Concept of Activation Energy in Unimolecular Reactions
R. G. Gilbert and I. G. Ross
Citation: The Journal of Chemical Physics 57, 2299 (1972); doi: 10.1063/1.1678584
View online: http://dx.doi.org/10.1063/1.1678584
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THE JOURNAL OF CHEMICAL PHYSICS
VOLUME 57, NUMBER 6
15 SEPTEMBER 1972
Concept of Activation Energy in Unimolecular Reactions
R. G. GILBERT AND 1. G. Ross
Chemistry Department, School of General Studies, Australian National University, Canberra, Australia
(Received 12 March 1971)
An exact expression is derived, valid for all pressure regions, for the activation energy of a unimolecular
reaction. Because of the reaction, the distribution of reactant molecules among vibrational levels deviates
from a Boltzmann distribution; this deviation is explicitly taken into account. In the high-pressure limit,
the expression reduces to a well-known form: E •• t = (EK)/(K)- (E), due originally to Tolman. In the
low-pressure limit our expression contains terms which do not appear in Tolman's formalism. These are
a term EM which depends on the deviation from a Boltzmann distribution, and a term EQ which arises
from the thermal average of energy-dependent vibrational cross sections. It is shown that the low-pressure
expression for E act does indeed yield an energy fairly close to the energy at which conversion
from reactant to product occurs. An approximate analytic expression for Ea• t is obtained for the case
that the cross sections for vibrational energy transfer can be approximated by a truncated harmonic oscillator model. A numerical study of the thermal decomposition of N 20, using SSH theory to evaluate collision rates, shows that both EM and EQ make important contributions to the final value of E a • t . It is also
shown that the steady-state approximation can lead to serious error in calculating low-pressure rate constants
and activation energies at high temperatures.
I. INTRODUCTION
It is well recognized that the activation energy of a
chemical reaction is essentially an experimental parameter, to which the height of a potential barrier is no
more than a first approximation. ' A preceding paper2
explored the meaning of the activation energy of a
unimolecular reaction in the high-pressure limit; the
conclusion was that while it need not necessarily correspond to the energies of potential maxima or crossing
points between potential surfaces, it did tell at what
energy the conversion from reactant to product becomes most effective.
In this paper we derive an expression for the activation energy of a unimolecular reaction at any pressure,
in an inert gas heat bath. In the approach we follow:
an approach initiated by Tolman,3 and further pursued
by Slater' ; the populations of reactant molecules in
different vibrational levels are specifically enumerated,
as are the probabilities of transitions between them due
to collisions with the inert gas. Slater made an important
assumption concerning the population of the active
levels of the reactant, namely the "steady-state"
approximation. The usual formulation of this approximation l ,4 involves the assumption that the population
of inactive reactant levels (i.e., levels with zero rate of
conversion to product) follows a Boltzmann distribution. (In a more recent paper,5 Valance and Schlag
adopt a somewhat less restricted definition of the
steady-state approximation, but it is Tolman's definition which is followed here.) In fact, when reaction
occurs, molecules are bled off from the higher inactive
levels, whose populations are then lower than the
Boltzmann population. This deviation is specifically
explored here: It is found to lead to an important contribution to the activation energy which goes to zero
only in the high-pressure limit.
We also take into account the temperature dependence of the thermal averages of the collision crosssections for vibrational excitation and de-excitation of
the reactant molecules. Consideration of this point is
found also to lead to a significant term in the activation
energy, although once again only at intermediate and
low pressures.
Recently Menzinger and Wolfgang6 considered the
second of these effects for bimolecular reactions, but in
a highly simplified way which amounts to the assumption that the reactants have no internal structure.
Attention has also been drawn7 to the fact that in practical kinetic runs thermalization of the heat bath may
need to be reckoned with; however, this effect can
always be minimized if the reactant is sufficiently
dilute.
Our treatment concludes with illustrative calculations, using the truncated harmonic oscillator model for
collision cross sections and followed by more detailed
computations on the unimolecular decomposition of
N 20.
II. GENERAL EXPRESSION FOR THE
ACTIVATION ENERGY
The key results of a general formulation of reaction
rate theory, at fixed temperature T, are briefly recalled
in the following Eqs. (1)- (8).
