A Random Walk among Accessible States S. H. Bauer Cornell University. Ithaca, New York 14853 The lnvltatlon I invite the reader t o stroll with me through the forest of ideas which make up chemical kinetics, and particularly to examine the central clearing, designated the "transition state," from which i t is postulated there is a direct path to the other end of the woods. In fairness, I warn you that the central clearine mav Drove elusive: indeed. it mav not be discernable. ~ i t h i s t i k ethe , propdsition &at for limited but significant class of kinetic orocesses use of conventional transition state theory ( T S T ~involves an artificial model must appear to be a futile attempt to by-pass the extremely popular "col" between reactants and products. I dare present such a proposition because a more satisfactory model is available. Its use entails no greater effort than do conventional computations and the results are nearly, but not quite, equal toTST. Besides complying with the Principle of Parsimony of Postulates, its principal advantage is the satisfaction of usine a model which olaces no strain on one's credulity. In the future, experiments may be sufficiently refined to oermit critical tests to determine which model more close& represents the data. Intrinsically, all reversible isomerizations comprised a class of reactions for which T S T is artificial; of immediate interest are intramolecular conversions over low barriers, generally designated as conformational changes. A few examples follow:
a
Rotations about stiff single bonds:
Inversions of nonplanar structures:
H-bond transfers:
Their common feature is the presence (in the multidimensional energy surface) of two or more potential wells, separated by relatively low barriers, considerably below the energy required for molecular fragmentation. For the examples cited the molecular enthaloies are comnarahle a t the notential minima so that reco&izably distinct species (thLy may have to he suitablv labeled-isoto~icallv or bv suhstituents) are in dynamic e&librium at ambient temperatures. EX: cluded from this discussion are chemical transformations which involve dissociations or displacement reactions (AB A B, or, AB C (ABCI A BC). These are inherently more complex, in that energy is shuttled between
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Figure 1. (a) Schematic of a potemial energy surface wHh a single deep well. The X and Y coordinates represent combinations of interatomic distances. while Z Is the potential energy, which increases when (X. VJ depart from lhe most stable configuration(231:157. In arbibary units). (b) A cut-away section of a potential energy surface wilh two adjacent wells. separated by a low barrier. (c) A potemial energy ourlace Mat has four adiacem wells of unequal depth. These are separated by a tetra-furcatad ~011.
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internal and external dynamic variables, vibration lation.
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We Load Our Knapsacks
A review of four basic concepts will establish the vocabulary for our discourse. (1) For any collection of n atoms which can be combinedto generate a molecule the number of discreet molecular states is enormous. Even when one restricts consideration to the lowest electronic energy configuration, for any specified bonding configuration (i.e., atom-atom connectivities) there still are many translational, overall rotational, vibrational, and for n 2 4 there may be internal rotational states. For intramolecular conversions we need consider only the last two types, the quantized internal states. Each of these states has a unique energy, but for any sizable molecule (say five or more atoms) there will be many near coincidences and close spacings between adjacent states. Hence it is convenient to represent all these states by a distribution function, g(E)AE, which specifies the number of states for which molecules AE. g(E) have total internal energies between E and E increases very rapidly with increasing E; this will be illustrated below. (2) Consider first an n-atom molecule for which there is a single deep well in its lowest electronic energy surface (Fig. la). That means: there is a single geometric configuration with specified bond distances and bond angles (i.e., a set of n(n - 1112 values for the internuclear distances between all atom pairs: r,(i j ) ) for which the electronic energy is lowest. Departure of r(ij)'s from these values can occur only when there is an increase in electronic energy. Thus the entire well serves as a potential function for vibrations of the nuclei. Whenever any [ r ( i j ) - r.(ij)l = 0, the vibrational energy of that oscillator is entirely kinetic, but when lr(ij) - r.(i j)l > 0 some or all of the energy becomes potential (i.e., electronic), to an extent that depends on the magnitude of the displacement. When the shape of the well is known (or oostulated) the laws of mechanics permit one to write a q u k t i t a tive description of the complex interatomic motions under the constraint uf the well walls (somewhat like the motion of a highly skilled skater demonstrating his virtuosity by skating w~ldlymound the inner walls of a large bowl). The oppormll) uncorrelated interatomic motionscan beexpressed as a weighted sum of a fixed number ( 3 n - 6) ((3n - 5) for linear mdecules) of nearly harmonic normal modes (nhnm). each with a characteristic fundamental freuuencv. The total number of these nhnm's is dependent only bn the number of atoms in the molecule, irres~ectiveof its eeometric structure; however, their characte&ic frequenies are