The early events of radiation chemistry

Int. d. Radiat. Phys. Chem. 1975, Vol. 7, pp. 83~93. Pesgamon press. Printed in Great Britain

THE

EARLY

EVENTS

OF

RADIATION

CHEMISTRY

A. MOZUMDER and JOHN L. M A o ~ Department of Chemistry and the Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556, U.S.A.* (Received 29 Apr/l 1974)

Almxaet--A critical review of the early events of radiation chemistry is presented with special reference to aqueous solutions and liquid hydrocarbons. It is concluded that a judicious combimttion of both direct and indirect methods must he used to get a complete picture of the early events. Proper interpretation of ~avenaing eaperiments plays a key role in the indirect methods. Contact reactions at very short times (~ 10-~ s) must also be comidered in concentrated solutions. The aim of the paper is not to develop any particular model in detail but to analyze the natural limitations of model construction. It appears that, for different reasons, there are "barriers" on the short time scale for both direct and indirect ways of inv~tilatiag early events. The proper use of indirect methods combined with early real-time measurements allows investigation of events behind these barriers.

I. INTRODUCTION INTERESTin the early events of radiation chemistry has been intensified by the introduction o f techniques for making measurements at times as short as 10 ps or so after energy deposition c14~. Observations at these early times make possible the more or less direct determination o f initial conditions for t h e radical diffusion processes. In turn, understanding o f the relationship of conditions at 10 tn to the pattern of initial energy deposition seems to be made more attainable. Our considerations here are concerned with a re-evaluation o f the possibilities for eluddation of the time period shorter than 10 ps. There are two general methods for obtaining information at early times. One is the real-time observation made possible by the pulse teclmiques of radiolysis; the other is the indirect method using presumed relationships between concentration and time to obtain information about early,time phenomena from measurements at much longer times. The limit on the real-time measurement may be imposed e i t h e r by the time o f deposit or the transit time of the analyzing light. F o r the purpose o f this consideration a minimal reaction cell can be taken as a cube 3 ram on a side. If the energy is deposited by a &pulse o f electrons travelling with the velocity o f light, the deposit require s 10 ps. The analysing light requires a slightly longer time to transverse the sample and inevitably record as simultaneous effects arising from energy deposits with a time-spread o f 10 ps o r more. It is unlikely that improvement o f experimental techniques will allow time resolutions o f less than a few ps for irradiation with electrons. We refer to this limit as the "picosecond barrier" to real-time observations. The most common indirect methods make use o f data taken essentially at infinite times after energy deposit to obtain information at early times ~t4~. These methods depend upon the conjugate relationship between concentration and time, and therefore have limitations of a different type. Systems with various compositions must be * The Radiation Laboratory of the University of Notre Dame is opera(ed hnder contract with the U.S. Atomic Energy Commission. Thisis AEC Document No. COO.-38-J)45. 83

84

A. Mozt~mn and JOHNL. MA(3~

used in the same study of early events to complete the analysis and so it may be difficult to ascertain the pattern of events for the system of interest (i.e. a pure component). The time scale measured is a "local" time and time resolution usually can be made finer than that obtained in real-time. The concept of the real-time measu~ment is that at various times after a deposit of energy all conditions are observed directly. One determines from such observations the transient intermediates which are present and conceivably all of the various reaction sequences can be worked out. A weakness of indirect methods is that reaction sequences must be known and so such methods must be used self-consistently. The assumed reaction ~ must agree with the ez4~erimental results. In such studies lifetime distribution of Species, such as electrons, are determined rather than time scales. Of course it is the reaction sequences and lifetime distributions of all intermediates which one would like to determine, not time scales as such. The most important i n ~ t e in many irradiated systems is the electron. Such electrons are created almost exclusively in ionization events at local times of 10-Is s or less. Their initial energies are very high compared with thermal but in much less than a picosecond they reach a local thermalization. In many organic media, such a~ liquid hydrocarbons, most of the electrons are recaptured by their parent ions and a small fraction escapes to form free charges. The most important problem to which the indirect m e t h ~ has been addressed so far is the time history of the recombinatiov of the electrons with their parent ions. This time history for the pure hydrocarbon i., the quantity of interest to us. Elementary considerations lead to the conclusion that the scavenger yield in an) system is related to the Laplace transform of the lifetime distribution of the electron., [see equation (8)]. The lifetime distribution can be obtained (provided certain con. difions as discussed in Section IH are fulfilled) by an inverse Laplace transform ant the latter requires experimental scavenger results from small to large concentrations This analysis uses a thermal reaction rate constant for electron with scavenger and the natural zero of time is therefore the time at which the electrons become thermallzed There is, of course, an earlier time history of events which is not susceptible to analysi~ of this type. Thus indirect methods have their own "time barrier" just as the real-time observations do. Other indirect methods can be used to explore the region of time behind this barrier (Section IV).