It is established that, within certain limitations, the
rate constant of a unimolecular reaction is the highest
eigenvalue of the matrix8 ,4,2
NA-K,
(1)
where N is the concentration of inert gas, K is the
diagonal matrix of microscopic rates of conversion from
reactant level i to product (the elements of l{ are
independent of pressure and temperature), and A is
the matrix of rate constants of collision-induced transitions between reactant levels. The elements of A are9
Aij=87rm-l/2(21rkT)-3/2/""
EQji(E) exp( -EjkT)dE,
Emin
(2)
2299
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2300
R.
G.
GILBERT AND I. G.
where m is the molecular mass of the inert gas, k the
Boltzmann constant, E min the minimum energy at which
the transitionj to i can occur (Emin=O if E;<Ej, and
= E i - E j otherwise), and Qj; is the cross section for
collisional transfer fromj to i. In Eq. (2), and throughout most of this paper, we have assumed that the
degeneracies of i, j are unity; inclusion of degeneracy is
quite straightforward. The microscopic reversibility
relation follows from Eq. (2)10.2
(3)
Aji=Aij exp[(E;-Ej)/kT].
ROSS
and SQ specifically depends on the variation with energy
of the cross sections Q(E):
0iiQ= -II: (i;jbj/b,+ I: (ij;1,
(17)
i>i
0;jQ=(i;h
(ij;b;/b J,
=
j<i
(i<j)
(i>j)
(iij=87rm-1/2(21rkT)-3/2i:in PQji(E) exp( -E/kT)dE.
It will be convenient to write
then Eq. (3) can be rewritten as
Aj;b;=Aijbj.
(5)
Also, from conservation of molecules in
system,
Aii=-I: A ji.
(19)
(4)
b;= exp( - EJkT);
do
nonreacting
(18)
where
It will be noted that the Boltzmann distribution of
molecules can be written as a vector b, whose elements
are defined in Eq. (4). Eact may then be expressed as a
sum of three terms. The first, associated with SM, is
(N /cTcR)
(6)
I: gi I:
j>i
IAji(Ej-Ei)[(g;/bi)- (gj/bj)]l
j;>fi
(20)
As the rate constant R is minus the largest (closest to
zero) eigenvalue of (1), we have
R= -gT(NA-K)g/gTg,
(7)
where g is the eigenvector, a vector whose elements
are the populations of levels i, corresponding to the
largest eigenvalue of (1); gT is its transpose.
Equation (3) may be used to convert A to a symmetric matrix B,8 whence
R= -cT(NB-K)c/cTc,
where
(8)
which is convenient to re-express (see Appendix) as
(c TcR)-1 I: Eil (gNb i ) (Kii-R)
+N I: AijbJ{(gj/bj )- (gJb i )]21,
where the convention is introduced that I:iI:i<i
implies i?, 2 (the lowest level being labeled 1). The
second term, from SQ, is
_ (N/cTcR)
I:
(g;jbi)/I: (ii;[(gib;/b,)-2g;]
i
c=Sg
(21)
j<i
J>i
(9)
and
J<i
S= diag(b;-1/2).
(10)
The third term is just
We now define the activation energy in the usual way:
aR /
E ac t=-k (T_l)
a
R.
(11)
Since NB-K is an Hermitian matrix, we may apply the
Hellmann-Feynmann theoremll to obtain
- (N /c TcRHkTgTS2Ag.
The sum of (21), (22), and (23) constitutes a general
expression for the activation energy at any pressure.
III. HIGH- AND LOW-PRESSURE LIMITS
aR/a(T-l) = - (cTc )-1. cT[a(NB-K)/a(T-I )]c,
whence
E act = (kN/cTcR)·cT[aBja(T-I)]c
= (kN/c TcR)·gTS2[aAja(T-l)]g
(12)
(13)
from (8)-(10). From the temperature dependence of
A [Eqs. (2) and (3)], aA/ a (T-I) can be expressed as a
sum of three matrices.
