A.
Mozumder
f Solvated Electrons in Dilute Solutions
olar Moiecules in Nonpolar Solvents 14.
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Department of Chemistry and fhe Radiation Laboratory, University of Notre Dame, Notre Dame, lndiana (Received May 7 , 1972)
46556
Publicstion costs assisted by the Radiation Laboratory, University of Notre Dame
A theoretical model is presented for the time required to form solvated electrons in a dilute solution of polar molecules in a nonpolar solvent when irradiated by ionizing radiation. The polar molecules are assumed to exist as monomers; however, this assumption is not seen as a serious limitation to the applicability of the model. In the system considered an electron escaping geminate neutralization following an ionization event will eventually find itself attached to a polar molecule, which is the first step in the formation of the solvated electron. This process is always very fast. I n the second step neighboring dipoles coagulate to the central negative charge. This process is relatively slow, consisting of dipole rotation and drift which contribute comparably to the so-defined microscopic relaxation time. The analysis presented here describes motions of electrons and dipoles in terms of drift velocities. The latter are obtained from instantaneous electric fields and h e a r and rotational mobilities. Concentration dependence of solvated electron formation time is evaluated and comparison with experiments is indicated.
In the radiolysis of polar liquids formation of solvated electrons is a common occurrence. I t is also a common belief OF assumption that there is some kind of dielectric relaxation process associated with the formation of solvated electrons in such liquids. However, there are considerable difficulties in the interpretation of the relevant relaxation mechanism and therefore also in the evaluation of the time required to accomplish the process. In the first place, it is d e a r that a microscopic relaxation mechanism may be different, either partly or wholly, from the corresponding macroscopic counterpart.2 In the second place, and this Consideration is by no means trivial, even the macroscopic dielectric relaxation is in many cases only poorly unders t ~ o d . The ~ , ~Debye theory, based on rotational Brownian motion, is the most commonly accepted model for macroscopic dielectric r e l a ~ a t i o n .Even ~ though in some cases calculations based on the Debye theory give relaxation times that are in fair agreement with experiments, the ~,~ foundations of the theory are somewhat o b s ~ u r e .Thus, 16, bas been argued that (i) a pure rotation of a polar molecule in a continuum is at best a difficult concept since most significant polar liquids are hydrogen bonded and energy required to break the necessary number of hydrogen bonds is not available either from the heat bath or from the external field; (ii) to keep the layer of the liquid next to the mcdecule at rest (relative to the polar molecule), it is necess w y to increase the molecular radius by at least a factor of 3 which increases the calculated relaxation time by more cban an order of magnitude. There are also other difficulties such as associated wit viscosity variation. Whereas the situation remains complicated and obscure with respect to macroscopic relaxation in neat polar liquids, the case of a dilute solution of polar molecules in nonpolar solvents is essentially simple. In the latter case the objections to the Debye theory which have just been described do not apply and the experimental situation is in good agreement with Debye's model.5 ;n an earlier attempt by the present author microscopic The Journal of Physical Chemistry, Vol. 76,
No. 25, 1972
relaxation was described in a manner similar to the macroscopic description except in an essential change necessitated by the occurrence of ionization itself, utz., the charge (or displacement) was kept constant rather than the field, the latter being the usual macroscopic restraint. This description results in a relaxation time shorter than the macroscopic value by a factor equal to the ratio of static to high-frequency dielectric constant. Such a n analysis is in agreement with experiments on water and some alcohols but in other cases theory predicts too high relaxation times. 7a Also, theory predicts a difference in the relaxation times of (say, for the sake of example) 1-propanol and 2-propanol which has not been experimentally observed. Thus, the case of dielectric relaxation remains somewhat obscure and largely not understood from both macroscopic and microscopic points of view. On the other hand, since macroscopic relaxation of dilute solutions of dipoles is a wellunderstood phenomenon, it is natural to expect that the formation of solvated electrons in such systems should similarly present a less complicated theoreticai problem. In this paper we calculate the formation time for solvated electrons in dilute dipole solutions through basically a charge dipole interaction. A similar model involving dimers K binding and has been used by Raff and Pohlm ~ O electron optical transition energy of solvated electrons in neat polar (1) The Radiation Laboratory of the University of Notre Dame is operated under contract with the U. S.Atomic Energy Commission. This is AEC document No. COO-38-844. (2) in the present context macroscopic relaxation refers to dielectric relaxation of the bulk medium in a (relatively) weak, external field. The raDid relaxation Drocess occurrina in reoions of molecular dimensions in the strong field surrounding an i h e d electron will be called microscopic relaxation. W. Kauzmann, Rev. Mod. Phys., 14, 12 (1942). A. von Hippel, J. Chem. Phys., 54, 145 (1971). P. Debye, "Polar Molecules," The Chemical Catalog Co., New York, N. Y . , 1929. W. Jackson and J. G. Powles, Trans. FaradaySoc., 42A, 101 (1946); see also H. Frohlich, "Theory of Dielectrics," 2nd ed, Oxford University Press, London, 1958, pp 120-121. (a) M. J. Bronskill, R. K. Woiff, and J. W. Hunt, J. Chem. Phys., 53, 4201 (1970); (b) L. Raff and H . A. Pohl, Advan. Ct9em. Ser., NO. 5 8 , 173 (1965).
