J. Phys. Chem. 1982, 86, 2279-2281
or C12 (10 f 2) x lo-'' cm3 molecule-' s-'), then the HgX(B) branching fraction must be approximately unity. Comparison of the k(HgX,B) values of Table I to the rate constants for alkali metal atoms and for metastable rare gas atom@ also suggests that the quenching rate constants must be nearly equal to the product formation rate constants. Considering that the HgX(X) potentials are bound by 0.5-1 eV, the similarity in the appearance of the XeX(BX)'O and HgX(B-X) spectra is very surprising. The explanation is provided by the large increase in r: relative to r/ for the mercury halides, the relatively weak binding of HgX(X), and the high product vibrational energy from reactive quenching reactions of both HgPP2) and Xe(3P2) with halogens. Several trends can be recognized from qualitative inspection of the spectra in Figure 1. The red shift for the lighter halogens is a consequence of the increased dissociation energy of HgX(X) as the halogen becomes lighter and the nearly constant electronic energy separation of the HgX(B) and HgX(X) states. The frequency of the main HgX(B-X) peak is given roughly by the potential energy difference at re) and this separation decreases with increasing D,(HgX(X)). The broad range of the emission arises because of the extensive vibrational distribution and because of the nature of the bound-free transitions, i.e., there is a large difference in X for transitions from the same u' vs. internuclear distance. As shown in Table I, the short wavelength onset of emission corresponds to the maximum energy allowed by the thermochemistry. The broad oscillations are a consequence of interference between transitions from the same (16) D. W. Setser, T. D. Dreiling, H. C. Brashears, Jr., and J. H. Kolts, Faraday Discuss. Chem. SOC.,67, 255 (1979).
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level state to lower levels with the same energy difference. Finally, the lesser importance of the oscillations for Br2 and 12,relative to C12,indicate that (fv(HgX))is reduced as the halogens become heavier, where (fv(HgX)) is the average fraction of the available energy present in HgX(B). Preliminary results12 from computer simulation of the spectra in Figure 1 show that the distributions strongly resemble those from Xe(3P2)+ X2,10bi.e., the distributions are two component in nature with the mean energy declining from (fv(HgX)) = 0.60 for C12to 0.46 for 12. The vibrational energy disposal for Hg(3P2)atom reactions with halogens therefore differs considerably from that of alkali metal atoms with halogens.16J8 The laser transitions for HgI, HgBr, and HgCl are close to the wavelengths of the maxima in the low-pressure spectra of Figure 1. These laser bands correspond to the transition from u' = 0 to the right-hand side of the lower state potential. Although the HgF(X) and HgF(B) potentials are not well e~tablished,'~ by analogy to the other mercury halides we can predict that the HgF(B-X) system should provide a laser transition near 650 nm. Acknowledgment. This work was supported by the U.S. Office of Naval Research under Contract N00014-80-C0346. (17) T.D.Dreiling, Ph.D. Thesis, Kansas State University, 1982. (18) B. E.Holmes and D. W. Setser in "Physical Chemistry of Fast Reactions", Vol. 2, I.W.M. Smith, Ed., Plenum Press, New York, 1980. (19) The short wavelength Rydberg-type electronic transitions have been observed for HgF (Y. K. S. C. Babu, P. T. Rao, and B. R. Reddy, Znd. J. Pure Appl. Phys., 4, 467 (1966); H. G. Howell, h o c . R. SOC. London, Ser. A , 182,95 (1943). These spedra yield values for we and wJe for HgF(X). Based on these results Gaydon ("Dissociative Energies and Spectra of Diatomic Molecules", 3rd ed, Chapman and Hall, London, 1968) has estimated D,(HgF) = 1.2 i 0.4 eV.
Dlffusion-Controlled Reactions Involving Steroid Molecules with Two Isolated Reactive Groups R. Kurt Huddleston' and Wllllam A. Mulac Chemistry Dhrision, Argonne National Laboratory, Argonne, Illinois 60439 (Received: February 19, 1982; In Final Form: April 15, 1982)
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Reactions of solvated electrons with steroid molecules having two electron accepting functional groups rigidly held 10 A apart were studied in ethanol at room temperature. The measured rate constants show that these difunctional molecules are considerably less than twice as reactive as monofunctional model compounds, so that diffusion processes involving the two reaction sites are not independent. An approximate model based on diffusion in spheroidal geometry gives reasonable agreement with the results.
