The reaction H+H→H2 in solution: Yields of ortho- and para-hydrogen

Radtat P h w ( h e m Vol 23, No 1 2, pp 187 192, 1984 Printed in Great Bntam

0146 5724/84 $3 00 + 00 Pergamon Press Ltd

THE REACTION H + H - - * H 2 IN SOLUTION: YIELDS OF ortho- A N D para-HYDROGEN BRIAN

BROCKLEHURST

Chemistry Department, The University, Sheffield, $3 7HF, England

(Recen,ed 26 January 1983, a~cepted 23 Mar~h 1983) Abstract It ~s proposed that the reaction. H + H--,H z, can only take place ff the atoms meet m a slnglet state Tnplet pa~rs of atoms will separate, but re-encounters are probable and the spm wave-function may change because of the electron proton hyperfine interaction the rate of th~s process can be altered by apphed magnetic fields A kmeuc model has been developed to study the effects on the reaction rate and on the ortho/para ratio of the hydrogen produced The predicted effects are small but should be detectable because of their dlstmctwe field dependence INTRODUCTION RECOMBINATION reactions of hydrogen atoms and hydroxyl radicals (1)--(3) are very i m p o r t a n t m the radlolysls of aqueous solutions

THE

(I)

H+H-~H2

kl=10×

10J°l mol

Is

(2)

H+OH--*H20

k2=25×

101°l mol

Is

(3)

OH+OH--*H202

k~=05×

l0 I°l mol

is

i

A range of values of their rate constants can be found in the hterature ~ some are d e n v e d from d~rect measurement, more from fitting spur models to experimental data, the values quoted are taken from R e f l b m which Boyd et al. showed that they are self-consistent insofar as they gwe the best fit to spur b e h a w o u r m a c o m p u t e r model One would expect all these reactions to be diffusion-controlled, there should be no need for activation Therefore, it is surprising that reaction (2) is the fastest of the three It has been suggested (2) that (1) is spln-hmlted while (2) and (3) are n o t - - t h e result of the very fast spin-relaxation of the hydroxyl radlcalJ~ 4) The s~mplest expression for a diffusion-controlled rate can be w n t t e n (4)

k = 4nao(Di + D2)p

where p IS the probability of reaction m each encounter T a k i n g p =¼ for (1) a n d 1 for (2) a n d (3) a n d using the recently reported diffusion coeffioents (D) for H and OH, ~s~ this gives reasonable values of the reaction radius, a0, of 3 4-3 8 A for all three reactions This result must be accidental to some extent, gwen the range of values m the llterature(td) however, ~t is clear that the ratio of the rate constants

of (1) and (2) is smaller t h a n one would otherwise expect The assumption is that two h y d r o g e n atoms meeting In a slnglet state will react if they are in a triplet state (probability = ~ for encounters at r a n d o m ) , they will separate again However, on separation, the chance of a re-encounter is high ( > 0 5) ~6.71before the second encounter, hyperfine interaction between electrons and nucle~ may change the triplet Into a slnglet spin state In turn, the probability of th~s depends on the orientations of the two nuclear spins relative to each other and to an external magnetic field leading to "'spin selectlon"--preferentlal r e c o m b i n a t i o n for some nuclear states In the case of two (d~fferent) radicals this a r g u m e n t ~s the basis of the radical pa~r theory of chemically induced polarlsatlon ( C I D N P and CIDEP). is/ F o r two hydrogen atoms, the consequence may be changes m the relative yields of ortho- a n d para-hydrogen ~9) The aim of this p a p e r ~s to develop a kinetic model for the re-encounter process in order to estimate the extent of the ortho/para differentiation a n d its dependence on magnetic field To the a u t h o r ' s knowledge, no effects of th~s type have been reported for reaction (l) m condensed phases The earher calculations ~9~were limited to zero and high field and employed very drastic simplifications THEORY

Spm-state evolution The n o t a t i o n of R e f 9 is retained ct and fl represent electron spin wave-functions as usual, to avoid confusion, ?' and 6 are used for the p r o t o n s with m j = + ~ Spin e v o l u t m n in a pair of well-separated a t o m s is considered first, the role of exchange is d~scussed later The 4 particles give rise to 16 spin

