Changing Conceptions of Activation Energy Philip D. Pacey Dalhousie University, Halifax. Nova Scotia B3H A13 Canada The concept of activation energy is fundamental to chemical kinetics. I t is one of the earliest concents a student learns. Even so, it is often mysterious and difficult to master. Part of this difficultv-mav.he caused hv the fact that activation enerev is not a single concept hut a series of related concepts. These are the Arrhenius activation enerev, the activation enerav a t absolute zero, the enthalpy of activation, and the thresxold energy, among others. Definitions of these concepts are found in many texts ( 1 4 ) . Diagrams have been prepared to compare certain of these quantities (1 b, 2,3c, d, 4a, 56,6). Despite this work, it has been pointed out in a recent review (7) that even ~rofessionalkineticists sometimes confuse these concepts. The purpose of this article is to compare these concepts in detail. All of the "activation energies" will he calculated for a single reaction. A diagram will be prepared illustrating five of these concepts. It will be seen that the concepts are related hut numerically different. The name of one concept will he shown to he misleading. A relationship commonly used to determine two of the "activation energies" from a third will he shown to be incorrect under many circumstances. The concent of threshold enerev will he seen to lose much of its signiiivanre fm quantum mrrhanicul systems. (:hemica1 kii desrrlbes the netirs is n rnvidlv . . drvt4ooinr . ..field. T h ~article new insights brought by modern kinetics to our conceptions of activation energy. The reaction chosen for detailed study has been
in the exponent in a common version of activated complex theory (Ic, 36,4c), as follows:
--
D+H2-DH+H
(1)
This gas reaction is simple and has been studied by a number of experimental and theoretical methods. Figure 1 shows a comparison of some of the "activation energies" for this reaction. The diagram is to scale vertically hut is schematic only horizontally. Starting a t the bottom, the diagram shows the potential energies of reactants, a t left, and of the activated complex, a t right. These energies are the minimum energies that the three atoms could have if the positions of the nuclei could be fixed. At the left. the two H nuclei would be seoarated bv the eauilibrium H; distance of 7.4 X lo-" m and the D atom would he a t infinite distance. The activated complex is the point of highest energy on the minimum energy path from reactants to nroducts. A auantum chemical calculation indicates that a t this point thk nuclei would he collinear, separated from their nearest neighbor or neighbors by 9.3 X lo-" m. The same quantum calculation haa provided the difference in these energies, the height of the potential barrier, V B ,which is shown in Figure 1and quoted in the table (8). Molecules also must have a t least the zero point vibrational energy, as shown by the next lowest lines on each side of the diagram. The subscripts i and j denote the individual vihrational modes of the reactants and activated complex, respectively. For convenience, the vibrational modes have been assumed to be harmonic and separable. It is here that isotopic substitution first affects the energies. For reaction (I), the single ui is available from spectroscopy (3c, 4d) and v;s have been calculated for the potential energy surface of Liu and Siegbahn (8,9). The zero point energies, 6.2 kcallmole for the reactant and 5.0 kcallmole for the complex, are significant by comparison with VB. The difference in potential energy plus zero point energy from reactants to complex, A E d , is the energy which appears 612
Journal of Chemical Education
(As an alternative, - V d R T could be placed in the exponent and the zero point energies could be included in the partition functions, Q t and Q, (46).) AEot is sometimes called the "activation energy a t absolute zero" (2,4a, 7). This name can be misleading, as will he discussed later. The value for reaction (1) is 8.5 kcallmole. Statistical Ouantities
The energies discussed to this point have been true malecular properties. Those which follow are statistical averages and depend on the temperature. The shaded regions on Figure 1 are intended to represent a thermal distribution of energies. At temperatures above 0 K, molecules have thermal translational, rotational, and vibrational energies. For a
Energies of
Actlvatlon lor D
+ HZ-
HD
+H Value