II. APPROXIMATE TIME SCALE OF EARLY EVENTS All events involved in the u t i ~ n of the energy deposited in a spur can b~ described on a single time scale; this is the simplest example of what we mean b2 "local time". The earliest time of significance is the energy deposit by the prima~ particle, and the uncertainty relationship (1)

A t A E ~ h ~ 6"6 x 10-16 eV s

can be used to obtain it. The m o s t probable energy of a spur in water is about 20 eV so we have At = 0-33 x 10-is s, and we often say that the r~

f i ~ is 10-as S.

The earlyeveat, of ~

ehemistry

85

Another aspect of the formation,of a s p a r is the uncertainty in position x of the energy deposit arising from the uncertainty relationship, wherep is the momentum of the particle (2)

Ap Ax ~ h,

which applies to the primary particle. Calling u the speed of the particle, from elementary relativistic mechanics we have (3)

Ap = AE/u

so that (4)

~

~, hu/AE,

and for a relativisticparticle'which loses 20 eV

Ax ~ I0 -e cm. Equations (1) and (4) are clearlyconsistent, just giving different aspects of the uncertainty. We can see this as follows : (5)

At ~ Ax/u ~. h/AE.

In the sub-picosecond region we can distinguish t h r e e ~ i m e domains. In the earliest (starting at 10-m s), :only eleCtrOnic proeesSeS m e possible. The time is too short for molecular motions of any kind. In the second (starting at I0 -It s or so) molecular vibrations become importmlt and ener8~' flows into them. In the third (starting shortly ~ I0 -is s) the lower-grade m o l ~ motions become excit~ and a local temperaturebecomes meaniilgful. (We. Can'note that'At = h / k r ~ 0-3 x 10- ~ s at T = 300 K.) Diffusion processesstart in ~ e picosecondtime scale when all degrees of freedom of the system become activated. Table I summarizes some current ideas on the early events in water. In the time scale of 10-le s, excited (HIO'), superexcited tHtO ~) and ionized (HtO+,H,O +') states of the water molecule are produced toge~er with the &rays and other (relatively) slow electrons as a result o f energy absorption. According to Platzman ct~ the positive ion of water has, on an average, 8 eV of excitation energy (HIO +'). The various excited and ionized states are not clearly discernible, since on this.t~e ~kle the energy uncertainty is = 10 eV. On the other hand, a s~tistical correlation problem is already evident and impormm at this time scale in relation to the spur theory. It may be stated as follows: given an ionization or excitation at a place, what is the probability of having another event (excitation or ionization) in the immediate vicinity conastentwith a given average ene~gyloss. Another way 0 f looking at the same problem is the distribution in species and space for spurs of given average energy. It has been assumed that the average nnmher of species nx(*) of a given type x in a spur of average energy, is given by a relationship

(6)

n=(Oo¢ C,(x),,

whe,e G(x) is the G-valua for production of the species. Very littleis actually known of the spatial distributions but it has usually been asmaned that they are gauuia~ about some center and that the maximum eoneen~atien~ofany tlmci'es is the u m e i n all t m e k entities. All of these ~ u s require further invatigetion.

A,~ M o z u s m e g and, J o ~ L. MAOEZ

86

TAaLe I. Su3et~y oF ~XLY ~Wm'S IN ~ 10-xe s

10-xu s

10-x4 s

Hie*

All electronic states discernible

HzO~ HlO+

H~O:~ ionizes Dry hole Ion-molecule (Hie +) reaction migrates by HlO + + HsO --* exact HsO+ + OH resonance takes place

IU~DIOLYSB OF WATEF.(LARGELYPOSTULATORY)

10-xs s

10-xg s

HsO* and HiO~ dissociate

H i O +*

e-

Spur electrons Spur electron fall to subacquires epiexcitation thermal energies energies

Electron thermalizes, is trapped and subsequontly hydrated. Prehydrated electron scavenging in highly concentrated solutions