Introduction Diffusion-controlled chemical reactions in liquids involving spherical or nearly spherical molecules are expected to be described well by the standard von SmoluchowskiDebye treatment.'p2 The more general problem of diffusion in nonspherical geometry is substantially more complex and has been considered for only a few special cases.3 An interesting special case, which has important applica-
tions for the analysis of data from studies of intramolecular electron transfer in s ~ l u t i o nis , ~that of a molecule with two reactive sites held at a fixed separation. Previously, kinetic results were presented for reactions of difunctional steroid molecules with trapped electrons (e;) in rigid g l a ~ s e s . ~The difunctional steroids were found to be considerably less than twice as efficient at e; capture as monofunctional model compounds. These reactions in rigid media occur by a long-range tunneling mechanism
(1) M. von Smoluchowski,Phys. Z.,17,557,585 (1916);2.Phys. Chem. 92,129 (1917). (2) P.Debye, Trans. Electrochem. SOC., 82, 265 (1942). (3)P. H.Richter and M. Eigen, Biophys. Chem., 2, 255 (1974).
(4) L.T.Calcaterra, J. R. Miller, and G. L. Closs, to be submitted for publication. (5) R. K. Huddleston and J. R. Miller, J.Phys. Chem., 85,2292 (1981).
0022-3654/82/2086-2279$01.25/0
0 1982 American Chemical Society
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The Journal of phvsical Chemistty, Vol. 86,No. 13, 1982
Letters
TABLE I: Data for es- + Steroids in Ethanol 10-9k, M-’s-l compd R,A A30 2.58 c 0.06 1.8 i. 0.2 A170 1.97 ~t0.04 1.4 c 0.3 ADO 3.90 c 0.34 5.58 f 0.38 AEO 3.9 i 0.5 AEOA 3.8 ? 0 . 5 5.41 i. 0.30 5.16 f 0.39 PEO 3.6 i. 0.5 PEOA 5.24 i. 0.27 3.7 i. 0.5 PDDO 6.94 ~t 0.27
light from a pulsed xenon lamp passed through the sample in a path perpendicular to the electron beam and then through a monochromator before striking a photomultiplier. The absorbance was monitored at 600 nm. The signal was digitized by a Biomation 8100 transient recorder and was sent to a Xerox Sigma 5 computer for storage and analysis. The decay curves were fitted to a single exponential function. Typically, three shots were taken on each sample, using fresh solution each time.
dm
0
/
Figure 1. Structural formulas for ketone and enone steroMs used as e,- scavengers: 5a-androstane3,17clione (ADO); 5a-ar~irostarr3one (A30); 5a-androstan-17one (A 170); 4,16-pregnadien-3,20dione (PDDO); 17&hydroxy-4-androsten-3one (AEO) and BP-hydroxy-S& pregn-l&m-2Oone acetate (EO). Also used were the acetate esters, AEOA and PEOA, of the alcohols AEO and PEO.
and are well described by a simple model which takes into account the overlap of the tunneling capture volumes of the individual reactive groups. Here we present experimental data for reactions of difunctional steroids with solvated electrons (e;) in liquid ethanol solution. It might be expected that, for a diffusive mechanism in which the step size is less than the separation, two reactive groups will act independently, in contrast to the tunneling mechanism. However, in solution the difunctional molecules are also found to be less than twice as reactive as monofunctional models, so that two reaction sites 10 A apart on the same molecule do not act independently. Thus, the reactivity of one site depends on the presence of the other site, i.e., the two sites are correlated.
