Donald G. Truhlar
University of Minnesota
Minneapolis, 55455
Interpretation of the Activation Energy
The Arrhenius activation energy E, is a phenomenological
quantity defined in terms of the slope of an Arrhenius plot [In
k(T) versus 1/T] as
tribution of such pairs of specified states of reactants
where k(T) is a rate constant and kn is Boltzmann's constant.
When E , isa constant (independent of T ) ,integration of eqn.
(1) leads to the Arrhenius fnrm of the rate constant
~ Egj
where the pair's internal energy E, is the sum of E Aand
d s in
~ the
and the term labelled a must be included d ~ ~times
sum of eqn. (9) because we have not included any degeneracy
factors here.
Let ( ~ ( V R ) ) denote an average of a function ~ ( V Rof)
initial relative speed VR over a Maxwell-Boltzmann distrihution P(VR, T ) of relative speeds
k(T) = A exp (-E.lkeT)
(3)
The Arrhenius activation energy is often interpreted approximately as the energetic threshold Eo for reaction but it
may be either larger or smaller than Eo. The Arrhenius activation energy for an elementary bimolecular gas-phase reaction step occurring at equilibrium does have a simple interpretation that was first given by Tolman (I). I t is the average
total energy (relative translational plus internal) of all reacting
pairs of reactants minus the average total energy of all pairs
of reactants. Although this is an elegant and useful result it
is not as well known as it should be; e.g., it is not mentioned
in most physical chemistry and kinetics textbooks, and some
books which do mention the result give a proof and discussion
which involve only relative translational energy Ere,(e.g., (2))
or only internal energyEint (e.g., (3)).Although several complete derivations, including both E,I and Eint, have been given
(e.g., 1,44), it is instructive to rederive the result (7) using the
lanauaae of modern collision theorv (8-11). i.e.. using stateto-btate reaction cross-sections. ~ h purpose
p
of this a.rticlr ii
to present such a derivation in the hope that it will lead to a
better appreciation of the result.
Review of Collision Theory: Deflnilions
The reader is referred to articles by Widom (12) and Boyd
(13) for discussions of equilibrium rate constants and to the
textbook of Gardiner (10) for the way equilibrium rate constants are expressed in terms of state-to-state cross-sections.
We summarize the results here in a convenient notation for
the proof which iollows. Consider the reaction of molecule A
in state i with molecule R in state i toaive molecule C in state
k and molecule D in state I . ( ~ h e s ~ e c jcase
a l of atoms is also
included, and the extension to more than two products will
be obvious.) Let (g(Ai))Tdenote an average of any function
g(A;) of states i of A over a Boltzmann distribution Pa,(T) of
states i
vE
J'
( y ) 3g(Vd%exp
1 2 (-g%l2k~T)dV~ (11)
2rkT
where p is the reduced mass for relative motion
= 4r
+
fi = m A m ~ l ( m A mB)
(12)
By g(E,d we denote the same quantity as g(VR) hut now
considered as a function of Ere,
E..I = gVV2
Then by these definitions
M V d ) GR = (g(E,d)lm,
(13)
(14)
(15)
-
Let dAi.
. .. B;.
,. E..I.. . Cb.
... DI)
. be the state-to-state reaction
cross-section for a fixed initial relative translational energy
E,,, and let o(n, E,,l) be its sum over states of the products
.. .
The state-to-state, fixed-velocity rate constant k(a, VR) is
simply the product V ~ o ( a VR).
