jones A. The summation potential parameters for BC affects only two terms, since there are two interactions, i.e., B-A and C-A. By use of the relations = bZ x z and z = tit, we can parameterize the interaction , ~ potential in the time variable. By introducing T ~ into eq 3 and by expanding each term in a power series, we extract out the [-dependent energy terms from eq 3. The result is expressed by
+
I
+
--_- _
+'
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v [~,b,t,Q,r(t)1 = 2 4 ( D / ~ ) [ ( u / r )~ ~'/z(~/r)'](& - Si)$ X
+ 8D(d/u2)[(42 cos2 0 -- 3)(u/y)l4 (12 e - 3/2)(u/~)~](sl~ + s2z)t (4) where the third- and higher-order terms in (d -+ [ ) / r are COS
0
COS*
-E--
-- -.-
1
I
I@%iww&
1
:!Ld+ [-'
I
'
I
L-- - - - - _ _ _
I
Figure 1. Collision model. Insertion shows the vibrational coardinatct.
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model for a general case of the BC A interaction defining the collision coordinates needed to describe the encounter. According to the sudden approximation, the probability of the vibrational transition m to n i8lO,Il
iv,b,e) = (nlexp Pidu,b,e,t)l l ~ ~ ) z
(1)
where the pliase shift is
Here .$ is Lhr: vibrational coordinate, b the impact parameter, v the relative collision velocity [E = ( 1 / 2 ) p u 2 ] , p the reduced mass of the collision system, 0 the orientation angle, a,nd V the potential energy which causes the transition. We use the linear trajectory approximation for the present high-energy collision system; then from Figure 1, we get the relations r2 = b2 82 and x -- u t , To see the effect of varying collision velocity, oriani,ation angle, and impact parameter on transition probabilities it is then necessary to find an interaction potential function which is simple enough Lo allow the integration of ecl 2 and yet complicated enough to be realistic in describing the dependence of transition probabilities on relevant collision variables. The assumed form of the interaction potential is9
+
2
U ( T l 7 Y 2 ) =-
2 0 2.=
1
[(u/rJ12
- (u/r1)6]
The Journal of Physical C'hhem{atry, VoZ. 76,N o . 7, 1971
(3)
neglected. For homonuclear diatomic molecules, the first part disappears since 81 = S z . For heteronuclear diatomic molecules such as hydrogen halides SI 4 x cm/sec, find < "'O sho.cv the dependence amf .&,I at, different rotation angles, we ChOOSC?
'"
1
,
,
,
,
,
2
,
,
,
_ . 3 4x106
v (cmi...) Figure 2. Dependence of the transition probability Pol on the collision velocity for b / o = 0.8. The chosen values of the rotation angle 6 are 0, 50, and 90". For this calculation, as well as those shown in Figures 3-5, the angle d, is set a t zero. The range of validity of the sudden approximation slightly changes as e varies; however, above v = 4 X lo5 cm/sec, the approximation is satisfactory.
V (W-1
Figure 3. Dependence of POIon v for b / u = 1.3 at different values of e. The dashed portion of the curve represents the range where the sudden approximation becomes invalid.
-+ 18.1 sin26' cos2 cp - 1.26)1O5/u
and
M
,I, I,/, ,
','
The Journal of Physical: Chemistry, Vol. 76, N o . 7 , 1971
the motion to be in the xx plane, parallel to the x axis (4 = 0). Although other values of 4 mould give significantly different curves, the calculation for 4 = 0 would furnish sufficient information about the velocity variation of Pol at different molecular orientations. As u increases, Pol sharply rises to the maximum value l/e, and then slowly decreases as I ) continues to increase; it should be noted that Pol is a decreasing function of u in high-velocity collisions. The maximum value shifts toward higher velocities as the angle increases from 0 t o a/2. The calculated transition probabilities do not exceed unity, in contrast to the result of the usual perturbation approximation.7 The curves shown in Figure 2 (also Figure 3) are hymrnetrical about (14) C. E. Trennor, J . Chem. P h y s . , 43, 532 (1965); 44, 2220 (1966). (15) w. Shin, Chem, Phys. Lett,, 3 , 125 (19693, (16) J. 0. Hirschfelder, C.F. Curtiss, and It. B. Bird "Molecular Theory of Gases and Liquids," Wiley, New York, K. I-., 1964, pp 1110-1111. (17) G. Herzberg, u, an increase in E counteracts mation. For b’tr = 1.3 the approximation is satisthe magnitude of the potential minimum and straightfactory above t* = 8 X IO5 cm/sec; for b / u = 0.8 the ens out the trajectory, thus leading to an increase in TO. corresponding velocli,y is as low as 4 X IO5 cm/sec. The transition probability for b > u decreases with inI n Figure 6, we show the numerical result of polfor creasing E (or velocity) as seen in Figure 6. Furtherthe two b values. S o t e that for both curves, the more, since the collision occurs near “the outer closest dashed portiton represents the region where the sudden approximation hecomes invalid ; we show these portions distance” rocfor b > U, the energy transfer can be very only for comparison. inefficient (the probability being order of IOw6 in the 0, Ar collision). On the other hand, when b < u, The curve for b/u = 0.8 raises to a maximum value the potential energy will be mainly repulsive (nearly at about IO6 cm/sec, then very slowly declines as the direct collisions). I n this case an increase in the colvelocity increases. For velocities up to IO6 cm/sec, lision energy decreases T O and straightens out the trathe probability Lontinues to rise as v increases. The jectory. Therefore, with b < u p the probability inopposite variation is seen for b / u = 1.3. For such creases for increasing v and values of the transition large-b collisions, Po, always decreases as v increases and probability are very large. Figure 6 shows such variaits magnitude is very small. ‘This is a noteworthy tions up to an intermediate velocity where Po, takes a feature of the collision process and will be discussed maximum value of about 0.24. At higher velocities below.
