Isotope Effects in Gas-Phase Chemistry - American Chemical Society

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Chapter 8

Kinetic Isotope Effects in Gas-Phase Muonium Reactions Susan Baer, Donald Fleming, Donald Arseneau, Masayoshi Senba, and Alicia Gonzalez 1

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Tri University Meson Facility (TRIUMF) and Department of Chemistry, University of British Columbia, Vancouver, British Columbia V6T 2A3, Canada The study of the reaction dynamics of muonium (Mu), an u l t r a l i g h t isotope of hydrogen ( Mu/ H = 1/9), provides a sensitive measure of mass e f f e c t s i n chemical reactions. The remarkable mass difference between Mu and the other hydrogen isotopes produces large k i n e t i c isotope e f f e c t s , providing a rigorous test of calculated potential energy surfaces (PES) and reaction rate theories. The low Mu mass also necessitates careful consideration of quantum e f f e c t s , i . e . tunneling i n the reaction coordinate. A review of recent r e s u l t s i n gas phase Mu chemistry i s presented, including comparison with relevant H chemistry and calculated PESs, where available. The magnitude and d i r e c t i o n of the k i n e t i c isotope effect i s shown to be a sensitive function of the PES, p a r t i c u l a r l y the height and p o s i t i o n of the saddle point. m

m

Kinetic isotope e f f e c t s have been extensively studied i n previous years to provide information about proposed p o t e n t i a l energy surfaces and theories of reaction rates on those surfaces. The study of hydrogen isotope e f f e c t s has proved p a r t i c u l a r l y i n t e r e s t i n g due to the large mass differences between the d i f f e r e n t isotopes, leading to correspondingly large differences i n rate constants; e.g., substitution of a deuterium atom for a protium atom i n a molecule w i l l have a much larger e f f e c t on the relevant v i b r a t i o n a l frequency than w i l l substitution of a C atom for a C . In addition, the p r o b a b i l i t y of quantum tunneling i s greatly increased by the low mass of H compared to heavier atoms and therefore the study of hydrogen isotope effects can provide insight into this fundamental process. 1

1 3

1 2

Current address: Loker Hydrocarbon Research Institute, University of Southern California, Los Angeles, CA 90089

0097-6156/92/0502-0111$07.75/0 © 1992 American Chemical Society

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

ISOTOPE EFFECTS IN GAS-PHASE

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112

CHEMISTRY

The three common isotopes of hydrogen are protium (Η), deuterium (D), and t r i t i u m (Τ), each possessing 1 proton, 1 e l e c ­ tron, and 0, 1, or 2 neutrons respectively. One novel i s o t o p i c form of hydrogen i s the radioactive species muonium (Mu), comprized of one muon (μ*) and one electron. The muon, which i s an elementary p a r t i c l e of the lepton family, possesses roughly one-ninth the mass of a proton, y i e l d i n g Mu/H mass r a t i o of 0.113. Although not an isotope by the textbook d e f i n i t i o n (variation i n neutron number), muonium behaves l i k e a hydrogen isotope i n that i t shares essenti­ a l l y i d e n t i c a l e l e c t r o n i c c h a r a c t e r i s t i c s and d i f f e r s only i n mass. This can be seen by the comparison between the properties of the hydrogen isotopes with Mu, given i n Table I. Despite large mass variations, the e l e c t r o n i c properties of these species, e.g. t h e i r i o n i z a t i o n p o t e n t i a l s , are v i r t u a l l y i d e n t i c a l , confirming that muonium can be treated as an u l t r a l i g h t hydrogen isotope. The l i g h t mass of the muon (ca. 200 times the mass of an elec­ tron) , however, raises the question of the v a l i d i t y of the BornOppenheimer approximation i n the atomic and molecular interactions of muonated species. It i s well known that this approximation begins to break down as the mass difference between the nucleus and the electron becomes small, and the comparisons i n Table I suggest that t h i s could be of concern i n the case of Mu. Theoretical calcu­ lations of one-electron problems, * however, (comparing for example, HD* and HMu* ), indicate that the Born-Oppenheimer approxi­ mation remains v a l i d for muon interactions. The main advantage i n the study of Mu reaction k i n e t i c s over that of t r a d i t i o n a l hydrogen isotopes l i e s i n the remarkable range and magnitude of possible isotope e f f e c t s i t affords. Within the Born-Oppenheimer approximation, i s o t o p i c species share a common potential energy surface; any differences that arise i n their respective reaction rates depend on mass e f f e c t s only. The true surface must be able to account for the behavior of a l l isotopes, no matter how l i g h t or how heavy. With the i n c l u s i o n of Mu, the a v a i l ­ able mass r a t i o range of hydrogen isotopes i s increased from 3 to 27. This unprecedented mass range therefore provides a uniquely sensitive probe of reaction dynamics and of the underlying p o t e n t i a l energy suface, p a r t i c u l a r l y near threshold. Perhaps the most propitious result of the low Mu mass i s the greatly enhanced p r e d i l e c t i o n for quantum tunneling of Mu r e l a t i v e to H, enabling observation of tunneling e f f e c t s at e a s i l y accessible tempertures (e.g. 100-200 K). This has f a c i l i t a t e d experimental observation of tunneling regimes i n some exothermic reactions where the experimental a c t i v a t i o n energy approaches zero,* i n d i c a t i v e of threshold tunneling. Another important advantage inherent i n the study of Mu l i e s i n the experimental technique. Muonium atoms are easy to form (provided a pion source i s available) and Mu events are i n d i v i d u a l l y monitored, thereby eliminating the r a d i c a l - r a d i c a l i n ­ teractions that often plague H experiments. * The experimentally obtained Mu rate constants may therefore be more accurate than those of t h e i r heavier atom counterparts and can, i n p r i n c i p l e , be used to predict H atom reaction rates, providing an accurate potential energy surface i s a v a i l a b l e . Although t h i s application c l e a r l y l i e s i n the future for most reactions, as c a l c u l a t i o n methods become 2

