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Proton-transfer reactions in ionized gases
Radiat. Phys. Chem. Vol. 26. No. 5. pp. 571-573, 1985
0146-5724/85 $3.00 + .00 © 1985 Pergamon Press Ltd.
Printed in Great Britain.
PROTON-TRANSFER REACTIONS IN IONIZED GASES W . STILLER, R . SCHMIDT a n d R . SCHUSTER
Central Institute of Isotope and Radiation Research of the Academy of Sciences of the G.D.R., DDR-7050 Leipzig, German Democratic Republic
differing from eqn (1) is obtained121:
INTRODUCTION ION-MOLECULE reactions play an important role in various radiolytic processes, e.g. gas-pulse radiolysis, environmental research. For a discussion of mechanisms rate coefficients have to be assessed. Here gas-phase rate coefficients of ion-(polar) molecule reactions are calculated using the ideas of interaction potentials, reactive cross-sections and distribution functions of the translational energies of both the reactants (ions I, molecules M). The potential function V(r) of ion-molecule interaction can be represented in many cases by the ansatz: A
(1)
B
C
D
aM
(3)
-
IZMCOS%
B
L(y),
qI/P'M y -- (kBTRr z)
Physically speaking, the model includes all the energetic configurations of the dipole molecule in the electric field of the arriving ion which are compatible with the distribution laws of the classic Boltzmann statistics. According to Turulski and Forys 131the potential 13), characterizing in a thermodynamic sense the internal energy, should be replaced by Helmholtz free energy for isothermal processes:
F
The quantities A, B, C . . . are coefficients depending on the model of structure used for the reactive ion-molecule pair. For greater distances r it is convenient to consider an ion as a point charge of the mass mi, charge ql and the velocity Vl. The polar molecule of mass mM and velocity VM can be described by the dipole moment IXM and the polarizability aM. Under such conditions it is in the simplest way possible to define the coefficients in the following manner: =
kB TRy
(L(y) = coth (y) - y-1 = Langevin function).
V(r) = 7 + 7~ + -~ + 7 + 7 + " " '
A
qiz
r4
--
r = ion-molecule distance.
(2)
VBR(r) =
(4)
VTF(r) =
1 a M q~
2
r4
kB TR In(sin h(y)/y).
Elements of relation (4) will be used to derive a "long-distance" reactive cross-section leading in combination with a suitable probability factor for proton transfer to rate coefficients for this reaction type.
= 0,
C = - to~Mq~,
D,F=
0
THEORETICAL
(~ = angle between the dipole axis of the molecule and the distance vector). This model named the locked-dipole approximation (LD) in the case ~ = 0, leads to rate coefficients which in general overestimate the experimental values. For this reason more sophisticated approaches were introduced(~): average dipole-orientation approximation (ADO) by Su and Bowers and a statistical theory (BR) by Barker and Ridge. Following the latter a potential
The starting point of our approach, directed especially to gas-phase proton-transfer reactions, is the idea that the rate coefficient k can be calculated as an ion-molecule capture-rate coefficient multiplied by a "steric factor" representing the probability for proton transfer. Mutual capture of the reaction partners within a possible reaction zone is caused by the physical interaction between an ion and a polar molecule on the basis of a "long-
571
572
W. STILLER et al.
d i s t a n c e " - a p p r o x i m a t i o n o f e q n (4):
T h e r e f o r e the r e a c t i o n rate for s u c c e s s f u l p r o t o n t r a n s f e r e v e n t s h a s the coefficient:
l qi
15)
Vid(r) -
×
2 1.4
k ( T ) = kldP
O~M + 3 k - - - £ - ~ / "
kBTR r2
T h e m i c r o s c o p i c c r o s s - s e c t i o n m0 o b t a i n a b l e via the c e n t r i f u g a l - b a r r i e r m e t h o d ~41 is (6)
O'ld(E)
=
O'L(E)
F,
F = (I + .~x) 1'2,
~/~M
(141
-
klF
PA(M) PA (X) + PA (M) PAIM)>
PA(X),
w h e r e the L a n g e v i n rate kt. is c o r r e c t e d by a l e c t o r F = (1 + :] .r) v:, w h i c h d e p e n d s on the r e d u c e d "polar energy" x (IX~/'aM)/kF~ T c o m p r i s i n g only p r o p e r t i e s o f the (polar) m o l e c u l e .