Some diffusion kinetics with time-dependent rate coefficient. Hydrated electron is produced and starts to react. Limit of laser and pulse radiolyfic experiments (the picosecond barrier)

* Excited molecule, tSupervxcited molecule. + Ionimd molecule, t + Ionized molecule in excited state. In the time scale (10 - ~ s) nothing of great chemica! s i ~ c a n c e happens. Thi,, time scale may be considered as an incubation period. According to Kalarickal and Magee m), the dry hole ( I ~ O +) can move by exact resonance on this t i m e scale Excited states can in principle also migrate if the resonant transfer time is shorter that the vibrational time. In the 10 -14 s time scale the radicals produced from the dissociation of HaO* anc H I e s i~itially have the same distribution as those of the excited states themselves The latter are, in principle, derivable from the distribution of energy deposition el the primary particle and the secondary electrons. In practice, however, this is largel~ unknown. The HsO + ion produced by the ion-molecule reaction is not hydrated a~ yet. Initial distribution o f O H radicals from this channel is the same as the'distribu. tion o f HsO +. Consideration o f thermalization (AE = 10-a eV) is not meaningful for times les: than 10 - ~ s, ~although it is conceivable that thermalization, trapping and hydratio! can follow in .quick succession. It is also plausible that the electron is never full: thermalized but is simply trapped followed by hydration. In this time scale radical diffuse a tittle [root mean square (r.m.s.)tfisplacement = 0.5 A]; however, they ma~ react in a conCentrated solution in the fashion o f static quenching (Czapski ant PeledCg)). On the other hand, the pre,hydrated electron can move significantly (r.m.s displacement = 30 A) and therefore has a significant probability of reacting in concentrated solution in competition with hydration and neutralization. The time scale o f 10 - ~ s (picosecond) signals direct observability (actually = 10 p in current experiments). In this:t~me Scale diffusion-kinetic reactions take place wit] time-dependent rate coefficients. The time dependence, sometimes referred to as th~ Smolnchowski transient, is a result o f the changeover, during the diffusion proces~

The early eve~ta 0f radiation chemis.~

87:

o f thediscontinuous concentration gradient at the reaction radius (at t ffi O) to smoothly varying concentration gradient given as a solution of Smoluchowski's equation (for t > 0). This stage has a short duration after which the regular, timeindependent rate coefficients apply. III. THE THERMAL TIME SCALE The Laplace transform method for information at early times has been used principally for obtaining the lifetime distribution for free electrons in liquid hydrocarbons. In such systems electrons are formed by ionization, become thermalized before they escape from the fields of their geminate ionsand recombine on the picosecond time scale. Various scavengers are used which can react with the electrons; the probability per unit time that an electron will react with a scavenger is (7)

-- [ d F ( t ) / d t ]

=

ks'[SlF(t),

where F(t) is the probability that an electron thermafized at t = 0 is stilluncombined at time t,ks' is the reaction rate coefficientand [5] is the scavenger concentration. The function F(t) is the desired information to be obtained by the scavenger studies. The total probability that the electron reacts with the scavenger before it is neutralized is equal to (8)

oJ(~) = ~

fo oexp ( -

At) F(t) dt,

where )t = kB'[S]. It is clear that if oJ(A) is measured experimentally for all scavenger concentrations from zero to or, F(t) can be obtained by an inverse Laplace transform. This method has been used quite extensively to study electrons in organic liquids and much is known concerning their lifetime distributions. There is a fundamental dit~culty in.the application of the Laplace transformation method in systems such as water whicli have second-order reactions. We illustrate this difficulty by consideration of the diffusion-controlled reactions of a species with itself in competition with a homogeneously distributed scavenger. The relevant equations are (9)

OCo/Ot = DV 2 Co- k' Co2

and

(m)

oc/ot = DW C - k ' ~ - k ; C, C,

where C0 and C are, respectively, the concentration of the reactant species in the absence of scavenger and in the presence of a scavenger at concentration Cs, D is the diffusion coefficient and k' and k 8' are respectively the specific rate constants for reaction with itself and with the scavenger. The total amount of scavenger reaction in a volume V is, with ~ ffi k 8"Cs, (11)

N(~)= ~ f:dt f:CdVffi ~[Laplace transform of fl(t)],

where

02)

f1(t>: ::C1d v

88

A. M o z u ~ m t

and Jom~ L. MAomz

and C = Cxexp(-At). However Cx is not the solution of equation (9), but instead is a solution of the following equation (13)