Solvated Electron Porr-iatinn
3825
media. In a certain sense the present treatment is complementary to their work. ode1
In the present model formation of solvated electrons in a dilute dipolar solution evolves in two stages. In the first stage the electron is drawn to the nearest dipole with a favorable orientation. Since the electron is far more mobile than the polar moiecule, the latter virtually remains stationary during this stage of motion.s In reality the electron sees the field at its own location contributed by all the surrounding dipoles. In practice, however, the field due to the nearest neighbor dominates because the charge-dipole interaction varies as P-3, i“ being their separation. In this paper we will consider only the charge-nearest neighbor force, the residual part being assumed as negligible. Also since we are dealing with dilute solutions, polar molecules will be treated as point dipoles. In the second stage of motion we consider the interaction of a dipole with a negative ion, the latter being just an electron attached to a polar molecule during the first stage of the motion. Again only the nearest neighbor dipole is considered, a certain amount of error in neglecting the other neighboring dipoles being explicitly acknowledged. However, in this stage, the ion and the dipole have comparable mobilities and we must consider their relative motion in terms of sum of their mobilities. Additionally, the torque acting on the dipole has time to orient the dipole along the radial direction and this motion competes with radial drift in producing the resultant polarization. The joining of dipoles in the second stage of motion will be called “dipole coag~lation,’~ this being the mechanism for the formation of solvated electrons in the present system. eale igure 1 shows the diagram for the electron-dipole interaction, The force on the electron may be resolved into two nts, F,, acting along the F direction and F I acting cular to it. The perpendicular force changes the orientation 0 even though the dipole remains stationary in the laboratory frame in this stage of motion. The electron motion will be described in terms of drift velocities, the drift velocity in any directioii being equal to electron mobility times the field acring on the electron along the same direction. We thus get
Polar plot of the curve r = A sin2 19, A being chosen arbitrarily in t h e present case. If A is set equal to ro/sin2 O0 where ro and 00 are initial separation and angle, respectively, then the curve will represent electron path in the attachment stage from the point (ro, 80). Motion is clockwise above the line 19 = 0.r and counterclockwise below it.
Figure 2.
and 8 may be obtained by eliminating t between (1)and (‘2) and integrating the resultant equation. With the initial condition, that a t t = 0, F = FO and 0 = 00, we then get
r = (ro/sin2 Bo) sin2 0
sin 80 f 09
13)
Equation 3 shows that, at attachment ( r -- 0 ) the electron arrives along the dipole direction (0 = 0). It also shows that the electron follows the lines of force of the dipole which is a consequence of using instantaneous drift veiocities. The situation is shown in Figure 2. However, eq 3 does not indicate the time required for the attachment ~ R X X S S . For that we substitute (3) in (2) and obtain
where
K = p e p sin8 00 /er0.O4 On the integration of eq 4, we get
K t = f(B) - f\Bo) where f(0) = cos 0 (a1 sin6 8
+ a2 sin4 0 + a3 sin2 0 + a.04)
(5) (6) (‘I)
and a1
= %,a2 = %5,a3 = %5,aa = %6
(8)
The attachment time; t’, for the electron may be obtained from (61, for a given initial position (FO, Bo), by letting 0 = 0. That is
and
~’(Fo,00) = 8tfg(@oo)
where t is the lapse time and p, p e , and t refer respectively to the dipole moment of the molecule, mobility of the electron ( i e . , drift velocity per unit electric field), and the dielectric constant of the medium. A relationship between r
-
t?