aA/a(T-I)
=-k-llsM+sQ-~kTAI,
(14)
where SM depends solely on the form of the microscopic
reversibility relation:
(15 )
A. High-Pressure Limit
In the region of N large, gi can be expanded by
perturbation theory in inverse powers of N: gi=b i b//N+···. We shall have no call actually to evaluate
the b/. Making this substitution in the general expression for E act , and letting N-HXJ, all terms go to zero
except the first term in (21). The demonstration is
elementary, save that in treating (23) it is necessary
to recall that A b = 0, and to use also the complementary
result b T S 2A=0, the proof of which uses Eq. (6). The
surviving terms then give
EactHP=
(L: b;E;Kii/L: biKii)- (L: b;Ei/L: bi)
i
i>i
i
i
or, using ( ) to denote Boltzmann-weighted sums,
(i<j)
(i>j)
(23)
(16)
(24)
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ACT I V A T ION ENE R G YIN U ~ I MOL E C U L A R REA C T ION S
Discussion of the content of (24) in terms of a model we
have been pursuing has already been given. 2
B. Low-Pressure Limit
Let n be the index of the highest inactive level, i.e.,
the highest level for which Kii=O. Then in the lowpressure limit gi=O for i>n.
Therefore the term Lg,2E iK ii/b i in (21) is now zero.
The remaining part of (21), together with (20) and
(23), yield five terms which we write as
EactLP=
(EA )g/ (A )g- (E)g+EM+EQ-!kT.
(25)
This form emphasizes a partial resemblance to (24),
and certain differences.
The first term in (25) comes from the second term in
(21), or rather that part of it with i>n, j~n. Sums
over the Aij, which represent the rate of reaching active
levels, then replace the K i • (the rate constants for
reaction from active levels) which appear in (24). This
is the expected result for a transition from high-pressure
to low-pressure kinetics. From Eq. (7)
cTcR= -NgTS2Ag= -.V L L gigjAii/b.=-N(A)g
i
j
(26)
(a positive quantity, since Ag is necessarily ~O). The
first term is then
(L Ei L Aijgl!bj)/ (A )g=- (EA )g/ (A )g.
(27)
i>n
The ( )g notation in (26) and (27) will be used for
various kinds of weighted sums involving actual populations gi. Note that because gi=O for i>n, the sums in
(A)g and (EA)g with unspecified limits actually terminate at i (or j) =n.
The second term in (25) comes from the R term in
(23), and is a new kind of average of the internal energy:
L
gi2 E./b i
i
L
gNb
i
=- (E)g.
(28)
Numerically, (E)g is close to (E) of Eq. (24) because
the main contribution comes from the lower inactive
levels, for which gi",b i.
The third term completes the contribution of (21) to
EactLP. It contains explicitly the difference between
exact and Boltzmann populations.
EM = - {L Ei L bjA,j[(gi/b i )- (gi/b j )]21 (A )g-l.
i
i<·i
(29)
The energy dependent cross sections are responsible
for W. From (22)
2301
Equations (25)-(30) constitute a complete expression for the low-pressure activation energy. If the levels
i are degenerate, the bi need only be redefined as including the statistical weights.
If in (25)-(29) we make the steady-state approximation gi=b i (i~n), gi=O (i>n), there results
EactLP~(EA)' /
(A)' - (E)+EQ-!kT,
(31)
where
(EA)' / (A)' = L biE i L Aj./L bi L A ji.
i>n
j~n
i>n
(32)
i'5,n
This expression may be compared with the following
EactLP~ (EA)' /
(A)' - (E)+!kT
(33)
which is Slater'sl formula (his p. 39, Eq. 114) generalized to admit the possibility that the Aji may not all
be the same. Slater spells out in detail only the "strong
collision" case, in which all Aji are equal for de-energizing collisions, in which case they cancel entirely in the
term (32). From (33) it is also apparent that the neglect
of the energy dependence of the cross sections Qji of
Eq. (2) amounts to the approximation W~2kT. That
this is so is also readily confirmed ab initio. The numerical examples given later suggest that this is a poor
approximation.