x - - kRT
RESULTS AND DISCUSSION with
(7)
m ~ E = -~-v ~,
m _ l = ml - I + m M ~,
a n d the L a n g e v i n c r o s s - s e c t i o n ~rL giving t h e n o n p o l a r i n t e r a c t i o n b e t w e e n I a n d M:
(8)
~rL(E) = ~rql (2etM/E) L~2.
A s s u m i n g M a x w e l l i a n d i s t r i b u t i o n s of the velocities o f b o t h t h e r e a c t a n t s a n d TR = T the c a p t u r e rate c o e f f i c i e n t s reads(5): (9) kla(T) = k ~
~---mk~T
ECrld(E) e - E / ~ " r d E .
On the b a s i s o f e q n (14) p r o t o n - t r a n s f e r rate c o e f f i c i e n t s w e r e c a l c u l a t e d for r e a c t i o n s b e t w e e n H C N [aM = 2.59 × 10 2 4 c m ~,I~M = 2 . 9 8 D , P A = (170 + 3) kcal-mol -~] a n d the positive ions Hi-, H30 + , HCO + , and N2H + . The corresponding p r o t o n affinities P A (X), X = H2, H 2 0 , CO, N : t a k e n f r o m M a c k a y et a U 61 are 10l _+ l, 165 _+ 3, 143 - 1, 117 _+ 1, r e s p e c t i v e l y . T h e q u a n t i t y x is for H C N e q u a l to 83.5955 a n d F = 5.3726. T a b l e I gives experimental values kc~ in c m 3 . m o l e c u l e ~.s- ] a n d t h e o r e t i c a l v a l u e s r e d u c e d to kexp in t h e L a n g e v i n v e r s i o n (Kt_ = kt,/k~xp), in the A D O a p p r o x i m a t i o n (KADO = k A D o / k ~ p , COS O m o d e l ) , in o u r p r o t o n - t r a n s f e r model (K = k (eqn (14))/k~p), a n d in the L D a p p r o x i m a t i o n (KLI) = kLD/kexp). E x p e r i m e n t a l values stem f r o m flowing a f t e r g l o w m e a s u r e m e n t s at T = (297 _+ 2) K by M a c k a y et al. m) in H e o r H2 with (total) p r e s s u r e s o f 3 3 - 6 0 Pa. T a b l e 2 c o m p r e h e n d s a similar c o m p a r i s o n of
P u t t i n g (6) into (9) a n d i n t e g r a t i n g leads to (10)
kid(T) = k L F ,
Ions
with t h e L a n g e v i n rate (11)
kL = 2"rrql(etM/m) I/2.
N o w t h e p r o t o n t r a n s f e r f r o m an ion I = X H + to a m o l e c u l e M is c o n s i d e r e d : (12)
XH + + M
k ) M H + + X.