3

3

5

6

7

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

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8.

BAER ET AL.

Table I.

Comparison of the properties of Mu with H, D, and T.

mass/m H TI

Mu

Kinetic Isotope Effects in Gas-Phase Muonium Reactions

reduced mass/m

e

IP(eV)

a)

a) Bohr radius/a (H) Q

0.1131

0.9952

13.541

1.0043

H

1

0.9995

13.598

D

1.998

0.9997

13.602

0.9998

Τ

2.993

0.9998

13.603

0.9996

1

a) Calculated from difference i n reduced mass, based on vapor values for the Η atom.

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

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ISOTOPE EFFECTS IN GAS-PHASE CHEMISTRY

faster and more accurate, the study of Mu r e a c t i v i t y may be used more i n this predictive capacity. The experimental technique also imposes a s i g n i f i c a n t con­ s t r a i n t , however, on the study of muonium isotope e f f e c t s . The muon i s a radioactive p a r t i c l e with a h a l f - l i f e of 2.2 με, l i m i t i n g the long term observation of muonium. Hence, very small rate constants are d i f f i c u l t to measure. On the other hand, measurement of very fast reactions can be limited by the formation time of thermal Mu (up to 100 ns at 1 atm i n some gases), and the time resolution of the detection technique (=1 ns). Therefore any chemistry we wish to observe t y p i c a l l y occurs within a 0.1-10 μβ time window. These time constraints l i m i t the observation of product molecules and secondary reactions of Mu. (Such observations can be f a c i l i t a t e d , however, using resonant techniques.) Most reaction studies to date have been of primary reactions of free Mu atoms, the subject of t h i s review paper.

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8

Experimental Technique The r e a c t i v i t y of Mu atoms can be monitored by the technique of muon spin rotation ^SR), which r e l i e s on the fact that the Mu i s formed 100% spin polarized. P o l a r i z a t i o n can be lost i n a number of ways including reaction to form some diamagnetic product. In a magnetic f i e l d , t h i s loss of p o l a r i z a t i o n and therefore the disappearance of free Mu atoms can be monitored. The μSR technique has been well described e l s e w h e r e and w i l l not be addressed i n great d e t a i l here. The following discussion i s intended to elucidate the basic features necessary to understand and evaluate the experimental results presented below. 9-11

Formation and Decay of Muonium Positive muons with high k i n e t i c energy (ca. 4 MeV) are produced 100% longitudinally spin polarized from the p a r i t y v i o l a t i n g decay of p o s i t i v e pions. Muon production therefore requires a source of pions. A l l the experiments described i n t h i s paper were performed at the TRIUMF accelerator i n Vancouver, Canada. When high energy muons enter a reaction chamber f i l l e d with some inert bath gas, they are thermalized by c o l l i s i o n s . During the thermalization process they undergo a series of c y c l i c charge ex­ change reactions with the bath gas, M, as indicated i n R l β , 10 , 11 , 13 12