188

B BROCKLEHURST

states, it lb assumed that all are equally p o p u l a t e d lmtlally (N B the energies are ~ k T even in a very strong field) Conversion between them results from hyperfine interaction between electrons a n d protons (~ '())This is most easily u n d e r s t o o d in the case of a very strong external field one can think in classical terms of the electrons precesslng r o u n d the field dlrecUon at a rate which depends on the sum of the external field and the local field of the p r o t o n If the p r o t o n spins are parallel, the rates are the same no change occurs If they are antl-parallel, then the electrons precess at different rates a n d their resultant angular m o m e n t u m oscillates between 0 and \. (2)(h/2~), ~.e the spin state oscillates between singlet and the M = 0 c o m p o n e n t of the triplet The other two c o m p o n e n t s T , and T c a n n o t give slnglet One can immediately predict that p a r a - h y d r o g e n formarion will be favoured at high field At lower held the problem is more complicated spin flips can occur, lnterconvertlng :~5 and [:17 states of each h y d r o g e n a t o m All three triplet c o m p o n e n t s become involved in q u a n t u m - m e c h a m c a l terms, the slnglet and triplet states of the pmr are stationary states of the system when the exchange Interaction, J, is p r e d o m i n a n t at small separat)ons they become n o n - s t a t l o n a r y - - a n d so the system oscillates when the a t o m s separate and J becomes negligible c o m p a r e d to the Z e e m a n (gtqB) and hyperfine (a) terms in the h a m l l t o m a n The superposltlOn principle is used to calculate the rate and extent of the oscillation Fortunately, the lnterconverslons of the 16 states are simplified by two hmltatlons Parity )s conserved it follows that triplet ortho converts only into slnglet para and vice versa (See Ref 9 a n d especmlly the corrigendum) Secondly, Em Is conserved ~:q';'( - T , 7 , ' ) is the only Z m = 2 state and c a n n o t change, there are 4 E m - I states but only T+ para ~c~(7(~-@) can convert into the slnglet ortho (~[~ -[~c~)7"), ( n o r m a h s a t l o n factors are omitted), the probability of an lmtlal triplet glvlng slnglet at time t ~s represented by p) The Z m = - 1, - 2 states behave m the same way There are 6 E m = 0 states 2 have odd parity T0 para - (~[] +/3~)(`/J - 67) a n d S ortho = (~[~ - [ ] e )('/(~ + (53,) (conversion probability p0) F o u r have even p a n t y T , T~) and T , ortho - ~ j , (~fl + fl~ )(`/ii + 6"/) a n d [~[~?`/ a n d slnglet para-- (:~/~ - [~)(`/~ - @ ) (Conversion probabilities p~, P2 and p~, respectively N B T~ j,5 and T 7`/ are essentially identical). Conversions between T~ and To(p~ = p , ) must also be considered The theoretical t r e a t m e n t given previously (~°) is readily adapted to the present p r o b l e m the e q u a t m n s needed are

(6)

/,() :

(1

l~2a2h: + a ~

cos at ),,'2

(7)

t),

(8)

p, -- {2h 4 ~ 2a~h '- + a 4 + a'~hZcosl t + a4cos'~l t 2/cosat

2a2h'~COSlt - a4cos21t}/414

(h: + a 2 c o s l t ) } / 4 1 ~

w h e r e / " - a ' + h e, h = M ~ B (the Z e e m a n energy) a n d a, h, I are taken to be expressed in radlans per second For a hydrogen atom, a is very large (1 4 2 G H z ) , g l i b = 2 8 0 6 G H z for a field of one Tesla (104G) Ele~ Iron e rchan~e l~or a t o m r e c o m b i n a t i o n there should be neither an a c t l v a n o n barrier n o r ,t stenc restriction then the reaction distance, a(,. will be the point at which the valency [brces (essentially the exchange Interaction, J ) become c o m p a r a b l e wlth /,7" Since a ~ k T ( ~ 6000 G H z at ordinary temperatures), there will be a range of distances for which J >> a In this region there will be no slnglet-trlplet conversion due to hyperfine c o u p h n g as J increases from zero, the oscillation becomes faster but the extent of the oscillation decreases b e c o m i n g negligible when J ~ a (sl~ A complete description of an e n c o u n t e r would require a full knowledge o f the dependence of .I on distance between the a t o m s (r) F o r hydrogen atoms, unhke ra&cals in general, this q u a n t i t y Js k n o w n precisely for the gas phase, ~ee e q u a t i o n (9)/~:~ (9)

J(r)/(rads

I)=2 1 x 10'~exp(-346r/A)

(10)

l(r)/(rads

1)= I 0 x [ 0 l ~ e x p ( -

l 71r/A)