~
where
and tf = 6ro4/8ppe
---
Charge-dipole interaction. The separation between the charge and the dipole is r, t h e vector being considered as positive when directed away from the charge. 0 is the angle between the negative end of the dipole and the radius vector. The symbol e stands either for t h e electron or for the negative ion of same charge.
Figure 1
19)
Measured electron mobilites in nonpolar, dielectric liquids are in the range -0.1 to -100 cm2/V sec. See, for example, (a) R. M. Minday, L. D. Schmidt, and H. T. Davis, J. Chem. Phys., 54, 3112 (1971); (b) W. F . Schmidt and A. 0.Allen, ibid., 52, 4788 (1970). Recently Freeman has found evidence for electron mobility in liquid methane as high as -300 cm2/V sec (see P. G. Fuochi and G. R. Freeman, ibid., 56, 2333 (1972)). By comparison, anion mobilities are of the order of crn2/V sec. If 00 = 0, then 0 is also zero at all times (see eq 2) and the radial equation is simply integrated from eq 1. The Journal of Physical Chemistry, Vol. 76, No. 25, 1972
3826
A. Mozumder
The quantity t f measures the time scale of the attachment process. To get a significant measure of t' we must average eq 9 over a random distribution of 00 and also over a distribution of ro for a given concentration of the dipoles. The first averaging is done for 60 = -x/2 to +n-/2, Le., for initial orientations that are favorable for electron attraction. The ) cause the residual initial orientations (00 = x/2 to 3 ~ 1 2will dipole to repel the electron initially. This repulsion usually makes r comparable to or greater than the mean separation between dipoles before the force on the electron becomes attractive by the necessary change of 0. It is reasonable to assume that about half the nearest neighbor dipoles are in the orientation 00 = 7 1 2 to n-12 and the rest with 00 = n-/2 to 3n-12. It is also reasonable to say that when an electron is repelled by the nearest neighbor because of unfavorable orientation it will find another neighbor to get attracted to. On these considerations the average attachment time is calculated here over a normalized population of dipoles with initial favorable orientations to the electron. We then obtain from eq 7 , 9 , and 10
In eq 12, 0' is a small angle approaching the,limit zero and z = cos 00. The integral appearing in eq 12 may be written as follows
I = a111 -"
(@I2
+ a313 + a414) + a415
where
11 = (Y2) In (1- 9)
- 22) I3 = 1h(1 - 22)2 14 = M(1 - z2)3 12
= %(l
( 5 / ~ d 2 / (1 29
+ P/32) In [(1+ z ) / ( l - z ) ]
-
(13f)
Substituting eq 13b-f in eq 13a and the last equation in eq 12 and evaluating the limitlo 0' 0 we get
+
+
(ygtf)(t')flg= ( a z / 3 ) + (1'7~3/90) (413~4/3780)
(5a4/16) In 2 (14a) Substituting the numerical values of the coefficients from eq 8 we get
(t')Bo= 1.994tt
tf = t:ro4/8pke
(15)
If the average dipole density be n per unit volume then, under random distribution, the probability of occurrence of a nearest neighbor dipole between distances r and r + d r as seen from any arbitrary point is given by11 w(r)dr
=:
47rr2n exp( -4ar3n/3)dr
( 16)
The average of ( t ' )over ~ ~ the distribution of ro is now given from eq 15 and 16 as follows
Concentration Dependence of Formation Times of Solvated Electrons in Dilute Dipolar Solutions
TABLE I:
Concn, M 0.03 0.02 0.01 0.007
0.005
0.002 0.001
Attachment time t l , psec
Coagulation time t 7 , nsec
8.9 15.2 38.4 61.8 96.8 328 827
4.648 7.836 19.136 30.297 46.735 152.15 371.58
Comments (1)
Physical parameters used: t: = 2, p = 2 D, and pe = 0.35 cm2/V sec
(2) Attachment time is
always negligible compared to t h e coagulation lime