If the p r o t o n affinity P A (M) > P A (X) the p r o t o n will b e t r a n s f e r r e d f r o m X to M with a p r o b a b i l i t y
(131
P =
TABI,E I. REACTIONS OE DIFFERENT IONS WITtt HCN
PA(M) P A (X) + P A (M) '
Hf H30* HCO ~ N2H*
kexp" 10 9
7.4 3.5 3.0 3.2
_+ 0.5 ± 0.5 ± 0.2 _+ 0.2
KL
0.31 0.32 0.34 0.32
KADO
0.88 0.91 0.95 0.89
K
KI.D
1.04 0.88 0.98 1.00
2.57 2.68 2.79 2.62
T A B L E 2. REACTIONS OF DIFFERENT IONS WITH
Ions H3+ H30 + HCO + NzH ~
kexp" 10 9
10 4.7 4.1 4.1
± ± ± ±
1 0.7 0.4 0.1
CHsCN
KL
KAD()
K
KI D
0.30 0.30 0.30 0.30
0.84 0.83 0.83 0.83
1.03 0.83 0.88 (I.97
2.45 2.43 2.43 2.43
573
Proton-transfer reactions in ionized gases kex~ with theory for reactions of the same ions with CH3CN (aM = 4.56 x 10 -24 cm 3, la,M = 3.92 D, PA = (186 - 1) kcal.mol -~, x = 82.1594 and F = 5.3279). As is already well known the Langevin approximation gives only values of a third of the kexp values for such ion-(polar) molecule reactions. On the other hand, the LD approximation (assuming locked dipoles) considerably overestimates the experimental data. With respect to the other models the average deviations from the measured rate coefficients are ~13% (ADO approximation, cos O model with C = 0.252~61), - 7 % (k according to eqn (14)). Thus the k values of both of these theoretical versions are in the experimental margins of error of about +_20% (HCN) and 25% (CH3CN). The simple expression (14) for k(T) can be recommended as a
first orientation for the computation of rate coefficients of ion-(polar) molecule reactions with proton transfer.
REFERENCES 1. M. T. BOWERS(ed.), Gas Phase Ion Chemistry, Vol. 1, p. 83. Academic, New York, 1979. 2. R. A. BARKERand D. P. RIDCE,J. Chem. Phys. 1976, 64, 4411. 3. J. TURULSKIand M. FORYS,J. Phys. Chem. 1979, 83, 2815. 4. H. S. JOHNSTON, Gas Phase Reaction Rate Theory. Ronald, New York, 1966. 5. M. A. EUASONand H. O. HIRSCHFELDER,J. Chem. Phys. 1959, 30, 1426. 6. G. I. MACKAY,L. D. BETOWSKI,J. D. PAYZANT,H. I. SCHIFE and D. K. BOHME, J. Phys. Chem. 1976, 80, 2919.
0146-5724/85 $3.00 + .00 © 1985 Pergamon Press Ltd.
Printed in Great Britain.
PROTON-TRANSFER REACTIONS IN IONIZED GASES W . STILLER, R . SCHMIDT a n d R . SCHUSTER
Central Institute of Isotope and Radiation Research of the Academy of Sciences of the G.D.R., DDR-7050 Leipzig, German Democratic Republic
differing from eqn (1) is obtained121:
INTRODUCTION ION-MOLECULE reactions play an important role in various radiolytic processes, e.g. gas-pulse radiolysis, environmental research. For a discussion of mechanisms rate coefficients have to be assessed. Here gas-phase rate coefficients of ion-(polar) molecule reactions are calculated using the ideas of interaction potentials, reactive cross-sections and distribution functions of the translational energies of both the reactants (ions I, molecules M). The potential function V(r) of ion-molecule interaction can be represented in many cases by the ansatz: A
(1)
B
C
D
aM
(3)
-
IZMCOS%
B
L(y),
qI/P'M y -- (kBTRr z)
Physically speaking, the model includes all the energetic configurations of the dipole molecule in the electric field of the arriving ion which are compatible with the distribution laws of the classic Boltzmann statistics. According to Turulski and Forys 131the potential 13), characterizing in a thermodynamic sense the internal energy, should be replaced by Helmholtz free energy for isothermal processes:
F
The quantities A, B, C . . . are coefficients depending on the model of structure used for the reactive ion-molecule pair. For greater distances r it is convenient to consider an ion as a point charge of the mass mi, charge ql and the velocity Vl. The polar molecule of mass mM and velocity VM can be described by the dipole moment IXM and the polarizability aM. Under such conditions it is in the simplest way possible to define the coefficients in the following manner: =
kB TRy
(L(y) = coth (y) - y-1 = Langevin function).
V(r) = 7 + 7~ + -~ + 7 + 7 + " " '
A
qiz
r4
--
r = ion-molecule distance.