However, as A d r i a n ('~ pointed out when he applied the concept of re-encounters to C I D N P , there will also be a longer range ln&rect interaction t h r o u g h the intervening solvent molecules Th~s is very difficult to calculate a n d A d r i a n ' s original estimate, eqn (10), is w~dely used These equations are plotted in Fig 1 (In a d a p t i n g A d r i a n ' s e q u a t i o n to the present problem, the "'molecular d i a m e t e r " has been taken to be 2 4 A, i e twice the van der Waals radius of the h y d r o g e n a t o m ) It will be seen that the range of distances for which J and a are c o m p a r a b l e is fairly small "1-o simplify the problem, the following a s s u m p t i o n is made J > ~ a when r < 7 5 ~ , J = 0 when r > 7 5 A , the effects of this a s s u m p t i o n are discussed later Therefore, the evolution o f the spin wave function of a triplet pair of a t o m s can be taken to start at 7 5 A they c a n n o t react unless they first separate so far Diffusion a n d re-encounters The simple case of conversion between two states,

The reactton H + H-~H 2 in solutmn ymlds of ortho- and para-hydrogen I

I

l

1

i

i

101/*

M o z u m d e r o4) Adapting their equation (9) for a function gives equation (13). (13)

<

189

R(t) = (4~zDt 3) '/2ao(1 - ao/ro) × exp { -- (r 0 -- ao)2/4Dt }

kT

"',,\

%

If there were no spin limitations, the overall reencounter probability,/32, would be gwen very simply by (14)

~oit

(14)

1013

/3~ =

f;

R(t) dt = ao/ro

So the yield of ortho-hydrogen is calculated from equation (15) m which p+ is the average value of p+ and P0 corresponds to P0 (15)

°°t t

+ p0/(1 - / 3 + p0)}/16

1010 OH

10 £

t

dp(ortho) = {3 + 2p+/(1 - 13 + p + )

J__

I

>

I

I

I

6

r/~

8

I

FIG 1 Dependence of direct electron exchange and exchange through the solvent on separatmn of two hydrogen atoms, from Adrian (Ref 13) Annotatmns--see text

For para-hydrogen production, the problem is more comphcated because four rater-converting states are revolved extensmn of the d e n v a t m n of the two pair case gwes (16) (16)

49(para) = (1 + [2p~/{1 -- 2p~p3/(1 --/3 + P2 + 2p3)}

+ 2p,p2/{(1 - 2ptP3)(1 --/3 +P2 the re-encounter probability and p is the probablhty of reaction (probabihty of finding the system m a smglet state) then a fractmn/3p will react m the first re-encounter, while / 3 ( 1 - p ) separate, a fraction fl2p(1 - p ) react in the second re-encounter and so on. If p were fixed, the overall reaction probability, q~, would be gwen by (11)

+ 2p3)}1/(1 - / 3 + p, + P3) + [Pz/{ 1 -- 2p~P3/(I --/3 + P , +P3)} + 2plm/{(1

-

2p~o~)(1-/3 +p,

+ P3)}]/(1 - / 3 + P2 + 2p3))/l 6.

Monte Carlo calculattons (11)

q~ =/3p(1 --/3 + tip).

In fact, p is an oscillating function of time, (cf. equations (5-8), to slmphfy the problem, equatmn (11) has been used with an average value of tip(p) calculated from (12) (12)

/3(t)R(t)dt

p = 0

where R ( t ) d t is the probability of a re-encounter (separation a0) occurnng between times t and t + dt. For the present purpose the initial distribution of separatmns is taken to be a 6-function at r0 ( = 7.5 A above). A convenient formulation of the radial diffusion of two neutral species is given by Abell and

The above treatment contains several assumptions and simphficatlons For example, part of the time between encounters is spent at pair separations between r 0 and a 0 where spin evolution cannot occur For the ~mportant region of short times the error may well be significant. The averaging of /3 and p can be avoided by making detailed numerical calculations A Fortran programme has been wratten using random numbers in conjunctxon with equation (13) to decide whether an encounter takes place in a gwen time-interval if it does, evolution of the wave-functxon up to that time is calculated and the probabilities of finding smglet pairs m ortho and para nuclear states are evaluated Exchange mteractxon alone cannot rater-convert

B BR()( KLFttVRSl

190

slnglet and triplet states However, it does alter the phase relationship between the smglet and triplet c o m p o n e n t s (~s and ~b:)I~* glwng a new wa',efunction

