Microscopic Reversibility for
Rates
of Chemical
out with Partial Resolution of the Product and
Reactions Carried
Reactant
States
.k
4.
James
L.
Kinsey
The Universiky of W'iseoansin Theoretical Chemistry Institute
Madison
W
is
e.
ogs
in
5
3
706
Applications
of
the principle of microscopic reversibility
are
considered for
rates
of reactions measured
in
experiments that
reveal
some of the dependence on the rotational, vibrational, andlor translational
states
of
the reactants or products.
a
charackeristfc
_set
of
quantum
states
for the
reactants
and products,
suggesting the removal
of
trivial
statistical factors (densities of
states) from the data to
retrieve
a
purely dynamical quantity
the state-to-state reaction
rate
suitably averaged over the initial and
final
sets
of
states
at
fixed total energy
E.
symmetric
with respect to the direction
of
the reaction--either the
foxward or
reverse
rate
coefficient
can
be obtained
by
multiplying
with the proper density
of
final
states.
Thus reflects the
intrinsic probability
of
the process, disregarding
the
statistical
bias introduced 'by the nature
of
the experiment itself.
accurate computation of state-densities appropriate to
several
kinds
3f
detailed kinetic measurements currently feasible
are
given.
Each type
of
measurement establishes
I
w
(E),
This quantity
is
fully
Methods
of
1
I.
Introduction
Despite
the impressive
successes
of
traditional chemical kinetics
in accounting for the gross features
of
chemical reaction
rates
such
as
their variation with temperature,
our
understanding
of
the
relative
importance
of
translationaf, rotational, and vibrational degrees of
freedom in reaction processes has remained highly speculative.
realized early,
to
be sure, that unimolecular processes require the
It
was
participation of internal degrees
of
freedom’, but the hiplieations of
this
fact
have been extensively exploited only in models
adopting
a
statistical viewpoint
at
the
outset to avoid consideration
of
fine-
grained details
e
The dffPicuSty in establishing experimentally the
2
effects of the different degrees
of
freedom originates
in
the indeterminacy
of the initial and final molecular
states
in
most experiments: The
reactant
states
are
usually unspecified except
as
they
are
weighed by
the Boltmann dbtribution, and the product
states
are
restricted only
by the conservation
of
energy.
Experimental techniques capable
of
resolving the
states
of
the
reactants andlor produets
are
needed
to
expose the more detailed aspects
of
the reactions. Although the reactant translational motion can be
well
controlled in molecular
beam
experiments and widely varied with the
use
of
special techniques
such
as
supersonic or chargeexchanged beams
,
there
is
no
useful method
at
the
present
time
for selectkg internal
states
of
the reactants over
a
wide range of internal energies, Results
have, howeverP begun
to
appear
on
partially resolved analyses
of
the
states
of the products of bimolecular reactions studied by
(a)
infrared
3
4
5
chemiluminescence
(b)
chemical
lasers
arid
analysis of the products
in
crossed molecular
(e)
velocity-
or
beam
reactions
.,
6
The reactions studied by these techniques
are
typically
exoergic,
so
that any
of
the contributions
to
Che
total energy
2
state-
quite
of
the
products
is
free
to
range
OVF,I
2
span much larger than
the
variation
allowed by
the
experimental
corxditions
in
the
total
energy
itself,
of
the infrared
chernilumfnesceace
reactions4',
C1+HT.+H@I+I
serves
as
an
example,
If
a
crude
line-of-centers model
is
used
to
account
for
this reaction's roughly
0,7
kcal/mole Arhennius
aeti17ation
energy
it
is
predicted
that
over
90%
of
the reactions
occur
with
reactcint
energies
(translational
plus
rotational)
in
the range
2
a
553
5
kcal/mole
e
Relative
to
the
product
-
ground
state,
which
lies
31.7 kcaE/mole below
that of the reactants, the energy
is
therefore
34.2&J.S
kcal/mole.
One
The principle of microscopic seversibil.ity7 permits the trans-
formation
of
data on produet state distributions
into
information
about
the dependence
of
the
rates
of
the
reverse
(endoergic)
processes
on
the division
of
the
available
energy among rotational,
vibrational,
and transXationa1 modes. This
is
not
merely
811
academic exercise
giving nothing
more
than
a
different representation of
the
same
data:
In
general
only
"atypical" reactions
(low
thresholds,
large cross-sections)
can be studied by these new teehniques.
thresholds
at
least
equal
to
their endoergicities,
are
much
more likely
to
be representative
of
the
majority of chemical processes.
Data
on
unusual reactions
are
thus
axehanged
for data
c3n
more ordinary ones.
The
+@verse
reactions, having
3
Each experimental technique
reveals
different aspects
of
the
distributions, some of them more detailed than others. In applying
the micro-reversibility principle to the best advantage,
it,
is
im-
portant to incorporate
all
the details produced by
a
given experiment
while excluding
all
those that
are
not.
The
purpose
of
this paper
is
to outline these considerations for
a
number of experimentally intergsting
cases, and to derive working equations
for
each. The fully resolved
infrared chemiluminescence experiments have already been analyzed
in
two papers by Anlauf, Maylotte, Polanyi
and Tardy9, but
will
also be included here.
8
and Bernstein and by PoLanyS
11.
Microscopic Revereibility
Symmetries in the equations of motion of
a
dynamic system
generate restrictions on the solutions of these equations usually
in the form of conservation
laws,
in their general ability to
limit
possible kinds
of
solutions or to
establish relationships between solutions
even
though none
of
the
aetua:
solutions may
be
known. When the Hamiltonian
is
invarianz to
time-
reversal"
a
second solution to the time-dependent Schrodinger equation
(or classical equations of motion), differing only in the direction
of
the motion, can be immediately obtained
from
any
given solution by the
operation
of
time-reversal on the original state-vector
(or
trajectory)
The expression
of
the consequences
of
time-reversal
symmetry
in
terms
of transition
rates
is
known
as
the "principle of microscopic reversib$lity".
Although such
a
principle can be formulated for any kind of process, our
The power of these relations
lies
11
,