Name Potential Energy Barrier Height Barrier Height Corrected far Zero Point Energies Internal Energy of Activation Enthalpy of Activation Arrhenius Activation Energy Experimental Threshold Energy ( S = 2.5 X m2) Quantum Threshold Energy
Svmboi
kcallmole
ve
9.68
+ 0.12
AEd
8.5
AFm
7.7 7.1
Aha300
Eh
6.55 i 0.04 6.0 0
Figure 1. Camprison of various "energies of activation." as defined in the text. The solid harirontal a c w e d lines represent potential and zero point energies for D M. at len. and D. . . H. . . H. at right. The shaded regions represent a
+
Boltzmann distrihution, the thermal energy per mole would he (10)
ETo - Eoo = JoTC,dT (3) where ETO is the internal energy a t temperature, T, and C , is the heat capacity at constant volume. Thermal energies for reactants and complex a t 300 K have been found by the methods of reference (10) to be 2.4 and 1.6 kcallmole, respectively, as shown in Figure 1. For the activated complex, no contribution is included from motion along the reaction ~ he the average energy of acticoordinate; that is, E T would vated complexes which have no component of velocity toward the reactant or product side of the harrier. , he the "inThe difference in energy shown, A E T ~would ternal energy of activation" in a thermodynamic formulation of activated complex theory (4d, 11,12), as follows: k =
e, h
eASrSlRe-AETilRTe-6
(4)
a
Here, A S T ~is the entropy of activation; with standard state of 1 molell; d is the number of molecules of gas in the activated complex minus the number in the reactants, and rn is the molecularity of the reaction. For reaction (I), d = -1 and rn = 2. The transmission coefficient, K , is often omitted (4d, 11, 12) from analogues of eqo. (4). Unless K is temperature independent, this omission would require a change in the defini. this is not pointed out. tion of A E T ~Usually T o calculate the enthalpy from the internal energy, one must add PV. For reaction (I),this is 2RT for the reactants and R T for the complex. The "enthalpy of activation" of activated complex theory, A H T ~is, 0.6 kcallmole less than A E T ~ a t 300 K. The only quantity on the figure which has not been discussed is E A , the Arrhenius or experimental activation energy, E* = -R- dlnk (5) d(lrn) The value shown is the experimental value for 300 K, 6.55 f 0.04 kcallmole (13). According to Tolman's theorem (14), which assumes that the reactant state populations follow a Boltzmann distrihution, E A is the difference hetween two statistical quantities. The tip of the arrow below E A is the average internal energy of the reactants (that is, the average energy of D atoms plus the average energy of HZmolecules). The tip of the arrow above E A is the average energy of complexes actually passing from reactants to products. This is equal to the average energy of newly formed products. If eqn. (4) is differentiated according to eqn. (5), one finds,
EA = A E T ~+ R [T - dln~/d(l/T)] (6) If K , the transmission coefficient, is unity or independent of temperature, as is often assumed (5c, 12), then eqn. (6) can he simplified to EA = AEd + RT (7) T h e interpretation of E A by Tolman's theorem and the definition of A E T ~seem very similar, so it is necessary to explain the R T tern which appears in eqn. (7). It has two causes. Firstly, real complexes do have a velocity along the reaction coordinate; this velocity was not included when calculating A E T ~The . average energy in this degree of freedom would he 0.5 RT. Secondly, the upper energy in E A is the average energy of the complexes forming products, not just the average energy of all complexes. Complexes with small velocities in the reaction cwrdinate s ~ e n ad lone time as complexes, contributing greatly to the pop;lation of complexes, but contributing little to the formation of moducts. Complexes with large . velocities in the reaction coordinate spend little time as complexes, contributing little to the average energy of complexes, but more to the average energy of newly formed products. Because of this, the newly formed products have an average energy ~
~~
greater than the average energy of all complexes by an additional 0.5 RT. According to eqn. (71, the upper arrow tip for E A should he 0.6 kcallmole higher than that for A E T ~Instead, . it is 1.0 kcallmole lower. Provided the theories and the data are not in error, this indicates that K is temperature dependent. Indeed there is mounting evidence, both theoretical ( 1 5 ) and experimental (16,17),that a temperature dependent K must he included in activated complex theory. Usually this dependence is attributed to quantum mechanical tunneling through the activation harrier. In the foregoing, it has been assumed that the various comolexes all have identical D. . . H and H. . . H distances. Fiecently, it has been shown (15) that the maximum internal enerev ".a t 0 K occurs for an unsvmmetrical configuration and has a value 0.2 kcallmole greater than the valie for a symmetrical complex. (See Fig. 1.) Threshold Energy