A sample calculation with t: = 2 , = 2 D, pe = 0.35 cm2/V sec, and a density corresponding to a 20 m M solution ( n = 1.2 x loi9 dipoleslml) gives tl = 15.24 psec. The calculated variation of t1 with the molarity of solution using otherwise the same physical parameters is shown in the second column of Table I. It is seen that electron attachment i s always a fast process. Dipole Coagulation In this stage of the motion we consider coagulation of the nearest neighbor dipole with the central negative charge ( i e . , electron attached to a polar molecule]. Figure 1 may still be deemed to represent the case, e now being interpreted as a negative ion. However, in this case, change of 0 may be brought about by either the torque acting on the dipole or by the perpendicular component of the force, FL. If these changes in the time interval d t be denoted by dBI and d&, respectively, then the equations for drift velocities may be written as follows12
and
+
Noting that d0 = d81 d02, the above equations may conveniently be put in a dimensionless form as given below
dp = - xcos BIp3
(19ai
dr
(1%)
In eq 19a and 19b p = r/ro, ro being the initial separation; r = t/to, where to = 8~1@$2/ep, a scaling factor having the dimension of time; a, the size parameter of the polar molecule; 7, the viscosity of the medium; e, the magnitude
-
(10)At 0' = 0, the various integrals appearing in eq 13a-f exhibit divergences of various orders. However the net result, as €" 0,is a neat cancellation of all divergences leaving only small finite terms.
To see this it is necessary to expand each integral in powers of 0 up to a nonvanishing, nondivergent term. (11) See, for example, S. Chandrasekhar, Rev. Mod. Phys., 15, 1 (1943). (12) Here generalized mobilities (6's) are defined as generalized velocities per unit generalized force (rather than electric field as was used in the electron attachment case). Thus 6 8 is angular velocity per unit torque exercised on the system.
The Journal of Physical Chemistry, Vol. 76, No. 25, 1972
Solvated Electron Formation 1
I I I
'-1
100
I 80
3
c
60 E
E
Q
0.41.
40
20 E
0
I
2
3
4
5
6
7
b0 -
I-
Evolution of p and @ in normalized time ( 7 ) for i-0 = 24 A and eo = 120'. See text for values of physical parameters used; time scale to = 0.483 nsec. Significance of the times shown , boomerang time when the dipole on the curve are ( i ) T ~ the , time for orienstarts retracting after an initial repulsion; ( i i ) T ~the tational relaxation; (iii) 7d, the beginning of the pure drift region, (@= 0);and (iv) T F , the final coagulation time.
0
Figure 3.
of the electronic charge; h = 16a2/3r02; and f(p) = 1 + Also, we have used13 Br = (3nqa)-1 and BO == (8n77a3)-1 The departure of f ( p ) from unity measures the relative effect of FL in changing 0. Computer calculations in thbeworst case (highest concentration) show this effect to be only a few per cent. 'This means that essentially the rotation i s effected through the torque only. The reason for this effect e found in the r-dependence (cf. eq 18a and lSbi To compute the coagulation time we start a t t = 0 with an initial separation ro and orientation 00. That is, eq 19a and 1% are solved numerically starting a t T =I 0 with p = 1 and B = 0,. Finally, arrival time ( T F ) is defined when p = Za/ro. However, sirice time is found to depend on a high power of distance (see later) the final distance is not critical as long as it is small compared to the initial value. Computer calcuPations show that in general both B and p change significantly during the coagulation period. In fact, if 80 is not too large, 8 0 s u b ~ t a n ~ before ~ a ~ ~the y time T F so that there is a region of pure drift, which is defined in