,
Then the equilibrium rate
constant k(T) is VRU(U,VR) averaged over Boltzmann distributions of initial internal states a and initial relative velocities VR
k m = ((v~~(a,Ed),T)l~,
(5)
(18)
(19)
where o(T, E,I) is sometimes called the excitation function
(14) and is defined by
where d ~is, the degeneracy, EA,is the internal energy of A,
and ZA,,,~is the internal partition funct~onof A
It is the reaction cross-section for a fixed E,.1 but averaged
= ( V R ~ TE.d)Z,,,
,
= ( Z A , ~I:
' d ~exp
, (-E~~lksT)g(A,)
= X d ~exp
, (-EaJksT)
ZA,"~
(6)
Similarly
d',
E d = (da,E,d),T
(20)
over initial internal states at temperature T. The corresponding rate constant k(T, E,,I) is simply VRU(T,E,.I). By
using eqns. (13) and (16), equation (19) can be written more
explicitly
Let ol denote the set (i, j ) of pairs of quantum states of A and
B and let (g(a)),Tdenote an average over a Boltzmann disVolume 55, Number 5. May 1978 /
309
Derivation of Tolman's Result
Substituting (21) into (2) yields
-
Xc, do,E r d exp (-E,lksT)
[:
g
E, exp (-E.lkeT)
(34)
e w (-E,IRT)]
Therefore
where
(24)
Using the two definitions (18) and (19) of k(T) then yields
-" .
the average internal energy of all pairs of reagents. This,
combined with eqn. (321, yields
which completes the proof.
Discussion
First consider the average value of E,.I for all A,B pairs. This
is given by
Next consider the average value of Ere]for all reactive A,B
collisions. This is given hy an average like eqns. (26)-(28)
except the additional weighting factor k(T, E,I) must be
added to account for the distribution of relative translational
energies at which reaction occurs
Substituting (28) and (31) into (23) yields
Thus if term (11) were zero [e.g., if o(a,E,I) were independent
of a],E, would he just the average E,.I of reacting pairs minus
the average E,,I of all pairs.
Next consider term (11).Using eqns. (14)-(16) and (21) we
rewrite this as
The Tolman interpretation of the Arrhenius activation
energy allows us to interpret the rate of change of the equilibrium reaction rate with temperature in terms of the distribution of reaction energies when the reaction proceeds a t
equilibrium at a given temperature. Experimentally, one
measures the so-called phenomenological or steady-state rate
constant, not necessarily the equilibrium rate constant (13).
But for typical atom-transfer and metathesis reactions with
activation energy characterized by a reaction cross-section an
order of magnitude (or more) smaller than the collision
cross-section, the measured rate constant is expected to equal
the equilibrium rate constant within experimental error (15,
16). One example where nonequilibrium effects may be important (and hence Tolman's interpretation may not apply
to the measured rate constants) is dissociation of diatomic
molecules in shock-tubes (17). The derivation of Tolman's
result did not actually require equilibrium. It merely required
what Boyd (13) calls "local equilibrium'' of reactants, in
articular a Maxwellian distribution of relative velocities and
Hu equilibrium distribution of the internal states of reactants
but not eauilibrium of reactant concentrations with oroduct
concentraiions. Snider (18)has shown that when t i e stateto-state reaction cross-sections are very small compared to the
state-to-state cross-sections governing transitions between
reactant states, then the steady-state rate constant is equal
to the one-way flux given by eqn. (18).The conditions under
which this local equilibrium assumption is valid have been
further discussed in Boyd's review (13).
Tolman ( I ) originally stated the result a different way. His
result is equivalent to saying that, at equilibrium, E. is the
average total energy of reacting pairs minus the average total
energy of colliding pairs plus % ~ B TThe
. version of the theorem used here was compared to Tolman's version by Fowler
and Guggenheim (4). The comparison requires the recognition
that while the average E,I for all pairs at equilibrium is given
by eqns. (26)-(28), pairs with high E,.I collide more often and
the average energy of colliding pairs a t equilibrium is given
by
The derivative can be evaluated using eqns. (20) and (9)
where k.,l(T, Ere1) is the equilibrium rate constant for collisions. Writing this as VRC~,,I(T,Erel)where rr,l(T, E-1) is the
collision cross-section yields
310 1 Journal of ChemicalEducation
T
(Edmi!idmg
pa"'
i-
E,I)EL exp ( - E , d k k a T ) d E , ~
J*4 T .