#

Mu + M

Rl

+

Below some threshold energy, which depends on the i o n i z a t i o n poten­ t i a l of M, Rl can no longer occur and the r e l a t i v e amounts of μ* and Mu are fixed. These f i n a l μ* and Mu y i e l d s depend reproducibly on the bath g a s . Therefore, when studies of Mu chemistry are desired, i t i s important to choose a bath gas, such as Ar or N , which produces a large amount of Mu. Since the μ* i s spin 100% polarized, muonium can be thought of as being produced equally i n two forms: " s i n g l e t " muonium, Mu, where the spin of the muon and the electron are paired; and " t r i p l e t " muonium, ^Mu, where the two spins are unpaired. » · The Mu state i s , i n fact, a super8 , 1 3

2

s

9

1 1

1 3

s

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

8. BAER ET AL.

Kinetic Isotope Effects in Gas-Phase Muonium Reactions

p o s i t i o n of [1,0> and [0,0> hyperfine states and thus i s quickly depolarized due to the strong hyperfine interaction (\> = 4463 Hz) between the μ* and the e" spins. C l a s s i c a l l y , i t can be considered as t o t a l spin S = 0 i n a weak magnetic f i e l d and thus as diamagnetic. I f thermalization takes place i n a short time period, the **Mu atoms w i l l r e t a i n their spin p o l a r i z a t i o n . Loss of p o l a r i z a t i o n w i l l occur i f the time spent as Mu during the charge exchange pro­ cess i s long r e l a t i v e to the c h a r a c t e r i s t i c period of the hyperfine interaction between the μ* and the electron i nMu (l/\> = 0.22 ns) . Since the time between Mu c o l l i s i o n s i s d i r e c t l y related to the c o l l i s i o n frequency, loss of spin p o l a r i z a t i o n can be avoided through use of s u f f i c i e n t l y high pressures (~1 atm) of the bath gas. Q

s

s

0

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s

Muonium Spin Rotation ^SR) When placed i n a transverse mag­ netic f i e l d , the muon spin precesses with a c h a r a c t e r i s t i c frequen­ cy that depends on both the strength of the f i e l d and the environ­ ment of the muon. Diamagnetic species such as μ* and MuH precess with a frequency of 13.6 kHz/G, while paramagnetic ^Mu precesses much f a s t e r — 1 . 3 9 MHz/G i n low f i e l d s . Regardless of i t s form, the muon decays with a mean l i f e of 2.2 με, emitting a positron prefer­ e n t i a l l y along i t s spin axis. Thus, a positron counter placed i n the plane of spin rotation displays a wave-like signal that varies i n amplitude with the c h a r a c t e r i s t i c frequency corresponding to the precession of the muon spin. An example i s shown i n Figure l a , f o r μ* stopping i n N at =1 atm pressure. This figure i s a histogram of m i l l i o n s of collected positron events (timed r e l a t i v e to when the corresponding muon entered the reaction chamber). The characteris­ t i c decay that arises from the 2.2 με lifetime of the muon has been substracted out for c l a r i t y . The s o l i d l i n e i s a computer f i t to the data. The frequency of the precession signal indicates the environment of the muon (μ*, Mu, e t c . ) , while the amplitude, often c a l l e d asymmetry, indicates the number of muons that are i n that environment. In an inert bath gas, l i k e N , the s l i g h t damping i n the amplitude of the precession signal over time i s simply due to some spin dephasing a r i s i n g from small inhomogeneities i n the mag­ netic f i e l d . The rate at which t h i s occurs i s c a l l e d the background relaxation rate, X . 2

2

Q

In the presence of a d i l u t e reactant, chemical reaction of the Mu atom can occur. When a diamagnetic product molecule i s formed, the muon spin begins to precess at i t s c h a r a c t e r i s t i c diamagnetic frequency. (Chemical s h i f t s are t y p i c a l l y too small to be resolved by the μSR technique.) Since t h i s frequency i s ca. 100 times slower than that of Mu, and, moreover since these diamagentic products are formed at random times and thereby possess no coherent phase, they are not experimentally observeable and the o v e r a l l Mu ensemble i s seen to relax e x p o n e n t i a l l y . This process i s i l l u s t r a t e d i n Figure lb where the i n i t i a l amplitude of the precession signal i s quickly damped as the reaction o c c u r s . The o v e r a l l relaxation rate, λ, i s related to the bimolecular rate constant, k, as shown i n Equation (1), where [X] i s the concentration of the reactant. 10

14

λ = λ

0

+ k[X]

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

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ISOTOPE EFFECTS IN GAS-PHASE CHEMISTRY

-0.25

1

1

0.0

0.3

a.