(2)
VBR(r) =
(4)
VTF(r) =
1 a M q~
2
r4
kB TR In(sin h(y)/y).
Elements of relation (4) will be used to derive a "long-distance" reactive cross-section leading in combination with a suitable probability factor for proton transfer to rate coefficients for this reaction type.
= 0,
C = - to~Mq~,
D,F=
0
THEORETICAL
(~ = angle between the dipole axis of the molecule and the distance vector). This model named the locked-dipole approximation (LD) in the case ~ = 0, leads to rate coefficients which in general overestimate the experimental values. For this reason more sophisticated approaches were introduced(~): average dipole-orientation approximation (ADO) by Su and Bowers and a statistical theory (BR) by Barker and Ridge. Following the latter a potential
The starting point of our approach, directed especially to gas-phase proton-transfer reactions, is the idea that the rate coefficient k can be calculated as an ion-molecule capture-rate coefficient multiplied by a "steric factor" representing the probability for proton transfer. Mutual capture of the reaction partners within a possible reaction zone is caused by the physical interaction between an ion and a polar molecule on the basis of a "long-
571
572
W. STILLER et al.
d i s t a n c e " - a p p r o x i m a t i o n o f e q n (4):
T h e r e f o r e the r e a c t i o n rate for s u c c e s s f u l p r o t o n t r a n s f e r e v e n t s h a s the coefficient:
l qi
15)
Vid(r) -
×
2 1.4
k ( T ) = kldP
O~M + 3 k - - - £ - ~ / "
kBTR r2
T h e m i c r o s c o p i c c r o s s - s e c t i o n m0 o b t a i n a b l e via the c e n t r i f u g a l - b a r r i e r m e t h o d ~41 is (6)
O'ld(E)
=
O'L(E)
F,
F = (I + .~x) 1'2,
~/~M
(141
-
klF
PA(M) PA (X) + PA (M) PAIM)>
PA(X),
w h e r e the L a n g e v i n rate kt. is c o r r e c t e d by a l e c t o r F = (1 + :] .r) v:, w h i c h d e p e n d s on the r e d u c e d "polar energy" x (IX~/'aM)/kF~ T c o m p r i s i n g only p r o p e r t i e s o f the (polar) m o l e c u l e .
x - - kRT
RESULTS AND DISCUSSION with
(7)
m ~ E = -~-v ~,
m _ l = ml - I + m M ~,
a n d the L a n g e v i n c r o s s - s e c t i o n ~rL giving t h e n o n p o l a r i n t e r a c t i o n b e t w e e n I a n d M:
(8)
~rL(E) = ~rql (2etM/E) L~2.
A s s u m i n g M a x w e l l i a n d i s t r i b u t i o n s of the velocities o f b o t h t h e r e a c t a n t s a n d TR = T the c a p t u r e rate c o e f f i c i e n t s reads(5): (9) kla(T) = k ~
~---mk~T
ECrld(E) e - E / ~ " r d E .
On the b a s i s o f e q n (14) p r o t o n - t r a n s f e r rate c o e f f i c i e n t s w e r e c a l c u l a t e d for r e a c t i o n s b e t w e e n H C N [aM = 2.59 × 10 2 4 c m ~,I~M = 2 . 9 8 D , P A = (170 + 3) kcal-mol -~] a n d the positive ions Hi-, H30 + , HCO + , and N2H + . The corresponding p r o t o n affinities P A (X), X = H2, H 2 0 , CO, N : t a k e n f r o m M a c k a y et a U 61 are 10l _+ l, 165 _+ 3, 143 - 1, 117 _+ 1, r e s p e c t i v e l y . T h e q u a n t i t y x is for H C N e q u a l to 83.5955 a n d F = 5.3726. T a b l e I gives experimental values kc~ in c m 3 . m o l e c u l e ~.s- ] a n d t h e o r e t i c a l v a l u e s r e d u c e d to kexp in t h e L a n g e v i n v e r s i o n (Kt_ = kt,/k~xp), in the A D O a p p r o x i m a t i o n (KADO = k A D o / k ~ p , COS O m o d e l ) , in o u r p r o t o n - t r a n s f e r model (K = k (eqn (14))/k~p), a n d in the L D a p p r o x i m a t i o n (KLI) = kLD/kexp). E x p e r i m e n t a l values stem f r o m flowing a f t e r g l o w m e a s u r e m e n t s at T = (297 _+ 2) K by M a c k a y et al. m) in H e o r H2 with (total) p r e s s u r e s o f 3 3 - 6 0 Pa. T a b l e 2 c o m p r e h e n d s a similar c o m p a r i s o n of
P u t t i n g (6) into (9) a n d i n t e g r a t i n g leads to (10)
kid(T) = k L F ,
Ions
with t h e L a n g e v i n rate (11)
kL = 2"rrql(etM/m) I/2.