our program when Isin B/ < From eq 19a 7 is proportional to p4 in this region. In any case a time of orientational relaxation ( T ~ ) can always be defined such that at this time cos 8 = P (1 - cos 80) exp(--1), i.e., 1 J e of net required orientational p o ~ a r i ~ a t ~still o n remains to be achieved. Figure 3 shows compute^ evolutba of p and 0 in a typical case using the same physical parameters as before. The orientational redrift region, and the final coagulation in this figure. Figure 4 shows the variation of ~ o a ~ time ~ with ~ a initial ~ ~angle o for ~ a fixed initial separation (24 A\. For 80 around f180", the dispersion of TF with $0 is also large which is a result of the fact that the dipole spends a lot of time in properly orienting itself for these starting angles before its separation from the negative ion changes s ~ ~ n i ~ ~ ~ ~ ~ t ~ y ~ As in the electron attachment stage a significant measure of the coagulation t i m e is.obtained only after averaging over 00 and TO. In the region of initial separations, 1632 A, the angular averaging is performed numerically over 32 angles, 8 in each quadrant, such that their cosines are equally spaced, In this region it has always been found that there exists a n angle close to 120"such that the arrival time initial angle ie equal to the angle-averaged arrival utside this region, then, we assume that the arrival
I
I
I
I
I
1
40
80
I20
160
200
240
L U 280
320
360
eo(degrees) Figure 4. Plot of normalized coagulation time ( T F ~as a function of initial angle (00) for a starting separation of 24 A. The time scale in this case is to = 0.483 nsec. See text for values of physi-
cal parameters used. The curve is symmetrical about 8 0 = f180";this symmetry can be seen from eq i 9 a and 19b by setting q5 = 2* - 8.
X/2p2.
-
~~~~~~~~~~
Figure 5. Variation of angle-averaged dipole coa-gulation time (nsec) with initial separation (A) (log-log plot). The data are well represented by t h e line fa = 1.9613 X 10-srro3.8646. See text for values of physical parameters used. time calculated for a n initial angle of 120" represents the sodefined average value. Figure 5 shows the variation of this time (ta) as a function of initial separation on a log-log scale. Same physical parameters are used here as applied before to convert the arrival time to absolute units. It is seen from this figure that the time-separation equation i s well represented, except perhaps for very low separations, by a power relationship ta(nsec) = Aro(w)m (204 (13)
BO is given by the Stokes-Debye equation (see ref 5). B, is taken equal to twice the value given for a moleCule of radius a by the Stokes-Einstein relation. This means that the linear mobilities of the ion and the dipole are taken eoual. The Journal of Physical Chemistry, Vol. 76, No. 25, 1972
A. Mozumder
3828
where in this case A
=2
1.9613 X 10-5and m = 3.8646
@Ob)
Note that if the rotational relaxation was always very quick, we would have expected m = 4.Averaging t, further over a distribution of initial distances for a given concentration n of dipoles is performed as before, i.e., averaging eq 20a over the distribution of eq 16. The result is (21a)
or, in our case t ~ ( n s e c ) 0.050752M-1.'''2
(21b)
where in the last equation the molarity of the solution replaces the number concentration n. Table I shows the variation of t z with M which is significant even if not severe. We also notice that the attachment time tl is always insignificant compared to t2, as perhaps expected. In fact, it is so short that mostly it will elude detection unless by specific design, i. e., low-electron mobility, large initial separation, etc.