1 0.6

1 0.9

I 1.2

I 1.5

I

1.8

I 2.1

I 2.4

L 2.7

3.0

Time in με

Figure 1. The μSR signal at 200 Κ for Mu precession i n a trans­ verse magnetic f i e l d of 7.7G for a) 1000 torr pure N and b) 1000 torr N i n the presence of 1% additional C D . The s o l i d l i n e i s a chi-squared f i t to the data, y i e l d i n g the indicated relaxation rates. (Reproduced with permission from Ref. 15. © 1990 American Institute of Physics). 2

2

2

4

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

8. BAER ET AL.

Kinetic Isotope Effects in Gas-Phase Muonium Reactions

By measurement of λ at a series of concentrations of X, the k i n e t i c quantity of i n t e r e s t , the reaction rate constant, can be extracted. Some addition reactions of Mu r e s u l t i n the formation of a paramagnetic, product (e.g., MuCjH^ i n the presence of a bath gas ). This type of reaction also results i n a decay of the Mu signal that can be described by ( l ) . The environment of the muon is s t i l l paramagnetic, but the coupling between the μ* and the elec­ tron i s greatly reduced r e l a t i v e to Mu due to the greater distance bewteen the μ* and the e" spin density i n the free r a d i c a l . Although such muonium r a d i c a l s are the subject of intense investiga­ t i o n i n t h e i r own r i g h t , » » only the k i n e t i c s of the addi­ t i o n step w i l l be considered i n t h i s paper. As noted above, the vSR technique monitors the reaction of each Mu atom i n d i v i d u a l l y . Each time a muon enters the reaction chamber a clock i s started, which i s only stopped after a positron has been detected or a fixed amount of time greater than several muon l i f e ­ times has elapsed. I f another muon enters the chamber during that i n t e r v a l , both muon events are rejected and do not contribute to the f i n a l histogram. In t h i s way, the p o s s i b i l i t y of d i s t o r t i n g the experimental r e s u l t s by competing Mu interactions i s eliminated. As mentioned above, t h i s has frequently been a problem i n k i n e t i c studies of the corresponding H reactions. » 1 5

1 6

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9

1 1

1 7

6

Muonium K i n e t i c

Isotope

7

Effects

Despite the growing sophistication of molecular beam experiments and measurements of state-to-state reaction cross s e c t i o n s , » the study of bulk chemical k i n e t i c s , as manifest by thermal reaction rate constants, remains important. F i r s t , rate constants provide some measure, a l b e i t i n d i r e c t , of absolute cross sections, r a r e l y reported i n beam experiments. Second, only thermal rate constants test the nature of the potential energy surface near the reaction threshold, thus usually providing the best determination of both the p o s i t i o n and height of the potential b a r r i e r . The measurement of k i n e t i c isotope effects i s of v i t a l importance to t h i s point, p a r t i c u l a r l y i n the case of the u l t r a l i g h t hydrogen isotope, Mu, with i t s pronounced s e n s i t i v i t y to dynamical mass e f f e c t s . This section addresses the s p e c i f i c effects of the low muonium mass on reaction rates; i . e . what kind of k i n e t i c isotope e f f e c t s (kvj /kjj) do we expect for Mu atom reactions? Consider f i r s t the r e l a t i v e energies for an atom-molecule reaction of either Mu or H. Since isotopic reactions share a common reaction potential surface, energetic differences must arise from differences i n translations or internal mode vibrations and rotations of the respective t r a n s i t i o n states. The lower mass of the Mu atom causes an increase i n the v i b r a t i o n a l frequencies and therefore the zero point energy of the t r a n s i t i o n state. The zero point energies of the reactants are, of course, independent of isotopic substitution. The r e s u l t i n g r i s e i n the reaction a c t i v a t i o n energy causes a decrease i n the muonium reaction rate; i . e . /ku < 1. This e f f e c t can be quite large, p a r t i c u l a r l y for an endotnermic reaction, such as Mu + H , » possessing a " l a t e " potential b a r r i e r . For such a reaction, the reduced mass of the t r a n s i t i o n state resembles the reduced mass of the products more c l o s e l y than that of the reactants, r e s u l t i n g i n a 18