(17)

number o f exteilS]OllS ate possible e g leactlon ,.~,flh ,i scavenger could be put into the klnehc model, most snnply b_~ supermaposlng ,m exponential deca3, on the re-encounter plobabfl]tm,, rathel than t r u n c a h n g the calculation s u d d e n h

¢, = ¢,, + ~ , e"' I)IS( fISSION

I f during a llOll-redctlve encounter. J was constant for a time l', then 0 would be .II' To improve the description o f an encounter, a distinction is made, between reactive encounters in which the pair reach a separation o f a, and nonreactive ones in which the minimum separatmn he,, between a, and r, An arbitrary but consistent treatment is as follows when r0 is reached, a fraction P is taken to reach a,, without separating again, one could regard this fraction as making a large diffusive ,lump from t,~ in to a 0 though the detailed dynamics are not relevant The non-reactive fraction ( 1 P ) is made to ,lump out the same distance, l e to 2r, - a,, t\w this fraction, (17) is used to recalculate the wave-function with a value o f 0 between 0 and rc chosen at l a n d o m (For most encounters Jr'>> ~z but It would be m o l e accurate to bias 0 towards small numbers to account lk)r the very distant encounters in which J Js very small) Putting P - ao:2r,~ retains the correct value o f fl, since P + (1 - P ) a o / ( 2 r o ao) = ao/r,, see equation (14) This use o f an enforced "'large j u m p " has the advantage of reducing the need to follow small ineffective excursmns beyond r,,, in any case. the r a n d o m walk treatment on which (13) is based is not valid for very short times and small distances For convenience, triplet reacUve pairs are also taken to separate to 2r, a, RESULTS Equations (15) and (16) have been used to calculate yields o f o r l h o - and p a r a - h y d r o g e n as a function o f magnetic field, the results are given in Fig 2 For comparison, if there were no re-encounters, the total yield would be one quarter with an o r t h o / p a r a ratio o f three and, o f course, there would be no field effect Numerical integrations were carried out on a Datalab DL 417 m i c r o - c o m p u t e r The mtegratlons were truncated at 10 ns this gwes a value o f f l (p = 1) o f 0.311 (cf ao/ro = 0 467) This truncation can be regarded as equivalent to scavenging o f the pmr by either an added scavenger or a third hydrogen atom. for a diffusion controlled reaction a scavenger concentration o f ~ 1 0 2 m o l l ~ would be required M o n t e Carlo calculations on the University o f Sheffield's Prime 750 c o m p u t e r have just been started, the first results are very close to those o f Fig 2, and various assumptions a b o u t the role o f exchange in non-reactive e n c o l l n t e r ~ h:~ve, t~n]',: ~ ~'m,~ll otq'oot A

I he effect ol electron ext.hange i,, to spilt the slnglet dFld l r l p l e t slate',, ol COtll'-.e, there will be a slnall i,lnge ol dl,,tam.es Ior which 1 and S ale neally degenelatc (.I - ,~/~B ) [hc effects ol 7 5, crossing ha',e been detected m ( l D N P hut it is omitted hcrc, ctmslstcntly ~lth the assumed sharp llSe in ,/ Rough eah.ulatlons ,,ugge,,t ,1 slight increase in the o t t h o p a / a ]atlo (towards 3) but the effect is likely to he stnall because o f the small langc ol distances m,,olved A pl opcr as',essrncnt of this and o f the el ror due to the .! ~'~ a ,lssumptJon l-, ',ely dlthcult ill the ,lbsencc ol det,ulcd lnfo~lnatlon about motion of the dIOllI'~

Ttlc piescnt wolk grcx~ out oI A study o f possible spin effect,, m spurs '''~ The M o n t e Carlo calculations ,ire part of ,I largel p l o g l a m m e aimed at simulating the behavlour of spurs c o n t , n n m g lv¢o radicals and two hydrogela atoms lit ,,uch spurs, the effects ot an CllCOUnter between one pair OIt subsequent reactions ,a~th other specms m the spur are n n p o r t a n t c,,, ko~ this reason, evohltton of the spin wave-funchon outside the encounter has been emphaslsed, while tleatmenl of the encounter ltsell has been sunphfied In theoretical studies related to ( 1 D N P , etc'~'" more sophisticated description', of the encounter process have been given, including the dependence o f I on , However they arc not easily adapted to the multiple p a n problem because ol the averaging procedures u,,ed For a full description one must calculate in detail lor huge numbers o f spurs and average only the final results An zdeal approach to both spur ,ind pan problems ~,otlld be to leturn to a detailed description o f random ,aalk dlsplacelnents the exchange and other interactions could be Incorporated correctly While this is not possible with the present lnodcl, ~t is probably adequate, given the uncertalntms about molecular m o h o n in liquids and the distance dependence o f .1 Figure 2 shows that there should bc a small efl'ect o f lield on the overall reachon rate not large enough to be easily detected The effect on the oil/to p a r a ratio appears to be worth lnvestlg,ltlOn Prehmlnar? experiments m the author'a laboratory' have not been successful Care will be necessary to avoid catalytic conversion, c g by oxygen, or by hydrogen atoms '~" The latter can be obviated by reducing the tempelaturc somewhat It must be emphaslsed that the