OCx/Ot = DV" C 1 - k' exp ( - At) Cxs,

which, in reality, does not describe the chemical reaction in any system. It is thus clear that the Laplace transform procedure used for aqueous solutions ~x°~can at best be a crude approximation valid in the limit that non-linear (second-order) reactions are negligible and in any case rather large errors are to be expected if such a procedure is used in a straightforward manner. It must be stressed that, in this context, the most important contributor to non-linear reactions is the electron-electron reaction. In the opinion of the authors this single reaction (or its equivalents) represents an essential difference between the radiolysis of water and liquid hydrocarbons. The consequence of this argument is that the application of the Laplace transform method to aqueous solutions remains to be a rather doubtful procedure. In a manner of speaking, even if F(t~) of equations (7) and (8) were available for an aqueous solution it would stiU not be possible to get the con'esponding function in the absence of the scavenger, since these two functions are not simply related when non-linear reactions are simultaneously present. In contrast, the electron-electron reaction is presumably unimportant in organic systems. Therefore the Laplace transform method is well applied in such systems tc reactions which are mathematically describable by linear equations (4-6a, b). Agai~ it must be stressed that the relevant question is the linearity of the partial differential equation (the S m o h ~ o w s k i equation) deseribing'the micgoscopic situation and no1 the overall order of the macroscopic chemical reaction. Thus ion-neutzalization, which in a chemical sense is formally a second-order process, can nevertheless tx treated through a Laplace transform method since the relevant Smoluchowslc equation is indeed linear. One of the authors (Mozumder ~sb~) has shown that ir hydrocarbons even the neutralization in multiple ion-pair cases can he treated throug~ a L a p l a ~ transform method. The Smoluchowski equation in this case can be writter as below* (14)

~P/~t = D[V s P - (kT) -1 V(PF)] - AP,

where P(r, t) is the probability density of existence of an electron around point r a time t, k, the Boltzmann constant, T, the absolute temperature and other quantitie~ [except F, which is defined in equation (16)] have their previous significanccs. W i t the transformation P(r, t) ffi P0(r, t)exp ( - At), equation (14) can be reduced to the case in the absence of scavengers as follows (15)

aP0/Ot -- D[ V~ P 0 - (kT) -x V(P0 ¥)].

In equations (14) and (15) the screened electrostaticfieldF is given by

10

l

* The principle is illustrated here taking spherkml geometry as an example. An exactl: analogous situation exists for cylindrical geometry.

The~cedy ©vea~ of l'~[mtion ~J~m~s~'y

where 0"7)

f:Po(r,

fs(t) ----

t):41rrSdr

is the number of electronssurviving neutralization at 6me t in the absence of scavenger if initially tiiere were n ion-pairs in,the entity. The net amount of scavenger reaction is given by

(18)

N(A) ----

t Afo°dfe '~P(r, t) d V

fi°d, xp(

OdV

-- A[Laplace transform of f~(t)].

It is important to real/ze that in deriving the Laplace transform relationship [equation (18)] in equations (14)-(17) P appears nowhere with a power higher thau unity. This justifies :the Laplace transform method even in the neutralization of multiple ion-pair entities although the field ¥ doesindeed evolve in a very complicated m a m a . This is indeed possible because of the absence of the ¢lectrea--ele~on reaotion. In its presence, as for aqueous solutions, even spurs containing initially two electrons cannot be adequately described through a Laplace transform procedure. The use of scavenger measurements at very long times as a function of scavenger concentration to predict fromindireet calculations the real-time observations cannot in any case yield valid results for times much shorter than a picosecond because thermal reactions do not take place at shorter times. This is a fundamental di$celty with such attempts to obtain information about early times; one cannot obtain anything about the precursors of the radicals one is observing in real time. There are, in addition, other serious problems with both the direct gad indirect methods. One is the time-dependence of reaction rate coelr~ents. This is an intrinsic difficulty. There are two sources of time variation of a reaction rate coefficient as pointed out by Schwarz(n). One is the relaxation time for the establishment of a steady-state concent~ration gradient about the reacting species. Although we know that such an effect nlUst exist, the case for an inhomogeneous distinction such a s found in a track has not been solved theoretically. It is presumed that the general magnitude of the effect is the same as that for the initially homogeneous system, i.e.