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Experimental Section The experiment involved observing the decay of the absorbance of e; in liquid ethanol at room temperature in the presence of steroid acceptors. Structural formulas and abbreviations for the steroids are given in Figure 1. The difunctional steroid molecules have ketone or enone functional groups at either end of a rigid hydrocarbon framework; the reactive groups are separated by 10 or 8.8 A, respectively, where these distances are taken from molecular models. Reaction rates for the monofunctional models were measured for comparison. Anhydrous ethanol was distilled from sodium borohydride under an atmosphere of argon. All of the steroids were specified by the manufacturer (Steraloids, Inc.) as showing one spot in thin-layer chromatography and were used as received. Solutions of typically three concentrations of the steroids in the range 10-3-10-4 M were made in ethanol to which 0.01 M Na20 had been added. The solutions were degassed by bubbling with helium and were kept free from air. Pulse radiolysis was carried out with 10-ns pulses of 15-MeV electrons from the Argonne linac. Analyzing N
Results The rate constanta shown in Table I were obtained from least-squares linear fits to plots of the reciprocal of the decay time vs. concentration of the acceptors. The stated error limits represent 95% confidence levels. The rate constants for the ketones A 3 0 and A170 are significantly different; that a faster rate is obtained for A30 agrees with the results in a rigid m e d i ~ mwhere ,~ A30 captures e; more effectively than does A170. Results for the compounds AEO,PEO, and their acetate esters all agree within the experimental errors. As expected, there is no evidence that e; reacts with either the hydroxyl or acetate functional groups. In analogy with the tunneling r e a ~ t i o n sthe , ~ relative reaction efficiency of difunctional vs. monofunctional compounds in solution can be defined as (1) 4 = k , , / ( k , + k,) where k12 is the rate constant for the reaction involving the difunctional molecule, and kl and k2 correspond to the two monofunctional model compounds. The case 4 = 1 corresponds to the two reaction sites of the difunctional molecule acting independently. If kl = k2, 4 is 1 / 2 in the case that the difunctional compound is no more efficient at e; capture than a monofunctional compound. For the diketone ADO, 4 = 0.86 f 0.08, so that its e; capture efficiency is somewhat less than that expected for an equal mixture of A30 and A170 each at the same concentration. For the dienone PDDO one obtains the even lower value 4 = 0.65 f 0.04, where average values for the rate constants for the two types of model compounds were used. If we assume diffusion-controlled kinetics and spherical geometry, encounter radii R (in A) can be calculated from the rate constant k (in M-’ s-l) for the monofunctional compounds from2 k = 4a(D,+ D,)N&/103 (2) where No is Avogadro’s number and De and D,are diffusion coefficients for e; and the steroid, respectively. These encounter radii are given in Table I. A value for De of (0.61 f 0.15) X lo5 cm2 s-l was used.6-8 To our knowledge, diffusion coefficients have not been measured for the steroids. These were estimated from the correlation suggested by Wilke and ChanggJo (6) H.A. Schwarz and P. G. Gill, J. Phys. Chem., 81, 22 (1977). (7) P. Fowles, Trans. Faraday SOC.,67, 428 (1971). (8)0. I. MiEiE and B. CerEek, J . Phys. Chem., 81, 833 (1977), give a
substantially higher value. (9) C. R. Wilke, Chem. Eng. Progr., 45, 219 (1949).
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982 2281
Letters
T(MX)l12 D = 7.4 X lo8 q vo.6
(3)
which has been found to hold to within 12% for a large variety of molecules. In eq 3, Tis the absolute temperature in K, 77 is the viscosity of the solvent (taken as 1.14 cP at 296 K), X = 1.5 for ethanol, and M and V are the solute molecular weight and molal volume (in cm3g mol-') at its boiling point. V was obtained to within f10% from a measurement of the density of 5a-androstane (d = 0.88 g cmT3at 210 O C ) and by using a table of atomic volumes" to account for the substituents. The reported error limits for R include the uncertainty in the rate constants as well as uncertainty in the values of De and D,.