IV. RELATION BETWEEN THE LOW-PRESSURE
ACTIVATION ENERGY AND THE ENERGY
OF THE LOWEST ACTIVE LEVEL
If the last two terms are ignored, the steady-state
approximation (31) is formally analogous to the highpressure result, Eq. (24), with Lj;;;nAji replacing K ii ,
and it is clear the first term will have a value close to
the energy of the first active level En+!' We seek here to
show that the general expression (25), whose physical
content is far from transparent, should also have a
value close to E n +1•
In an actual unimolecular reaction, thousands of
vibrational levels will be involved, and even if one
could calculate the cross sections accurately, manipulation of the complete A matrix is impractical using
present techniques. It is therefore customary to introduce a system of condensing the actual levels with a
certain range of energy into a single "level" with appropriate degeneracy,12 thereby reducing the size of the
collision matrix A. Examples of this method include
the graining schemes of Tardy and Rabinovitch 13 or of
Olschewski et al.,14 or the 'pseudolevel' technique discussed in an earlier paper. 2If the A matrix is sufficiently
coarsely grained (i.e., if there is a sufficiently large
energy gap between the "levels") it will be tridiagonal,
i.e., only nearest-neighbor transitions need be considered. It is quite likely that an actual reaction cannot
adequately be described in terms of nearest-neighbor
transitions between condensed levels, each containing
many individual levels. Nevertheless the use of this
approximation enables us to reduce our complex formulas to a comparatively simple expression, which
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2302
R.
G.
GILBERT AND I. G. ROSS
TABLE I. First interation corrections to the populations of the
topmost inactive level. Truncated harmonic oscillator model.
ex
n
0.1
0.5
0.75
To force this still complex expression into a form
which recognizably relates EactLP to the energy of the
first active level (E n+1 ), we may be tempted to retain
only the leading term in the first sum (i=n). Then
since gn/L:gi«l, and setting Enr::::;En+1, (36) reduces to
EactLPr::::;En+l(gn-l/bn_l)- (E)g-!kT+W
Initial vector:
10
0.99
0.99
0.997
20
30
g(O),
Eq. (39)'
0.96
0.96
0.98
and if the level spacing is coarse enough, gn-l will not be
drastically less than bn - 1• Thus EactLP does apparently
emerge as closely related to the energy of the first
reactive level, if EQ is ignored. The effect of the neglected terms in the summation in (36) is in fact to
make up the difference between bn - 1/ gn-l and unity.
To see this, however, we must turn to specific models for
the rates of collision. We undertake a full numerical
study of the complete formula (25) later in Sec. VI;
for the present we turn to a simplified model for the
elements of the collision matrix A.
0.96
0.96
0.99
Ini tial vector: Boltzmann vector b
10
20
30
0.90
0.90
0.90
0.64
0.66
0.66
0.55
0.56
0.56
• Tabulated entries are values of Eq. (41).
b Tabulated entries are values of Eq. (42).
provides insight into the physical meaning of the lowpressure activation energy.
In the tridiagonal approximation the activation
energy equations (25)-(30) simplify considerably, to
E Mt LP=En+l l0.
bn
+
±
gi_l)2
Ei bi- 1 (~ _
Ai,i-l
i=2
gn bi bi- 1 A n+l ,n
- (E)g-!kT+EQ.
(37)
(34)
The last term we are still neglecting. The two terms
preceding it are typically only a few kcal/mole, and
may also be neglected for the present. The most important terms in the summation will then be those for
which i equals or is only slightly less than n. If gi= bi,
the sum in Eq. (34) is identically zero, and the steadystate expression is obtained. If, however, the deviation
from equilibrium in the upper levels is large (gn«b n ),
then the "difference" term in (34) (i.e., the summation)
will become dominant. Thus the calculation of the
activation energy depends on a knowledge of the
deviation from equilibrium of the populations of the
inactive levels.
We may further reduce (34) by using the eigenvalue
relation for a truncated tridiagonal collision matrix,
together with Eqs. (5) and (6). These yield
Note that this expression depends on the collision rates
only through the populations of the inactive levels and
through EQ. (This statement holds for the tridiagonal
model only.)
V. TRUNCATED HARMONIC
OSCILLATOR MODEL
The use of our low-pressure activation energy formula
requires knowledge of the actual population vector, g.
The steady-state approximation, or even cruder approximations to g, can give quite accurate values of the
low-pressure rate constant, if used in formulas based on
iterative methods (see, for example, Valance, Schlag,
and Elwood, I;; who derive upper and lower bounds for
the low-pressure rate constant, though without treating the temperature dependence). However, the difference term in Eq. (34) clearly shows that such approximations cannot be used in our expression for the lowpressure activation energy.