(19)

k'(t)

=

k®'{1 + [R/(~rDt)~]},

where R is the reaction radius, D the diffusion constant and k®' the reaction rate coefficient at long times. The second source of time van'ation is the relaxation of the ion atmospheres and applies to reactions between charged species, particularly in water. This problem has'not been considered explicitly fortracks. An oth~..:intrin~ di~culty is the possib/I/ty o f a "sta~":reaction. Attention has been called to this problem by CzaI~eki and Peled(s). A transient intermediate such as a hydrated electron may be formed within a reactive radius of a scavenger. Clearly the r e a c ~ n o f such a pair will net be dmcribed by a ~ ' m m eq[imtioL ."Be/'~ the Laplace mumform method can be used in a n y . ~ - c a s e it must be demm~trated:that these ~ ditliculties do not invalidate the application~

90

A. M o z u t m ~ mad JOHN L. MAO~

IV. THE PRE-THERMAL TIME SCALE

In the Laplace transform method the scavenger reaction is competing with the recombination of thermalized electrons and geminate ions. The earliest possible time which can be involved is ~ 10- ~ s, the thermalization time. In fact, the time t = 0 in the Laplace transform method is 10- ~ s after energy deposition. If earlier times are of interest, another method must be used. We have noted that in this earlier time scale scavenger reactions are not thermal, but involve species at higher energy. They are "hot" or "epithermal" processes. Hamill(Z~,In) has suggested that in the earliest time scale the epithermal electrons (he calls them "dry" electrons) react with scavengers in competition with the hydration process according to the following mechanism. (20) (21)

e- + S e-

k'

k.' • S-,

> et -

> eaq- ,

where in (21) • t - indicates a state which irreversibly goes to hydration. The reaction rate coefficients are to be taken as the appropriate averages (ov~ where ~ is the cross-section and v the electron velocity; a trap concentration also appears in k'. In this simple mechamsm the fraction of reaction in the scavenger channel is (22)

k,'[S] A = k,'[S] + k ' "

The ordinary form of expression used in the analysis is 1

(23)

k'

~ = I q ks,iS],

and one expects to obtain a linear plot for f - x vs [S]-1. In (23)fa is proportional to G(s), the measured yield at long times. The competition described by equations (20) and (21) is usually called a Stern-Volmer mechanism. Sawai and HamilIm) and Ogura and Hamill(m have shown that in many concentrated solutions in water and alcohols the experimental results for the yields of hydrated electrons seem to be in agreement with this mechanism. If assumptions are made that the epithermal electron hops about from a molecule to neighbouring molecules, that the trapping process (21) can occur at any site and that reaction with S occurs on every encounter, the time scale can be established. The analysis sets the trapping time at about 10- ~ s in both water and alcohol. A similar analysis can be made for the motion of the epithermal H~O÷ ion. According to Ogura and Hamill(1.), in concentrated chloride solutions the reactions (24)

HsO++C 1_

(25)

I-l~O++HtO

k~" , HIO+C1 ' ~' > HaO++OH,

are in competition, followed by reaction of the Cl atoms with chloride ions: (26)

CI + CI-

> Cls-.

Measurement of the yield of CIt ~-gives the effectiveness of the scavenger reaction (24). Here it is remoaable to assume that the I-gtO+ hole jumps between neighbouring molecules randomly and reacts on each encounter with C1-. The results for the ratio

The early evonts of radiation chemistry

91

k'/k a ' was found to be 14. This means that the I~O + jumps on the average 14 times before reaction (25) occurs. Estimates of the jump time~s~ are 10"~s and t h e fime scale of the ion-molecule reaction (25) is10 -14 s. Another unresolved problem exists with respect to the pre-thermal period. The Stern-Volmer mechanism of Hamill and his associates requires an electron reaction which is proportional to the scavenger concentration because of the competitive nature of the depletion [equations (20) and (21)]. His experimental results seem to be compatible with this mechanism. On the other hand, Hunt et al.W~ have reported an exponential depletion of electrons, i.e.