27

u

1 9

2

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

2 0

117

118

ISOTOPE EFFECTS IN GAS-PHASE

CHEMISTRY

maximum difference i n zero point energies and thereby enhancing the rate of H abstraction r e l a t i v e to Mu abstraction. A second muonium isotope e f f e c t , mentioned previously, i s the enhanced p o s s i b i l i t y of tunneling. Since tunneling p r o b a b i l i t y de­ pends exponentially on the square root of the mass, Mu, with oneninth the mass of H, has a much greater chance of tunneling at a given temperature, » leading to a corresponding increase i n the muonium reaction rate constant; i . e . k^u^H > 1. Quantum tunneling i s most important for exothermic reactions, such as Mu + F ,* which tend to exhibit "early" b a r r i e r s . A t h i r d mass e f f e c t , a r i s i n g from t r a n s l a t i o n a l motion i n simple c o l l i s i o n theory, i s that the average v e l o c i t y of the reac­ t i v e p a r t i c l e at a given temperature i s inversely proportional to the square root of i t s mass. This implies that Mu has three times larger v e l o c i t y than H, causing a three-fold increase i n the Mu encounter frequency. For c e r t a i n reactions t h i s increased encounter frequency translates d i r e c t l y into an enhanced reaction rate, causing an isotope r a t i o of up to kw /kjj = 2.9. A further e f f e c t of t h i s enhanced v e l o c i t y i s that the duration of the Mu c o l l i s i o n i s shorter than i t s H counterpart. This effect can be important i n reactions where s t e r i c e f f e c t s and/or molecular orientation play a role since the Mu atom may not have time to " f i n d " the proper orien­ t a t i o n . This effect therefore tends to decrease the r e l a t i v e muonium reaction rate: k^ /kjj < 1. The magnitude of the f i n a l k i n e t i c isotope e f f e c t , kw /kjji i s an interplay of a l l the above, and a sensitive function of the potential energy surface; i n p a r t i c u l a r , the p o s i t i o n of the saddle point on that surface. A wide range of k i n e t i c isotope e f f e c t s has been observed: from k / k = 0.08 for the reaction of Mu(H) with H, to k / k = 100 at low temperatures for the reaction of Mu(H) with F .* These reactions are discussed below. 2l

22

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2

u

u

u

M u

H

19

2

M u

H

2

T r a n s i t i o n State Theory The effect of muonium substitution on reaction rates can be expressed i n terms of t r a n s i t i o n state theory.* » » » » * The expression for the thermally averaged rate constant, k(T), for the reaction A + Β -> products, i s given i n Equation (2), where kg i s Boltzmann's constant; h i s Planck's constant; I" i s the transmission or tunneling c o e f f i c i e n t ; Q , Qa, and Qg are the p a r t i t i o n functions for the t r a n s i t i o n state, A, and B, respectively; and E i s the v i b r a t i o n a l l y adiabatic barrier to reaction. (The v i b r a t i o n a l l y adiabatic b a r r i e r equals the difference i n energy between the ground v i b r a t i o n a l state of the t r a n s i t i o n state and that of the reactants.) 15

2 0

2 3

2

t

v a

k ( T )

, r

kT _B_ R

t

t ^ _

-E„ /k T va Β

n

a

e

n

( 2 )

The k i n e t i c isotope e f f e c t , k / k , can then be expressed as Equa­ tion (3) where Δ Ε i s the difference i n the v i b r a t i o n a l adiabatic b a r r i e r s , and the prime denotes the muonated species. M u

H

ν &

k

H

r

" t

QMU

Q

t

Θ

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

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8.

BAER ET AL.

Kinetic Isotope Effects in Gas-Phase Muonium Reactions

By expanding the relevant p a r t i t i o n functions and expressing AE i n terms of the v i b r a t i o n a l frequencies of the t r a n s i t i o n state, s (excluding the reaction coordindate) , the k i n e t i c isotope effect can be written as (A), where m equals mass, μ equals reduced mass, and 1^, Ig, and Iç refer to the moments of i n e r t i a of the t r a n s i t i o n state. v a

**L u H

.

_V Γ t

k

3n-7

/

2

3

(

1

i

e x

,Wc ΛΙ/2 T L L ABC

2

A

, [-h/2k T Σ (υ* - υ*)] i +

(

1

/

Λ t ' μ μ

1 v

. ,-h\i.* /k-T* ( ~ P * ) l-exp(-h\>: /k_T)

n

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3

J V riLMu

f

exp

+

(A)

R

D

This expression can be s i m p l i f i e d by use of the Redlich-Teller product theorem, Equation (5), where m^ are the masses of the atoms comprising the molecule with mass M. -

wι

111^1 '

/ 0

3 / 2

(4-)

(

1 / 9

)

1

/

2

n m.' , = π (-^-) i i m

W c

/ 9 3 / 2

3n-6 π i

υ.· -±i

(5)

After factoring out the imaginary frequencies \)*, corresponding to the reaction coordinate and substituting u^ = hxj^/kgT, k / k j j can be writen as Equation (6). Mu

k

H

" ^tT

\>*

u

i

i sinh(u|/2

W

)