Discussion The present results show that for the reactions involving difunctional steroid molecules the quantity 4 defined by eq 1 is less than unity. That is, the reaction efficiency of the difunctional molecule with two reactive groups held at a fixed separation of -10 A is less than that of two reactive groups at random positions in the solution. Thus, two closely spaced reaction sites do not act independently in a diffusion-controlled reaction. This result may be interpreted in terms of a physical picture in which some of the incoming diffusive flux toward one reaction site is intercepted by the other site. A more precise statement is that the steady-state concentration field around the difunctional molecules results in a reduced reactive flux over that for two independent sinks at an infinite distance apart. The smaller value of 4 measured for PDDO than for ADO is consistent with the larger rate constant and smaller separation of the enone functional groups in PDDO, since a larger fraction of e; diffusing toward one site will be intercepted by the other site. An estimate of the reactivity of difunctional molecules can be made from a treatment of diffusion-controlled reactions in spheroidal geometry presented by Richter and Eigen.3 Their results can be applied to the present experiments under the assumption that the difunctional steroid molecules can be approximated as prolate spheroids. The rate constant for a diffusion-controlled reaction involving a spheroidal target molecule can be written
k = 4rDN&g(B/A)/1O3 (4) where D is the sum of the diffusion coefficients for each species, and for a prolate spheroid (A > B ) g(X) = In [(I + (5)
2d=/
d 3 ) / ( 1diF?)] -
is a function of the lengths of the semimajor and semiminor axes of the spheroid, A and B, respectively. Substituting eq 2 and 4 into eq 1 yields
4 = (B/R)[g(B/A)/(2B/A)I (6) If we assume that B can be equated to the encounter radius of a single functional group, eq 6 relates unique values of 4 and the ratio B / A . Also taking the reasonable value A = B + a/2, where a is the center-to-center spacing between the functional groups, leads to predicted values of 4 which are compared with the observations in Table 11. The spheroidal model overestimates 4 by 12% for ADO and by 6% for PDDO. It is expected that, in general, the (10)C. R. Wilke and P. Chang, AIChE J.,1, 264 (1955). (11)J. H. Perry, 'Chemical Engineers Handbook",McGraw-Hill, New York, 1963,pp 14-20.
TABLE 11: Observed and Calculated Values for the Reaction Efficiency @ @
compd
observed"
spheroidal modelb
ADO 0.86 * 0.08 0.96 f 0.15 PDDO 0.65 r 0.04 0.69 * 0.10 " Defined by eq 1. From eq 5 with B = R and A = B t al2.
spheroidal model will overestimate 4, since it effectively uses a larger average encounter distance than for two separated spheres, but it can be concluded that the model gives a reasonable representation of the difunctional molecules. Apparently, the molecule PDDO with larger encounter radii for each enone group compared to their separation is better described by a prolate spheroid than is A30. In order to simplify the analysis, we have assumed diffusion-controlled kinetics to apply. Although the nondiffusion-controlled case is complex, a few conclusions are possible. The ratio 4 is expected to approach one as the reaction probability upon an encounter with a reaction site tends to zero. This is so because the probability of the e; surviving the encounter with one site will be close to unity and thus will not affect the probability of reaction at the other site, except for possible steric effects. As the rate approaches the diffusion-controlled limit, 4 will decrease below one. This is consistent with the observations in that PDDO, with larger rate constants for the enone functional groups, has a smaller 4. Thus far we have neglected the effects of rotation of the steroid molecules. Rotation rates are not expected to be extremely fast for the relatively large steroid molecules. In general, rotation will increase the reaction rate, since it will increase the probability that a species diffusing near the steroid molecule will be within a short distance of the reactive site. Some evidence that rotation is not important is provided by the relatively small reaction radii observed for the monofunctional compounds. A general theory of diffusion involving correlated reaction centers is desirable. Work on a solution to the diffusion-controlled case is in progress.I2 In principle, measurements of rates of reaction of difunctional molecules contain information about the interesting question of whether slow reactions in solution correspond to small encounter radii or low reaction probabilities upon an encounter. The reaction rates can be used to relate the encounter radii for the individual reactive sites to the known spacing between the sites. An analysis using the spheroidal model to calculate the encounter radii from the measured value of $I and the spacing a yields values for R for both the ketone and enone steroids which are -25% larger than the measured encounter radii for the model compounds, which provides some evidence for a reaction probability less than unity. Unfortunately, these values are sensitive to the experimental uncertainties, and it is difficult to assess the error introduced by using the spheroidal model.
Acknowledgment. We thank Robert Clarke and Patricia Walsh for technical assistance and John R. Miller for very useful discussions. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, U S . Department of Energy under contract No. W-31-109-ENG-38. (12)R. K.Huddleston, to be submitted for publication.