(27)

Ge[S] =- G° exp ( - [S]/[S~]),

where Gee is the observed yield in theabsence of scavenger and ~Sa~] is the concentration of scavenger for which the initial yield is reduced t0 e-x (i.e. 37 per cent). Hunt et al. have not proposed a mechanism which would lead to this result, but Czapski and Peled~°~have argued that it is in agreement with the reaction of hydrated electrons with ions in contact at the times of formation. They derive an exponential fonction of the type of equation (27) using Poisson statistics for the probability for formation of contact pairs. Although one can criticize the use of Poisson statistics, the basic result remains valid: in concentrated solutions, the formation o f contact pairs is to be expected. This situation is discussed in the Appendix. Because of the difficulties with ionic reactions in the ~ ha:tier time period, Ogura and Hamill ~xe~looked for a system which would be free ofthe problem. Ttiey found that solvated electrons in ethyl alcohol react very slowly (k' = 10T dins mol-X s -:) with benzene, whereas there is asubstantialreduction ofelectronyields at ~ 1 moldm -~concentrations. This appears to .be a case i n which there is a pre-thermal reaction uncomplicated by the thermal barrier problems. The Stern-Volmer mechanism is in good agreement with the experimental results, and it appears that the trapping time [equation (21)] is about 10 - ~ s. Another kind of relatively short-time (= 60 ps) radiolysis experiment has been done by ll¢~k and ThomasClV~ in investigating ionic and excited states in liquids. They found tha~ the sotvated ele~ron was formed in ethanol, i~ 2-5 In but took ~ 50 ps to form in 1-propanoL A l s o the excited state formedrapidly in cyclohexane but it decayed slowly.

i

V. DISCUSSION

It is clear that phenomena which occur on the sub-picosecond time scale can only be investigated by indirect means. I n order to obtain a complete description of the processes of an irradiated system, one must conceptually use at least two types of measurements. The very earliest processes require rather large scave1~gerconcentrations and the results should be analyzable using the Stern-Voimer method of Hamill eta/. Of course, there may be complications arising from contact reactions of~olvated electrons and uncertainties in the time-dependence of early thermal ~ . The times immediately following thermalization can perhaps be analyzed by the Laplace transform method and the latter results should be consistent with real-time observations on the sub-nanosecond time scale. T h e ~ c a l l y itls possible to imagine various scavengers with which the required measm'ements can be made, but in practice not

92

A. MOZOMDI~ mad JOHN L. M.~3EB

manY such scavangers are known. Ogura and Hamillcle~ say that benzene in ethyl alcohol is an example as we have seen in Section IV. In the large-scavenger fimit used to explore the short times, the scavengers must be effective on thedesired time scale at concentrations small enodgh so that they do not make substantial changes in the energy deposit patterns. The changes they make in the subsequent events must be subject to systematic analysis. In the small-scavenger limit used to explore the post-thermalization time scale, the system must be linear. It has been implicitly assumed that electron scavenging in a wide variety of organic liquids by many scavengers satisfies this condition and analysis of the experimental reselts seems to verify the applicability. The Smoluchowski equation which has been widely used to investigate initial distributions and relate them both to lifetime distributions and free ion yields only applies to geminate pairs. The geometrical structure of tracks is quite well known and gem~inate pairs comprise less than half the ionization in any sysmm of interest. One of the authors (A. M . teb)) has considered the track entities with larger numbers of pairs but how to apply the diffusion ¢qtmtion to these entities is still not well known beyond the prescribed diffusion approxivmtion. However, these entities can be considered as linear in the seine re~luized for apl#ication of the Laplace transform method in all cases in which the electrons do not react with electrons. On the most 8eneral grounds one can argue that pre-thernmlization reactions of eleOtrons can occur. Electron capture-reaSons areknown to occur at higher energies in the gas phase and the same mecbank-m~ in ¢omdemed phases cannot be ruled out. Also, on generglgrounds in concentrated ~ s o k t ~ contact reactions and anomalously fast early theamal processes are possible. Evaluation of the relative importance of these mechanisms is necessary before the pre-thermal time period is understood. DEDICATION

This paper is dedkated to the memory of tht late Professor R. L. Platzman whose interest and insight in the early events of radiation chemistry have inspired experimemtalists mad theoreticiam alike for over a quarter of a century.