Using the harmonic o s c i l l a t o r approximation, the i s o t o p i c frequency r a t i o , \ r / \ J , can be written as (u*/u* ) ' where μ refers to the reduced mass of the reaction complex at the geometry corresponding to the t r a n s i t i o n state. The rate constant r a t i o can therefore be written as Equation (7). (

1

h^.lxL H ~ t k

F

(

μ* μ*^

1/2

3n-7 i

u

2

u\ s i n M u , /2) i sinh(u!/2)

U

)

This equation i s composed of three separate r a t i o s , which can be considered independently. The f i r s t i s the strongly temperature dependent r a t i o of the transmission c o e f f i c i e n t s , Γ£ /T , which i n ­ cludes contributions from both b a r r i e r recrossing and quantum tun­ neling. Although rigorous quantum formulations e x i s t , transition state theory i s e s s e n t i a l l y a c l a s s i c a l theory and does not a - p r i o r i include quantum tunneling. The transmission c o e f f i c i e n t i s therefore introduced as an ad-hoc factor to incorporate quantum tunneling into the formalism. There has been s i g n i f i c a n t debate on the methods of c a l c u l a t i n g these transmission c o e f f i c i e n t s ; and measurements of Mu isotope e f f e c t s have been important i n evaluating the accuracy of these m e t h o d s . * * The second r a t i o , (μ /μ ' ) , i s the square root of the reduced mass of the two t r a n s i t i o n states. It i s important to emphasize that this term depends on the geometry of the t r a n s i t i o n state and therefore on the position of the reaction b a r r i e r . This term, therefore, contains information about the reaction dynamics and i s often called the t

2 5

22

26

27

1 / 2

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ISOTOPE EFFECTS IN GAS-PHASE CHEMISTRY

120

2 8

primary isotope e f f e c t . For an endothermic reaction with a corres­ pondingly late b a r r i e r , the reduced mass of the t r a n s i t i o n state resembles more closely that of the products and the r a t i o (μ /μ ' ) approaches 1. For an exothermic reaction, on the other hand, with an early b a r r i e r to reaction, the reduced mass of the t r a n s i t i o n state resembles that of the reactants. In t h i s l i m i t , the r a t i o (μ /μ ' ) ' approaches 2.9, the v e l o c i t y difference men­ tioned above. The t h i r d r a t i o , n(u [sinh(u^/2) )/(u^ sinh(u|/2) ) , appears com­ p l i c a t e d at f i r s t , but can be simply considered as containing a l l the information about the internal modes of the t r a n s i t i o n state other than the reaction coordinate (e.g., for a bimolecular exchange reaction, the symmetric stretch and bend modes). For this reason, i t i s often c a l l e d a secondary isotope e f f e c t . For an early b a r r i e r , where these internal modes have very low frequencies, the value of this r a t i o approaches 1. For a late b a r r i e r , on the other hand, v i b r a t i o n a l modes have much higher frequencies and are strong­ l y affected by isotopic substitution. The secondary isotope e f f e c t approaches zero and thus can have a dramatic e f f e c t on the kinetics. In conclusion, for an endothermic reaction with a late b a r r i e r : the r a t i o of the transmission c o e f f i c i e n t s should approach 1 since tunneling i s not expected to be important (barrier recrossing could reduce this r a t i o ) ; the primary k i n e t i c isotope e f f e c t approaches 1; and the secondary isotope e f f e c t approaches 0. The net e f f e c t can therefore be a pronounced negative isotope e f f e c t : k^ /kjj MuH + H, where quantum tunneling may play an enhanced r o l e , i s of considerable interest. This has led to several theoretical studies of the Mu + H system i n recent y e a r s , » » as well as 2

2 9 - 3 1

29

32

34

34

2

29

3 2

3 5

2

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

8.

BAER ET AL.