We may derive a good approximation for the lowpressure population distribution g, and thus the activation energy, for a simple model: the "harmonic oscillator" collision matrix. s,16 This model assumes that the
molecule is an harmonic oscillator with dissociation
occurring at levels i>n.
For the form of the potential assumed by Landau and
Teller,16 the thermally averaged cross sections for vibrational transitions induced by collisions are, to a good
approxima tion,
Ai,i+m=iX
(m= 1)
=0,
(m>l)
where X is a constant. Equations (5) and (6) give the
remaining elements.
The dissociation truncates this matrix at i=ll. The
eigenvector problem for this matrix has been studied
by several authors. The exact solutions are known,S but
are far too complex to be manageable in Eq. (36). We
have, however, come across a simple analytic approximation for the eigenvector, which will be shown to give
excellent estimates of the exact value, and which permits
the series in Eq. (36) to be summed. Since the energy
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ACT I V A T ION ENE R G YIN U N I MOL E C U L A R REA C T ION S
If we calculate (E)g from (39), then (43) becomes
levels are equidistant, and nondegenerate, we have
exp[(E1 -E2 )/kT] = bdb1 = b3/b2 = ••• =b,,/bn - 1 =a,
(38)
where
a< 1. Our approximation to the eigenvector is
gi:::::;b i ( 1_an+i- 1 )
= ai-1( 1-an+i- 1 ),
(39)
(where the normalization of h is such that hI = 1). We
may show analytically that the error involved in this
approximation is small, by finding the next approximation to the eigenvector in an iterative technique. Form
the matrix17
l+A/Z,
where
Z= max(-Aii)=Ann=(n-1+na)x.
(40)
The eigenvalues of l+A/Z are all less than unity,
and it is well known that, because g is an eigenvector
of A, it is also that eigenvector of l+A/Z whose eigenvalue is largest in magnitude, and closest to unity.
Thus if g(O) is an approximate eigenvector, then g(l) =
(l+A/Z)g(O) will be a better estimate (power method
of finding the largest eigenvector) .18 If we renormalize
g(l) so that gl(O) = gl(1), then gn (I) / gn (0) will be the correction to gn (0), which is the most sensitive element of the
eigenvector to this procedure. We find in this case that,
using expression (39) for gi,
g,,(1) / gn(O) = (n-1) (a+ 1)/ (n-1 +na).
(41)
If instead we had used the Boltzmann vector hi as our
initial eigenvector, we would have obtained
gn(l)/gn(O) = (n-1)/(n-1+na).
(42)
Values of the two functions are given in Table I for
representative values of nand a. It can be seen that
Eq. (39) is quite a reliable estimate for the eigenvector,
and the Boltzmann vector is not. The deviations from
unity of figures in the lower half of the table are minimum measures of the inadequacy of the steady state
approximation. If Eq. (39) for g is substituted into
(36) or (25)-(30), we obtain, omitting terms of order
an,
EactLP=
AE{n+ (1- 2a)/ (1-a) 1- (E>a-!kT+J§J,
(43)
where AE is the energy spacing. In the computations of
our earlier paper,2 the value of a for which the matrix
becomes approximately tridiagonal is ",-,0.5. Neglecting
J§J, this gives the expected result that the activation
energy is approximately E n+l.
Note that (43) remains bounded as a~1, i.e., for
small grain size, for it can be shown that, in the limit as
~1, the right-hand side of (43) becomes
nAE- (E)a-!kT+J§J.
2303
(44)
EactLP=
(n+1)AE-!kT+W,
(45)
which can also be obtained directly by substituting (39)
into the expression for the low-pressure rate constantl
R LP =
L: gi L: Aji/L: gi
i
i>n
(46)
i
which can then be used in Eq. (11).
We may extend the model by assuming that the
levels have degeneracy di , when (41) becomes
gn(1)
gn(O)
(n-1)(1+a)
(47)
{n-1+nad n+1/d nl'
In such a model it is harder to establish a general
connection between the activation energy and E n+1• If,
for example, we assume that the molecule consists of s
degenerate oscillators, then
=
di = (s+i-1) !ji!(s-l)!.
If we substitute this into (44), and use the representative value of 0.5 for a, then g,,(I)/g,,(0»0.9 if s<n/3.