Discussion The present model only underlines the essential basics of the theory without going into details of calculation or into an elaborate comparison with the few available experiments. No great accuracy is claimed but it is believed that the results are correct within their respective orders of magnitude. To keep the mathematical complexities to a minimum we have (i) assumed point dipoles, (ii) neglected bulk neu.tralization, and (iii) also assumed that in the dilute solution dipoles exist primarily as monomers. Improvements of the theory must be based on relaxing these simplifications to more realistic descriptions. Also, a more realistic field in the second stage of motion is indicated through the selfconsistent mutual interaction of all the neighboring dipoles. Modifications due to these effects are difficult to make but they are highly desirable for comparison with experiment. At a first sight it may appear that bulk neutralization will pose a lower limit on the dipole concentration for the practicability of the experiment. Taking a dose -1018 eVjml and using a G value for escaped electrons -0.1, we compute the initial concentration of electrons in volume as cg -i015/ml. Using the Debye equation for bulk neutralization (Le., k = 4 ~ D r , )with D = 8.8 X 10-3crn2jsec (from measured elec3 tron mobility in cyclohexane) and rc = 300 A, we get k x 1 W 7 mljsec or that the first half-life of neutralization = ( k c o )- l -3 nsec. However, the electron-polar molecule attaehment time for a concentration of 30 m M is -10 psec (see Table I). Hence, it is clear that the bulk neutralization must be between the positive ion and the solvated electron. For the latter process, it may be argued that the positive ion i s the more mobile species as some experiments require the existence of a mobile positive ion for interpretation.14J5 Takingls &/D+ 17, we get k+to e$ 1.76 X lo-' ml/sec and t 1 / 2 50 nsec. 'Thus, the half-life for bulk neutralization is a t least an order of magnitude greater than dipole coagulation time (see Table I) at the smallest experimental concentration. A high degree of coagulation should, therefore, set in before volume neutralization becomes significant. A crude estimation for lower concentration of dipoles from the neutralization point of view may be obtained by setting v x' time for single coagulation = t1/2 for neutralization, where u is the number of dipoles required to approximate the structure of the solvated electron. Taking u = 4 (more or
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-
-
-
The Journal of Physical Chemistry. Vol. 76, No. 25, 1972
less arbitrarily) and using tl/2 = 50 nsec as before, we get coagulation time = 12.5 nsec which gives us a concentration, from eq 21b, -14 mM. Experiments are usually done a t much higher concentrations. We then come to the conclusion that the lower limit of concentration is not imposed by neutralization; it is more likely that the intensity of absorption of the solvated electron, i.e., Gt, imposes this lower limit. Actually our calculation puts the electron attachment time scale 30 mM. For this situation the polar molecules will scavenge not only the escaped electrons but some of the geminate fraction also. An uncertain feature of solutions of' polar molecules is polymerization or aggregate formation. Existence of polymers, up to octamers, has been argued from experiments with ultrasonics and nmr and also on the basis of thermodynamics. The basic question here seems to be the following: does the electron coagulate the monomers to form the solvated electron or does it simply get attached to a fairly large-sized entity already existing in solution? Our analysis shows that probably the first alternative applies in dilute solutions but it does not rule out the second possibility for concentrated solutions. In any case it may be safely stated that existence of large aggregates is not a prerequisite for the observation of the solvated electron. On the other hand, if they do exist then the yield (of the solvated electron) should exhibit a sharp concentration dependence in a certain region. This conclusion derives from the fact that the field of a higher order pole varies inversely as a high order of distance, the orientation being unimportant due to inherent angular averaging. Electron attachment to a single polar molecule in the gas phase does not occur if the dipole moment is less than a critical value, -1.6 D. On the other hand it may be argued that in the gas phase the electron can be preferentially ejected into the vacuum. In the solution, however, it can only be thrown into the bulk to be interacted upon electrostatically by a neighboring polar molecule. In this manner the concept of electron attachment by default evolves. Tn our model an absolute attachment in terms of negative energy is not a strict requirement. The model will work satisfactorily if the electron can be held near a polar molecule for sufficiently long time such that a significant coagulation can occur. If we assume for the sake of simplicity that, only translational partition function is a relevant consideration and that only monomers and dimers exist in abundance in the dilute solution then we can simply calculate the monomer to dimer ratio. With )r = 2 D, a = 2 A, T = 300"M, and mass of polar molecule = 4 x 10-2$g, we then obtain the upper limit of dilute solutions as -50 mM. At this concentration monomers outnumber dimers by six to one; however, this is barely high enough concentration for most experimental purposes. Also, the calculation just referred to tends to give a low value for the dimer fraction because of the use of dipolar model of a hydrogen bond. In principle it would be better to base the discussion on the free energies of hydrogen bond formation which, however, is not done here for the complexities and uncertainties involved. Comments on Currently Available Experiments. At present only a few experimentsl~lgrelate to the observa(14). M. Kondo, M. R, Ronayne, J. P. Guarino, and W. H. Hamill, J. Amer. Chem. Soc., 86, 1297 (1964); J. B. Gal!ivan and W. H. Hamill, J. Chem. Phys., 44, 2376 (1966); P. W. F. Louwrier and W. H. Hamill, J. Phys. Chem., 72, 3878 (1968). (15) A. Mozumder,J. Chem. Phys., 55,3026 (1971). (16) T. J. Kemp, G . A. Salmon, and P. Wardman in "Pulse Radiolysis," M. Ebert, J. P. Keene, A. J. Swallow, and J. H. Baxendale, Ed,, Academic Press, London, 1965, pp 247-257.