Kinetic Isotope Effects in Gas-Phase Muonium Reactions

experimental measurements of the reaction rate constants for Mu + H and Mu + D i n t h i s l a b o r a t o r y . » The Mu + H reaction i s endothermic by the difference i n the zero point energy between MuH and H : 31.8 kJ/mol (for Mu + D , 38.5 kJ/mol). Therefore, both high temperatures and high pres­ sures of H (D ) were necessary i n order to observe the reaction within the muon l i f e t i m e . A comparison between the measured reaction rates and Arrhenius parameters for Mu + H and H + H i s given i n Table I I . The most s t r i k i n g r e s u l t i s the large negative isotope e f f e c t : ^ / k ^ =0.08. This value i s temperature independent. » » As mentioned above, t h i s can be q u a l i t a t i v e l y understood i n terms of a late potential barrier for the endothermic Mu + H reaction, which results i n an increase i n the zero point energy of the t r a n s i t i o n state. Since the reactant energy i s unchanged, t h i s leads to a con­ siderably larger a c t i v a t i o n energy (E ) for the Mu + H reaction, and correspondingly much slower reaction rates. The roughly threetimes slower reaction rate of Mu + D r e l a t i v e to Mu + H occurs for the same reason: an increase i n the zero point energy of the tran­ s i t i o n state r e l a t i v e to the r e a c t a n t s . This same e f f e c t i s observed for H + D versus H + H (Table I I ) . A comparison of the Mu + H and Mu + D experimental results with theory i s shown i n the Arrhenius plot from ca. 500-850 K, given i n Figure 2. The dot-dashed l i n e refers to the v a r i a t i o n a l t r a n s i ­ t i o n state theory calculations of Garrett and Truhlar using the least action ground state approximation to compute the tunneling correction. The b a r r i e r height of the potential energy surface has been adjusted s l i g h t l y i n the c a l c u l a t i o n . As can be seen from the figure, t h i s method works very well at high temperatures, but starts to break down s l i g h t l y at lower temperatures for both Mu + H and Mu + D . Tunneling i s predicted to become important for these reactions around 500 K , » so the f i t t i n g problems i n the tran­ s i t i o n state theory calculations at lower temperatures are l i k e l y due to inadequate treatment of tunneling e f f e c t s . (As noted above, Mu r e a c t i v i t y provides an extremely sensitive measure of the correct tunneling path to be used.) This disagreement between the experi­ mental data and the t r a n s i t i o n state theory calculations would be expected to become increasingly pronounced at lower temperatures where tunneling begins to dominate. The slowness of the reaction r e l a t i v e to the μ* l i f e t i m e , however, prohibited i t s measurement at these temperatures. The dashed l i n e i n Figure 2 refers to the "exact" coupled states 3-dimensional quantum c a l c u l a t i o n by S c h a t z , which shows excellent agreement with the data over the entire temperature range. (There are no adjustable parameters i n t h i s calculation.) This suggests that under conditions where tun­ neling can be important, such as i n Mu reactions and/or at low tem­ perature, the quantum calculations are s i g n i f i c a n t l y superior to even sophisticated t r a n s i t i o n state theory calculations. It would be interesting to see the comparison between Mu and Η reaction rates extended to more recent quantum calculations of the Η + H system. 19

2

20

2

2

2

2

20

2

2

2

2

k

u

19

20

29

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2

a

2

2

2

20

3 1

2

2

2

2

26

2

2

3 2

3 5

32

2

3 3

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122

ISOTOPE EFFECTS IN GAS-PHASE CHEMISTRY

Table I I . Comparison between Mu + H

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Reaction l)

a

2)

a

3)

b

4)

b

2

H + D

2

at 745K.

k(cm molecules

0

Ε (kJ/mol) a 35.6±2.1

4.2

39.3±1.3

14.4 χ 10

AH(kJ/mol)

H + H

and H + H

-14 40 χ 10 -14 -14

Mu + H

2

31.8

55.6±0.8

3.3 χ 10

Mu + D

2

38.5

61.5±1.7

1.0 χ 10

k./k. = 0.08 3 1

s )

-14

k./k = 0.07 4 2 0

a) Data from Ref. 31 b) Data from Ref. 19

ICVT/LAG Theory

1.0

1.2

1.4

1.6

1.8

2.0

2.2

1000/T (K) Figure 2. Arrhenius plots for Mu+H and Mu+D . The s o l i d l i n e s are f i t s to the experimental data from Ref. 19. The dashed l i n e gives the result of 3-dim. quantum mechanical c a l c u l a t i o n s . The dot-dashed lines refer to v a r i a t i o n a l t r a n s i t i o n state theory calculations on the same s u r f a c e . (Reproduced with permission from Ref. 19. © 1987 American Institute of Physics). 2

2

32

26

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8.