N o w t h e p r o t o n t r a n s f e r f r o m an ion I = X H + to a m o l e c u l e M is c o n s i d e r e d : (12)
XH + + M
k ) M H + + X.
If the p r o t o n affinity P A (M) > P A (X) the p r o t o n will b e t r a n s f e r r e d f r o m X to M with a p r o b a b i l i t y
(131
P =
TABI,E I. REACTIONS OE DIFFERENT IONS WITtt HCN
PA(M) P A (X) + P A (M) '
Hf H30* HCO ~ N2H*
kexp" 10 9
7.4 3.5 3.0 3.2
_+ 0.5 ± 0.5 ± 0.2 _+ 0.2
KL
0.31 0.32 0.34 0.32
KADO
0.88 0.91 0.95 0.89
K
KI.D
1.04 0.88 0.98 1.00
2.57 2.68 2.79 2.62
T A B L E 2. REACTIONS OF DIFFERENT IONS WITH
Ions H3+ H30 + HCO + NzH ~
kexp" 10 9
10 4.7 4.1 4.1
± ± ± ±
1 0.7 0.4 0.1
CHsCN
KL
KAD()
K
KI D
0.30 0.30 0.30 0.30
0.84 0.83 0.83 0.83
1.03 0.83 0.88 (I.97
2.45 2.43 2.43 2.43
573
Proton-transfer reactions in ionized gases kex~ with theory for reactions of the same ions with CH3CN (aM = 4.56 x 10 -24 cm 3, la,M = 3.92 D, PA = (186 - 1) kcal.mol -~, x = 82.1594 and F = 5.3279). As is already well known the Langevin approximation gives only values of a third of the kexp values for such ion-(polar) molecule reactions. On the other hand, the LD approximation (assuming locked dipoles) considerably overestimates the experimental data. With respect to the other models the average deviations from the measured rate coefficients are ~13% (ADO approximation, cos O model with C = 0.252~61), - 7 % (k according to eqn (14)). Thus the k values of both of these theoretical versions are in the experimental margins of error of about +_20% (HCN) and 25% (CH3CN). The simple expression (14) for k(T) can be recommended as a
first orientation for the computation of rate coefficients of ion-(polar) molecule reactions with proton transfer.
REFERENCES 1. M. T. BOWERS(ed.), Gas Phase Ion Chemistry, Vol. 1, p. 83. Academic, New York, 1979. 2. R. A. BARKERand D. P. RIDCE,J. Chem. Phys. 1976, 64, 4411. 3. J. TURULSKIand M. FORYS,J. Phys. Chem. 1979, 83, 2815. 4. H. S. JOHNSTON, Gas Phase Reaction Rate Theory. Ronald, New York, 1966. 5. M. A. EUASONand H. O. HIRSCHFELDER,J. Chem. Phys. 1959, 30, 1426. 6. G. I. MACKAY,L. D. BETOWSKI,J. D. PAYZANT,H. I. SCHIFE and D. K. BOHME, J. Phys. Chem. 1976, 80, 2919.