Equation (39) is then still a fairly good approximation
for gi, and (44) is still valid.
This treatment is as far as we have been able to go in
establishing a connection between the low-pressure
activation energy of unimolecular reactions and the
energy of the first active level.
VI. THE LOW-PRESSURE ACTIVATION ENERGY
OF THE THERMAL DECOMPOSITION
OF N 2 0
We now turn to an evaluation of the low-pressure
activation energy for the thermal decomposition of
N 20, using the complete expressions derived in Sec. II.
We will not be so concerned with reproducing experimental results as in obtaining an idea of the contributions that the various terms in (25) make to the
activation energy in this real system.
In our previous paper2 a graining system was devised
whereby the "pseudolevels" were expressed as levels
having the average quantum numbers of all the actual
levels over a given energy range, and the collision
matrix was calculated by treating these as a single
level, with appropriate degeneracy. The grained system
approaches the real system as the number of actual
levels per pseudolevel approaches unity. Using the SSH
theory19 to evaluate the cross sections, it was indeed
found that the process converged for sufficiently small
grain size. Using the methods of Ref. 2, the evaluation
of the quantities in Eq. (25) is quite straightforward.
It is interesting to compare the values obtained by this
method with those obtained by approximating the
popUlation of inactive levels by their equilibrium values
[Eq. (33)].1.3
Some results of such calculations are given in Table
II. They assume a model of the N 20 decomposition in
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2304
R.
G.
GILBERT AND
r.
G.
ROSS
TABLE II. Calculated low-pressure rate constants and activation energies for the thermal decomposition of N.O.··b
Complete expression Eq. (25)
900 0K
17000K
Steady-state approximation Eq. (33)
4000 0K
900 0K
17000K
4000 0K
1.4XI0'
5.9X109
5. 5X 1014
67.66
67.66
67.71
3.58
-4.36
-2.68
6.76
-9.20
-5.07
15.91
-25.39
-11. 93
64.20
60.15
46.30
Rate constants (cm S mole-I'secl )
9.1X101
1.6x109
4.7XlO l '
Components of the activation energy (kcal mole-I)
EM
21.05
41.79
37.90
45.47
6.81
-4.36
-2.68
18.71
12.73
-9.15
-5.07
3.94
26.33
-17.47
-11. 93
(EA
(EA )ul (A)u
EQ
- (E)u
-ikT
)'1 (A )'
2kT
- (E)
-~kT
Total activation energy (kcal mole-I) C
EaetLP
66.29
59.01
38.77
R.e,LP
• Corresponding terms in the two equations used are set out in the same line. Concerning the kT terms in the steady-state approximation, see text following Eq. (33).
b For a description of the model used, see text. The lowest active level was at 64.3 kcal mole-I.
e Activation energies calculated from pairs of rate constants (using the values obtained from the complete expression) are 63.5
and 46.9 kcal mole-I for the intervals 900-1700° and 1700-4000 0K, respectively.
which reaction proceeds by a radiation less transition
from the ground state, described by a potential-energy
surface which is harmonic in NO and NN stretching
displacements, and in the degenerate NNO bending
vibration. The radiationless transition is to a linear
excited state (a triplet state) which is dissociative
along the NO stretching coordinate. In Ref. 2 this
model is described as "one-dimensional," and the
numerical parameters are given. The lowest active
level is at 64.3 kcal mole-I. The results in Table II
were obtained with a grain size of 500 cm- I (1.4 kcal
mole-I) between pseudolevels. Essentially similar
results were obtained with finer graining.
It can be seen from Table II that at relatively low
temperatures the activation energy approximates quite
well the energy of the first reacting level. At higher
temperatures, however, EactLP decreases, and the decline
is more marked when the complete expression (25) is
used. In the physical picture represented by the steadystate approximation this decrease in EactLP is primarily
attributable to the increase in the thermal energy
terms (E)+!kT. The reaction term (EA)' / (A)' remains
essentially constant. For the complete expression, the
component terms disclose a much less simple picture.