The reaction H + H ~ H 2 m solution yields of ortho- and para-hydrogen I

0.068

-/

I

I

I

I

I

\

xN

I

(a)

\

I I

191

"-

L

~

0-198

0.066

-

-

I

I

I

I

I

I

o-192

I

",\

- 0.265

(b) 3-0

-

\\

)' \\\

0.260 2-9

I

I

I

0-2

I

I

0 -4

I

0-6

I

I

0.8

B / TESLA FIG 2 (a) Calculated yields ofortho-, - - , (b) their ratio, - - ~ and their sum, - -

and para-hydrogen, - - - - , as a function of magnetic field, The overall H + H encounter rate is taken as unity

Uon between two hydrogen atoms; formation of hydrogen by abstraction reactions or recombination of aqueous electrons should show no effects Radlolysls o f acid soluUons may form a useful approach, scavenging effects should be studied because slmdar effects but in the opposxte direction should result from reaction (1) reside spurs, if there is any mmal excess smglet character in the hydrogen pairs ~9~ The form o f the field dependence should be specific to the H + H reaction Quahtatwely slmdar results should be obtained from D atom recombination while the H + D reaction would give striking (but transient) C I.D N P. effects,~m future calculaUons will include these systems Because o f ~ts specificity,

th~s p h e n o m e n o n would be a very useful tool m radiation chemistry It would also throw hght on the details o f the diffusion process, these are still a matter for debate ~7~ and studies o f re-encounters w~th large molecules are c o m p h c a t e d by stenc factors

A~knowledgernent--The author wishes to thank the Science and Engineering Research Council for an eqmpment grant REFERENCES 1 (a) M ANBAR FARHATAZIZand A B Ross, Selected specific rates of reactions of transients from water In aqueous solution II--Hydrogen atom, NSRDS NBS51, National Bureau of Standards, Wash-

192

2 3 4 5 6 7 8

B BROCKLEHURST

lngton, 1975 ( b ) A W BOYD, C WILLIS and G C LALOR, Canad J Chem 1972, 50, 83 B BROCKLEHURST, Radtat Phy~ Chem 1983, 21, 57 N C VERMA and R W FESSENDEN. J Chem Phy~ 1976, 65, 2139 B BROCKLEHURST, J Chem Soc Faradai' Tran.~ 11, 1979, 75, 123 V A BENDERSKn,A G KRIVENKOand A N RuKm, High Energy Chem 1980, 14, 303 R M NOYES. Progr React Km 1961, i, 129 B STEVENS, J Ph),s Chem 1981, 85, 3552, 3555 R G LAWLERand H R WARD, In Determination o! Orgamc Structure~ by Ph)wlcal Methods (Edited by F C Nachod and J J Zuckerman), Vol 5, p 99 (Academic Press, New York, 1973), Chemtcally Induced Dynamu Nuclear Polartaatlon (Edited by A R Lepley

9 10 11 12 13 14 15

and G L Closs) Wdey-lntersclence, New York, 197~, Chemwally Induced Magnetw Polamsatton (Edited by L T Muus, P W Atklns, K A McLauchlan and J B Pedersen) D Reldel, Dordecht, Holland, 1977 B BROCKLEHURST, J Chem So~ Farada~ Tran~ IL 1982, 78, 751, 1791 B BROCKLEHURST, J ('hem Soc Farada) Tran~ l/ 1976, 72, 1869 P W ATKINS, Org Magn Resort 1973, 5, 239 C HERRING and M FLICKER. Phvs Ret 1964, AI34, 362 F J ADRIAN, J Chem Phv~ 1972, 57, 5107 G C ABELLand A MOZUMDER,J ('hem Ph~ 1972 56, 4079 A FARKAS,Ortho-hydrogen, para-Hydrogen and Heal'i ttydrogen Cambridge Umverslty Press, 1935