A related quantity, not shown on the diagram, is the threshold enerw, .. which is the minimum relative translational energy that the reagents require in order to react (30, 4e, 50). The relationship of this to the energies shown depends on the definition and-method used to determine the threshold. Thresholds may be determined in hot atom (18)or molecular beam (19) experiments or in classical (20) or quantum (21) calculations. Normally the probability of reaction per collision is calculated or measured for a series of relative translational Usually results are expressed as total reactive energies, Ere]. cross sections, S (3a,4e, 22), which are products of the prohability of reaction per collision times the collision cross section, a u ~The ~ threshold ~ . could be defined as the smallest value of Ere, for which S is large enough to he detected. Alternatively, data could be extrapolated to estimate the largest value of E,.I for which S would he zero. An experimental estimate ( 1 8 )for reaction (1) was found by producing energetic D atoms, or "not atoms," by photolysis of DI. Reaction could be detected provided that D and H2 had a relative translational energy of at least 8 kcallmole. At this enerm, calculations indicated that S was 2.5 X 10-21 m2. The estimated dependence of S on E,.I is shown as the shaded region in Figure 2. Karplus (20) studied reaction (1) by simulating collisions of D with H2 on a computer using classical mechanics. He
~
EreI [kcal mol
-'I
Fioure reactive cross sections for reaction 11) as a function at relative - - 2. ~ Total vans ationa energy Snsaed reg on. estomatea trom e x p mem 1 18) from clessocal trafecton, calm atcons(201 e,from a oostonea-wavequanlm ca Culation (21). ~~
-
~
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Number 8 August 1981
613
assumed that the interaction between the atoms was such that V g would he 9.1 kcallmole and AEot would be 8.3 kcallmole, different from the hest estimates given in the table. When the Hz molecules were given an initial vibrational energy of 6.2 kcallmole (%hs,), . . the cross sections were found to he in excellent agreement with the experiment, as seen in Figure 2. The results aave a true threshold with S=O a t 6.7 kcall mole. Tang and Choi (27 ) used the same harrier as in reference (20) hut simulated the collisions using quantum mechanics. They considered only Hz molecules which were not rotating initially. They found a finite probability of reaction even for energies as small as 4.6 kcallmole. In quantum mechanics, tunneling occurs, allowing these low energies to contrihute. Reaction (1)is slightly exothermic. A small probability of reaction should oersist down to E.,I = 0 (23). All of the cross sections in Figure 2 are approximate, hut certain conclusions can he drawn. The experimental threshold is clearly dependent on the detection sensitivity and on the minimum relative translational energy accessible. The quantum cross sections approach zero asymptotically a t low enereies. Onlv the classical calculations show a simple threshold. Other Reactions For other reactions, the conclusions of this article would he different quantitatively hut similar qualitatively to those for reaction (I). Barrier heights could vary from ahout zero to ahout 100 and the other energies are not necessarily less than VB. For himolecular gas reactions, A E T ~is likely to he less than AEOt,because of a loss of translational degrees of freedom and a decrease in C, on activation. For other classes of reaction, A E d could he ereater than or less than A E d . Again, for himolecular gas reactions, P V for the reactants would he greater than PV for the activated complex hy R T : AHT' would he less than AETt by the same amount. For termolecular e a reactions the difference would douhle to 2RT. For unimolec&r reactions there would he no difference. For reactions in condensed phases, PV would become very ~... m . -. ..~ l l ~
Equation (7) relating E Aand AErt is true only for reactions which exactly ohey activated complex theory with a temperature indevendent K . This rules out reactions in which tun-
.