Solvated Electron ~ o ~ ~ i a ~ ~ Q n tion of solvated electron absorption spectra in solutions of polar molecules in nonpolar solvents. Of these, the work of Kemp, et a1.,I6 seems to be the earliest. They use methanol dissolved in cyclohexane or T H F in the concentration range 4% and up, the entire concentration range being high These experiments are probably not owever, certain general features are already evident. They are (i) slight red shift of peak with dilution, (ii) half-width (but, note, not the shape) and lifetime independent of dilution, and (iii) Gc (and, therefore, by implication G) falling rapidly with dilution. We do not agree with the authors t h a t their experiments indicate that polar aggregates must exist. On the other hand, since in some cases the yield is ca. six times greater than what would be obtained on mole fraction basis, it is reasonable to assume that a significant amount of coagulation has taken place in the presence of the electron. In the experiments of Magnuson, et al.,17 a fully developed spectrum is seen at the end of a 33-nsec pulse of irradiation. They further observe (i) small but systematic red shift of peak with dilution and (ii) spectral shift with concentration stated to be related with polymer formation. Their lowest concentrations fall in the category of dilute solutions in our terminology and their conclusion that trapping i s determined by nearest dipole interaction is consistent with our findings. However, dipole coagulation may not be safely neglected in any of these experiments. use fairly slow pulse The experiments of Brown, et (3.5 met) and a solution of ethanol in n-hexane at 22". No spectral change is geen over the range of concentrations, 5 mol 70of ethanol Their smallest concentration is too in our terminology. Faster pulse and more dilute solution are clearly indicated; however, as they are, these experiments indicate B fairly long life for the so-formed solvated electron. In the experiments of Kenney-Wallace and HentzlS solutions of alcohols (61-612) in cyclohexane are used in the concentration range 0.1-0.5 M . After a 5-nsec pulse of iree a fully grown spectrum which is invariant shape. This finding is consistent with our
3829
description (see Table I and eq 21b). In comparatively dilute solutions a red shift of the absorption spectrum is seen. The concentrations used in these experiments are rather high in our terminology but it is not seen as a serious obstacle in extrapolating eq 21b in the region of some of their lower concentrations. In this paper we have only computed the formation time of solvated electrons. It is reasonable to expect that attachment followed by gradual coagulation will shift the absorption spectrum of the solvated electron to the blue. However, there is as yet no simple a priori way to calculate this effect quantitatively, Values of physical parameters used in this work are for a typical polar molecule in a typical hydrocarbon at room temperature. However, from the experiments we desire to know the dependence of formation time and of the absorption spectrum on the concentration of the solution. We also expect to see a little of the evolution of the spectrum with time. With these factors in mind we should look into media of high viscosity and high yield for escaped electrons. The last item facilitates observation since in a zero-order approximation the yield of solvated electrons i s equal to the yield of escaped electrons. With respect to time scale of ob10 nsec from eq 21b €or a mediservation we calculate t2 um of 11 -0.2 P ( t z 0: 11) and a concentration of 0.1 M . Such a system seems promising as it will put the experiment in a convenient time scale.
-
Acknowledgment. This paper is dedicated to Professor Milton Burton on the occasion of his seventieth birthday. A part of the material of this paper was presented at the Radiation Chemistry Conference held at the University of Notre Dame during April 4-7, 1972. The author benefited from numerous discussions with members of the Radiation Laboratory. In particular, he would like to thank Dr, C. KenC. Abell. ney-Wallace, Dr. Pierre P. Infelta, and Dr. @;. L. B. Magnuson, J. T, Richards, and J. K. Thomas, int. J . Radiat Phys. Chem., 3, 295 (1971). B. J. Brown, N. T. Barker, and D. F. Sangstar, J . Phys. Chem., 75, 3639 (1971). G.Kenney-Wallace and R. R. Hentz, private communication.
The Journal of Physical Chemistry, Voi. 76, No. -75, 1972
