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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