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Pol(v,blf?,c$)sin e dB dd, (11)
a
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The Journal of Phvsical Chemistry, Vol. 76,No. 7 , 1971
HYDROLYSIS OF FER.RIC ION
929
where polvery slowly decreases, multilevel transitions (0 + 2, 0 -+ 3, etc.) would compete with the 0 1 transition, t,hus leadling to smaller values of pol. The result also sh xvs that for a given 1~ the probability normally decr3ases with increasing b. I n general the present calculratfon of the average transition probability
correctly reflects the physics of molecular collisions at differentvalues of b.I8 Acknowledgment. I wish t o thank Dr. Young 0. Koh of the Computer Center, University of Kevada, for assistingwith the programming*
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(18) B. Widom and 8. H. Bauer, J . Chem. PhUs., 2 1 , 1670 (1953).
Kinetics af Hydrolysis of Ferric Ion in Dilute Aqueous Solution
by Paul Hemmes, Larry D. Rich, David L. Cole, and Edward M. Eyring* Department of Chemistry, University of Utah, Salt Lake City, Utah 8411!8 (Received August $1, 1970) Publication costs assisted b y the A i r Force Ofice of Scientific Research
Dilute aqueous solutioiis of ferric perchlorate have been studied by the electric field jump relaxation kinetic b2
technique. A t 25” and ionic strengths less than 3 x M the specific rates of the reactions FeOH2+(aq) Fe(QII),+(aq) H+(aq) were found to be k z = 6.1 x lo4sec-l and k-2 = 8.0 x 109 M - l see-’. A preceding hydrolysis step Fea+(aq) FeOH2+(aq) H-k(aq) was found to reach equilibrium too rapidly for rate measurements by this method.
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Introduction The hydrolysis of metal ions in aqueous solution is an important process ’in many areas of pure and applied chemistry. In some cases, the normal solution chemistry of an element io a given oxidation state is not the chemistry cf the aquo ion at all but rather that of a hydrolyzed form of the ion. A classic example of such behavior is the vase of aqueous ferric ion.’ While for many years thermodynamic studies of aqueous metal ion hydrolysis have been made,2especially in Scandinavia, the kinetics of hydrolysis have been susceptible to study only since the advent of relaxation techniques. Kinetic investigations of the hydrolysis of aqueous m d a l ions
+ PlzO :z NOH2+ + H+ 1, 1
313+
*K’ = k’/k-l
(1)
hn
+ H 2 0z?M(OH)z++ H+
i\40H2+
*Kz
=
kz/k-Z
(2)
have revealed a surprising sameness of the hydrolysis rate constant k, g. 10j sec-’at 25” and nearly zero ionic strength for aqueous trivalent a l u m i n ~ r nchromium,s ,~ scandium,6 and indium ions.’ As would be expected from Debye’s equations for the specific rate of diffusioncontrolled ion recombinat>ion reactions, the values of k-1 also do not differ markedly for these metals lying as
they do between -4 X loe and -1O’O M-l sec-’. A possible case in which jG1 could differ significantly from -1Oj sec-I would be aqueous iron(II1) since for this ion p”K1 is variously reportedg~’Oas 2.2 to 2.5 (at zero ionic strength) in contrast to 5.02 for aliamin~m(I1I)~ and 3.98 for chromium(III),6 for example. The electric field jump relaxation method3 kinetic study of dilute aqueous ferric perchlorate reported below confirms this expectation. Experimental Section The ferric perchlorate was reagent grade (G. F. Smith Co.). Stock solutions were analyzed volumetrically (1) F. A. Cotton and G. Wilkinson, “Advanced Inorganic Chemistry,” 2nd ed, Interscience, New York, N. Y . , 1366, p 859. (2) L. G. Sillen and A. E. Martell, “Stability Constants,” The Chemical Society, London, 1964, p 39 ff. (3) M. Eigen and L. DeMaeyer, “Technique of Organic Chemistry,” Vol. VIII, Part 11, S. L. Friess, E. S. Lewis, and A . Weissberger, Ed., Interscience, New York, N. Y . , 1963, Chapter 18. (4) L. P. Holmes, D. L. Cole, and E. M. Eyring, J . Phys. Chem., 72, 301 (1968). (5) L. D. Rich, D. L. Cole, and E. M . Eyring, i b i d . , 73, 713 (1969). (6) D. L. Cole, L. D. Rich, J. D. Owen, and E. R f . Cyring, Inorg. Chem., 8 , 682 (1969). (7) I?. Hemmes, L. D. Rich, D. L. Cole, and E. M Eyring, J. Phys. Chem., 74, 2859 (1970).
(8) P.Debye, Trans. Electrochem. Soc., 82, 265 (1942). (9) A. B. Lamb and A. G. Jacques, J. Amer. Chem. Soc., 6 0 , 1215 (1938). (10) R . M. Milburn and W. C. Vosburgh, ibid,, 77, 1352 (1955). The Journal of Physical Chemistry, Vol. 76, N o . 7,1971