BAER ET AL. Mu + X

Kinetic Isotope Effects in Gas-Phase Muonium Reactions

2

4

The reactions of Mu with the halogens ( F , C l , and B r ) are also simple abstraction reactions l i k e Mu + H but d i f f e r markedly i n their reaction energetics. As shown i n R2-R4, a l l three reactions are very exothermic and therefore very d i f f e r e n t k i n e t i c isotope effects from those i n the Mu + H reaction are expected. 2

2

2

2

2

Mu + F

> MuF

2

Mu + C l

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Mu + B r

2

2

+ F

ΔΗ = -369.0 kJ/mol

R2

> MuCl + CI

ΔΗ = -155.2 kJ/mol

R3

> MuBr + Br

ΔΗ = -142.3 kJ/mol

R4

These experiments were performed i n a similar manner to those for Mu + H described above, except that, due to the exothermicities involved, high pressures and temperatures were not required. In each case, the reactions Mu + X were measured from ca. 150-500 K, depending on the vapor pressure of X , and at ca. 1 atm N pres­ sure. The measured rate constants and a c t i v a t i o n energies for R2-R4 are given i n Table I I I , along with the k i n e t i c isotope e f f e c t s at 298 K. The k i n e t i c isotope effects for these reactions are a l l £ 2.9, as predicted for an exothermic reaction with an early b a r r i e r . The early p o s i t i o n of the b a r r i e r along the reaction co­ ordinate has been confirmed by calculations of HX p o t e n t i a l energy surfaces. » As discussed above, the k i n e t i c isotope e f f e c t for a highly exothermic reaction consists of a simple mass contribution of 2.9 and a tunneling contribution given by the r a t i o of the trans­ mission c o e f f i c i e n t s (barrier recrossing e f f e c t s are not s i g n i f i c a n t for exothermic reactions). The extent of the tunneling contribution can therefore be gauged by dividing the experimentally determined k i n e t i c isotope effect by 2.9, y i e l d i n g the r a t i o of the transmis­ sion c o e f f i c i e n t s i n Table I I I . These values are a l l greater than one, demonstrating that quantum tunneling i s indeed enhanced for Mu r e l a t i v e to H, even at room temperature. At 250 K, t h i s r a t i o i s further enhanced, equaling 8.0 for Mu + F and 2.1 for Mu + C l . Further evidence of quantum tunneling can be found i n the com­ parison of the experimentally obtained Arrhenius a c t i v a t i o n energies E (Mu) and E (H). The Arrhenius a c t i v a t i o n energy can be defined by Equation (8) , 2

2

2

2

2

21

36

2

a

2

a

3 7

Ε

a

=

-

(8)

where i s the average energy of those c o l l i s i o n s leading to reaction and i s the average energy of a l l molecules. C l a s s i ­ c a l l y , for an exothermic reaction, E i s expected to depend only s l i g h t l y on temperature and/or on isotopic substitution, » as observed for H + X . ~ The Arrhenius plots for these reactions y i e l d e s s e n t i a l l y straight lines over the measured temperature range. This c l a s s i c a l behavior i s not observed, however, i n the reactions of Mu with the halogens, as shown dramatically by the a

4

3 9

38

4 1

2

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ISOTOPE EFFECTS IN GAS-PHASE CHEMISTRY

Table I I I . Comparison between the reactions of Mu and H with the naiogens.

k g(Mu)^(10

1 1

2g

k g(H) (10

1 1

29

k /k M u

F

/ Γ

Mu Μ α

( 2 9 8 K )

Η

(250K)

Η

Ε (Mu) a Ε (H)

a) b) c) d) e)

3

cm molecules *s

(298K)

H

Γ /Γ

3

cm molecules *s

e

9

(kJ/mol) (kJ/mol)

ci

b

c

2

Br

2.62±0.06

8..5010.1A

56.,010.9

0.16±0.01

2..1010.10

6..510.5

16.A

A,,0

8..6

5.7

1,.A

3,.0

8.0

2,.1

A,.2

3.110.3

2,.710.2

-0,.A010.08

9.210.3

5,.010.A

5,.610.5

F

2

H data from Ref. 39. H data from Ref. AO. H data from Réf. A5. Mu data from Ref. A. From Arrhenius f i t s over 250-500 K.

Kaye; Isotope Effects in Gas-Phase Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1992.

2

8. BAER ET AL.

Kinetic Isotope Effects in Gas-Phase Muonium Reactions

Arrhenius plots for reactions R2-R4, given i n Figure 3. Both the reactions with F and C l show a pronounced curvature at low temper­ atures. (Mu + B r displays quite d i f f e r e n t behavior, as discussed below.) In fact, the rate constant of Mu + F appears to approach a temperature independent regime below ca. 150 K. This dramatic decrease i n the apparent a c t i v a t i o n energy of the reaction i s indicative of Wigner threshold tunneling, which has been hypothe­ sized to occur when the r a t i o of the de Broglie wavelength of the tunneling p a r t i c l e to the thickness of the b a r r i e r i s much greater than l . This can be expressed i n terms of temperature, T, as given by Equation (9), 2

2

2

2

5

2

Τ