E,i)E,l
ocOl(T,
=
(39)
exp (-E,ilk~T)dE,i
In the special case where a,l(T, E,.I) is independent of E,I
(e.g., if it equals r D 2 where D is the hard-sphere collisiondiameter so that (kCol(T,Ere,))&, hecomes the familiar
hard-sphere collision rate constant (8rkTlp)'/2D2 ( l o b ) )then
(E&gding is ZkeT, which exceeds E,I for all pairs by the
"o;.~
v"..-
%kaT in Tolman's statement. This has sometimes been
misstated in the literature (19). Thus Tolman's statement
involves an avvroximation for the collision rate. The version
of the theorem proved here does not involve any such assumption and is completely general.
Many chemists consider the Arrhenius form of the rate
constant as just a useful functional form for fitting experimental data. It is hoped that the derivation presented here will
lead to a greater appreciation for the Tolman interpretation
of the Arrhenius energy of activation.
Acknowledgment
The author's research is supported in part by the National
Science Foundation. I am grateful to Dr. Robert K. Boyd for
helpful comments.
Literature Cited
(1) Tolman.R. C., J Amer Chem. Soc.. 42.2506 11920): Tolman, a. C., "Statistical Me-
chsnies with Application. to Phyaies and Chemiatri." Chemical Cetalag Company,
New York, 1927. pp. 260-270.
12)
. . Laidler. K . J.."Theoriesof ChemiealhadionRates."MeGraar-HillRookComoanv.
. ..
New L o r k , ~ e wYork. 1969. pp. 6 7 .
(3) Bennon, S. W., "ThermochemicalKinstics," 2nd Ed., John Wiley &Sons, Now Yark,
1916. pp. 15-16. Note: iE)..,,*ngshould
he (E1.u in the result given on p. 16.
(4) Fowler. a.H., and Gwgcnheim. E. A,, "Statktical Mechanics," MacmiUan. New York,
1939. pp. 491-506.
(5) Johnston, H. S.,"Gas P k a Rsadion RateTheory." Ronald Press,New York, 1966.
pp. 215-217.
(61 Levin*, R. D.. and Bematem, R 8.. "MolecdmReeethn Dynamics."Oxford Univenity
PIPBB,
Near York, 1974.p~.108-110.
(7) firplus, M., Porter, R. N., and Sharmq R. D.,J. Chem. Phys.. 43,3259 (19651.
(8) Eliason, M. s.,and Hirsehfelder, J. 0.. J. Chem. P h w , 30,1426 (1959).
(9) Grwne, E. F.,and Kuppermann, A,, J. CHEM. EDUC., 45,361 (1968).
(10) (a) Gardiner, W. C., h.,"Rates end MechanismsofChemiurl Reaefions."W. A. Benjamin, Menlo Park, 1969. pp. 75-85: (bl PP 87-88.
inGme%andPlaomas."
(11) Ross, J., Light. J.C., andshuler, K . E., in "KinetiePra-a
(Editor: Hochstim, A. R.). Academic Preaa, New York, 1969, p. 261.
1121 Widom, B.. Seienca, 148,1555 (19651.
(131 Boyd. R. K.,Chom. Re"., 77,93 (1977).
(14) LpRoy,R.L.. J P h y s . Cham.. 73.4538i1969l.
(15) Shizgal, B., and Ksrplua, M., J. Cham. Phya., 52,4262 (1970);54,4357 (1971).
(16) Widam, B., J. Chrm. Phys.. 61,672 (1974).
(171 Johnston, H., and Birks, J..Acc. Chsm. Re*., 5,327 (19721.
(18) Snider, N. S., J . Chem. Phys., 42.548 (1965).
(19) Menzinger, M., and Wolfgang, R., A n e w Chem., Int. Ed., 8.183 (19691.
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