,
...
by diffusion (Je). These effects almost always lead to lower ~ values of E a... For liauid . nhase . reactions. values of A H T ,and AETf are often calculated from En usingeqn. (7) (4d, 5c, 11, 12). Similar assumvtions are made in calculatina- ASr' . from experimental pre-exponential factors (4d). Recause of the limitations on eon. (7). it should onlv he used with caution. The pattern observed for the qiantum cross se&ion in Figure 2 should he quite general for reactions with activation
614
Journal of Chemical Education
harriers. For reactions which are exothermic a t 0 K. the true threshold energy would be zero (23);for reactions endothermic at 0 K, it would equal the chance in zero voint enerev -.vlus . potential energy from reartants to products. At very low temperatures, thermal energies would vanish: E A should approach the theshdd energy, not AEJ. It is misleading, then, to call AEJ the "activation energy a t ahsolute zero" (2,4a, 7). Preferred names would he "the harrier height corrected fi~rzero point energies" (Ib) or "the adiabatic, ground-state energy change on prt~ceedingto the saddle point" (1.5). Conclusion In summarv. it can be seen that new develovments in kinetics are changing our understanding of hasic k c e p t s . The different concepts of "activation energy" are related to each other, hut are numerically distinct. For reaction (1)a t 300 K, for examvle, E A is onlv twwthirds of Vu. Thermochemical , AS;^, cannot he deterquantities, suchas AH;^, A E T ~and mined easily from experimental data, unless one can he certain that activated ctrmplex theory, without tunneling, works exactly for a particular reaction. The concept of threshold energy is difficult to define, especially for quantum mechanical processes. For such processes, the activation energy a t 0 K would equal the endothermicity for reactions endothermic a t OK and zero for reactions exothermic a t 0 K; it would not equal AEaX. The concepts which emerge with the simplest and most general significance are E Aand V g . These quantities should he emphasized in future teaching. Acknowledgment The author wishes to thank Prof. M. Niclause, E.R.A. No 136, CNRS, 1, rue Grandville, Nancy, France, for hospitality while most of this work was completed. Literature Cited 0 1 Canliner. W. T.. "Rster snd Mechanisms ofchemical Reactiune." Benjamin. N Y.. 1969.p. 86: lbl P. 10'2: lci p. 104: Id1 pp. 122-112; let p. 187. 121 iohnatun. H. S.. " h a P h s ~ eReectVm Rate Theory," Ronald Press. N. Y.. 1966, pp. 169-170. 1x1 la1 1.sldler. K. .I.. "Theories ~,TChsmicslReactionRa(er: McCrsw~Hill,N. Y., 1961. pp. ,Si; 11,) p. 48: lei p.87.88:(dl p. 164. I41 la1 Mulrahs.M. F. R.."Cas Kinetin." Nelron.Lnndun. ,973. p. 10: lbl p. 24:lel p. 26: Id1 UP. :l:i.l.l: (el pp. 192,198: 11) pp. 1%-204. w ~ t ~H. n E.. . and Schwam,H.A.. "chemid ~ i n e t i n . "~ r e n t l r e ~ ~ n~ gl l~. e w ~ a l (51 Cliffs. N..l.. 1972, p. 68: Ihl p. 9% lc) pp. 108, 109. IRI Wullrum. J.. Rnr Hunr.unper.phjsck Chem .RO.H92 11978l. 171 Truhlar, U. C.. and Wyatt. R. F L A d r . Chem Phyr .36. I41 119771. IHI Liu. 8 . s n d Sieyhshn, P., J. Chem Phys.68.2457 11978). 191 1. Shnuitt.J rhrm Phyr.. 49,6048 119681. (101 R m w n . S . W.. "Th~rmnchemicalKinetics." 2nd M.. Wiley. New York. 1971, pp. '16. :38. (111