REFERENCES 1. J. W. HUNT, R. K. WOIWF,M. J. BI~O~Xff.L, C. D. JONAH, E. J. HART and M. S. MATt~SON, J. phys. Chem. 1973, 77, 425. 2. R. g . WOLFF,M. J. BtONS~__I~, J. E. ALDIU~WIand J. W. HUNT, J. phys. Chem. 1973, 77, 1350. 3. C. D. JONAH, E. J. HART and M, S. M X ~ N , J. phys. Chem. 1973, 77, 1838. 4. A. Hutman~,.I. chem. Phys. 1968, 0 , 4840. 5. S. J. R z ~ , P. P. I ~ T A , J. M. W,~.J~N and R. H. S¢~.nam, J. chem. Phys. 1970, 52, 3971. 6. (a) G. C . A~LL, A. MOZU-U~ and J. L. MAO~, J. chem. Phys. 1972, $6, 5422. (b) A. M o ~ , J. chem. Phys. 1971, 55, 3020, ~3026. 7. R. L. Pt~TZMAN, in Radiation Research, ~ b y G. Sham, North-Holland, Amsterdam, 1967, pp. 20-42. 8. S. ~ g . ~ , Ph,D. Dissertation, University of Notre Dame, 1959. 9. G. Czs~s~l a n d E , ~ , J.plrys. Chem. 1973, 77, 893: 10. T. I. B ~ t , J. H. FeNDl~t and R, H, ~ l ~ J t ~ J. phys. Chem. 1970, 74, 4497. 11. H. A. ScI~MtZ, J. ehe.m. Phys. !971, ~ , 3647. 12. T. SAWAIand W. H, ~ , d. phys; Chem. 1970, 74, 3914. 13. W. H. HtmLL, d. phys, Chem. ~969; 73, 1341,

The early events of radiation chemistry

14. 15. 16. 17.

H. R. H. G.

93

O o u t ~ and W. H. HAMILL,J. phys. Chem. 1973, 77, 2952. K. WOLFF, M. J. BItOI,~JaLL and J. W. HUNT, J. chem. Phys. 1970, 53, 4211. OGUP.Aand W. H. ~ , J. phys. Chem. 1974, 78, 504. Beck and J. K. THOMAS,J. phys. Chem. 1972, 76, 3856.

APPENDIX: CONTACT REACTION OF THERMALIZED ELECTRONS As Czapski and Peled c'~ have pointed out, the problem can be analyzed in terms of the probability that the electron will be thermalized (hydrated) in a small volume v which does not contain a reactive scavenger molecule. If the entire system contains N such molecules in volume V, the probability that none of these N particles placed at random in V will fall in the small volume v is = ( l - v l V ) N. Since N i s very large (~10 le) we have 1~ -- exp ( - Nv/V),

(a)

and this is the probability for survival of the electron against contact reaction. In fact, however, the chemical solution has a very'definitestructure and the scavenger molecules, especially if they are ions, are not arranged randomly. Let us consider the same problem from a more rigorous point of view using a grand ensemble. The required ensemble has the independent variables V, T and N. (It is not the grand canonical ensemble~of Gibbs which has the independent variables V, T and/~, where ~ is the free energy per particle.) Our treatment is elementary. The ordinary partition function of the system is called Z(N, V, T). We consider a grand ensemble which has representative systems with volumes which differ by varying amounts of v, where Z(N, V - v , T) is the partition function for those systems which have the volume V - v . In this ensemble the probability that a representative system chosen at random will have the volume V - v within the interval dv is Z( ~'- v) dv y(v) dv -- i: o

(b)

Z(V-v') dr'"

where we have dropped the arguments N and T because they do not vary. Although the partition function is diff~ult to obtain, we know the relationship (c)

In Z ( V - v ) -- - A / k T ,

where A is the work function or Helmholtz free energy and k is the Boltzmann constant. Since A is an extensive property and v is a very small quantity, the e x p a ~ i o n (d)

i n Z ( V - v ) = InZ(V)-v(~-~--Z)~v,~,

can he used. We also know

(a!nz

(e)

=

from elementary thermodynamics where P is the pressure. Substitution of (d) and (e) into (b) and performlng the integration gives (f)

~,(v) dv

=

exp ( - P v l k T ) d(PvlkT).

We can now give the probability r(v) that the small volume is at least equal to v as the integral (g)

r(v) =

/:

(P/kT) exp ( - P v ' / k T ) dr" = exp (-.Pv/kT).

In this formula P is the osmotic pressure of the scavenser which is to he excluded from the small volume v. This consideration allows many hierarchies of refinemeoat, such as effects of structure, ion atmospheres, etc; We can note at this time that in the lowest approximation the ion behave~* as a perfect gas, P = NkTIV, and the Czapski and Peled approximation of equation (a) is obtained. On the other hand, it should be emphasized that improvements on the basic formula are clearly possible with a knowledge of the structure of the solution.