Thus Fig. 1 shows the ratio of actual populations gi
to equilibrium populations bi, as a function of energy
and for a particular temperature (17000K). The fall off
towards the energy at which reaction first occurs is
pronounced, and of course becomes still more marked
at higher temperatures. As a result (E)g increases less
rapidly than (E) as T increases. Figure 1 also includes a
few illustrative values of a quantity
AJ=
L: Aji
(48)
i>n
which measures the rate of reaching all active levels
from an inactive level i. The probability that a collision
0.5
2.5x'O
Aii{6.1 x10' 8
3.4 x10 2
o
20
40
60 64.3
Energy (kcal mole-1)
FIG. 1. The ratio of exact to equilibrium populations, gi/bi,
for the low-pressure region of the N.O decomposition, at 1700oK.
The collision matrix is truncated 64.3 kcai/mole above the zero
point leveL The spacing between pseudoleveis is 500 cm- I (1.4
kcal/mole). Also included are numerical values of collision rates
from these inactive pseudoieveis into the active region, namely
the Aii' of Eq. (48), for a few of the higher inactive levels.
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ACTIVATION ENERGY
IN
will be followed by reaction is overwhelmingly greatest
for those levels for which gi most deviates from bi. The
expected consequences are perhaps best seen by referring to the approximate equation (37); here the terms
(EA )g/ (A )g+ Elf which jointly replace (EA)' / (A)' of
the steady-state approximation have been reduced to
En+1(gn-I/b n- 1), which decreases with increasing temperature. The figures in Table II show that this is
indeed so.
Finally, comparison of corresponding values of EQ
and 2kT show that the values of Eact as calculated by
the two methods would have been much more unlike
had the energy dependence of the collision cross sections,
neglected in the steady-state approximation, been
neglected also in evaluating the complete expression.
The extent to which the two calculations give comparable results is thus to some extent dependent on the
particular method used-here, the SSH approximation-to find EQ. But, fortunately perhaps, EQ is not a
large term at the lower temperatures where most
kinetic studies are conducted.
The experimental results for N20 20 appear to indicate
a decrease in the activation energy above 2S00oK,
although these results must be viewed with caution.
Also, it is possible that this decrease in the low-pressure
activation energy at high temperatures, due to deviations from equilibrium populations, could partly
explain the anomalous value for the low-pressure
activation energy for the thermal decomposition of CO2.
The experimental value for the activation energy
appears to be very much less than the minimum energy
for reaction,21 a result which the present theory has
successfullyaccommodated.22
ACKNOWLEDGMENTS
We thank the Australian Research Grants Committee
for support of this work. R. G. G. acknowledges the
award of postgraduate scholarships by r.C.r.A.N.Z.
Ltd., and the Australian National University.
APPENDIX
Expression (21) is here derived from (20), a transformation which is needed in order to obtain a determinate high-pressure limit. It will be convenient to
write
(AI)
Then (20), apart from a multiplicative factor, is
N
L: g. L: Aii(Ei-Ei ) ('Yi-'Yi)
i
(A2)
3>i
Equation (A3) uses the summation convention introduced in the text after (21). Invoking microscopic
UNIMOLECULAR
REACTIONS
230S
reversibility, Eq. (S), it becomes
N'L Ed'L giA;j("(j-'Yi)
i
i<'
+ L: bi'YiAii('Yi-'Yi) I.
(A4)
3>'
Now
i
from the fundamental definition of the rate constant R
[cf. Eq. (7)J, whence
(g;/N)(Kii-R)
= L: Aijgi
i
= L:
(Aiigj-Ajig.)
:i;#i
from Eq. (6)
which may be substituted in the second term of (A4)
yielding
'L E,{gi'¥i(Kii-R)+N L
i
(gj-'Yibi)Aii('Yi-'Yi) I·
~i
Using (Al) again, and restoring the constant factor,
gives the required result (21).
1 N. B. Slater, Theory of Uninwlecular Reactions (Cornell U. P.,
Ithaca, N.Y., 1959).
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3 R. C. Tolman, Statistical Mechanics (Chemical Catalog Co.,
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(1966) .
• W. G. Valance and E. W. Schlag, J. Chern. Phys. 45, 4280
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6 M. Menzinger and R. Wolfgang, Angew. Chern. Intern. Ed.
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Chern. Phys. 51, 2885 (1969).
22R. G. Gilbert, Chern Phys. Letters 11,146 (1971).
1.
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