very sensitive to the structure. Another a a v of expressine this is: the (3n - 6) values for the fundament& a r e ' d e t e d n e d by the shape of the electronic energy well. The complete set of states for any molecule is merely a superposition of all possible harmonics and combinations of these (3n - 6) fundamentals. That set is designated g(E). Example: CH30NO has 15 fundamentals which range in frequency (for syn) from 153 cm-' (the torsion about the CH3-0 bond) to 3032 cm-', characteristic of C-Hstretching vibration. For both conformers the d E ) functions have been calculated by direct state count. opth he syn isomer thereare 19 stares in the intervnl0 500 cm-I: ROstares in the interval 500-1000 cm-'; 266 states in the interval 1000-1500 cm-'; 721 states in the interval 1500-2000 cm-'; 1794 states in the interval 2000-2500 cm-', etc. The g(E) function is treated as continuous, but actually it consists of adense forest of spikes with crowding which very rapidly increases as E increases. (3) Most chemists talk as though they track the time history of a single molecule. Of course, this is an abstraction but the consequences of accepting seriously such an unrealistic model leads to a gross misconception of how reactions occur. There is no experiment that measures the behavior of
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a single molecule. Even with modern ultrasensitive techniques whereby single events can be counted, the final result depends on an accumulation of a large number of such events. All chemical reactions involve a v e r s large number of molecules; a femto-mole represents 6 X lo8molecules; in a single drop of liquid, 1mm in diameter, there are -5 X 10'9 molecules. This vast conglomeration is in constant motion with the molecules perpetually colliding and exchanging energy. Durina these interactions some vibrators are slowed down while others are speeded, or the molecules rotate faster. In liquids and gases they also lose or gain translational energy. How can one describe such a dynamic, chaotic sys-
"....
tom?.
Heraclitus (535-475 B.c.) believed that there was no permanent reality except the reality of change - that the only possihle "real" state was the transitional one of becoming. The 20th century chemist's model of matter is not inronsistent with this conceot, . . but extends bevond it. Whereas anv single molecule is continually changing from one state to another due to its being buffeted by its neighbors, the time scale for the transition is considerably less than the time of sojourn in any state. However, the ~ r o ~ e r t i of e sthe svstem do not depend on which molecule~isin any specifieatate; rather, they depend on how many molecules are in each of the accesiible states. The continuous chaotic redistribution generates change on a molecular scale but allows the svdtem properties to remain unaltered as long as the popuiation distribution remains unaltered. The details of how transitions between states occur is an interestine" soecial Dart of -chemical kinetics, but generally chemical kineticists are concerned with drifts in state ~ o ~ u l a t i orather ns than with the mechanics of transfer fromAst'ateto state. More than a centurv azo durine the develooment of the kinetic theory of gases, ~oltzm&n recognizei the significance of the population distribution function and marshalled argum&& to show that there was a unique function Neq(E) (the number of molecules that have a total energy between E and E AE) which is stable with time; i.e., all other distributions drift and eventually attain this ~ a r t i c u lar distribution due to the collisional ehcounters and consequent energy transfers between the molecules which comprise every sample of matter. The basic Maxwell-Boltzmann distribution is of great significance for i t rests on the plausible assum~tionthat no state has a ereater chance of heins ~~~~occupied than any other state. ~ t a t h i c a arguments l base: on the condition that the total enerw -" for a snecified number of molecules ~
~
.~
~~
~~~
~
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~
~
remains fixed lead to the following relation for the fraction of molecules which have the total energy between E and E + a
This is indeed a remarkable statement in that the condition for statistical equilihrium for any system depends only on: (1) the universal Boltzmann constant k (= 1.380 X 10-'6 erg molecules-' degree-'); (2) the magnitude of the temperature; and (3) the molecular density of states (g(E)).The sum 2& g(E) exp(-PE), for which the symbol Qt(t) is often used, is called the (total) partition function, which for any species of molecules depends only on the temperature. However, it is also dependent on whether the reactants are in the gas phase or consensed. because for a dense assemblv of . . molecule, even in a so-called "neutral solvent" intermolecular forces modify the intramolecular potential function to
some extent and consequently affect the g(E) distribution. Consider what happens to a system in statistical equilihrium at T, when by some means, such as a shock wave, the random relative translational energies of the molecules are suddenly increased by ~E,..,I (per mole), so thatthe increase in translational temperature is: 6Tt,.,d = 6Eea,1/1.5 R. Due t o collisions some of 6Etr.,d will be converted to increase the molecular rotational energy. Then 6Tt,.,l will decline until 6'1',,,,,, = 61', = 6EI3R. On a somewhat longer time scale (after many more collisions) some of the injected energy will induce greater amplitudes in molecular vibrations. The temperature will further decline until 6Grm,, = 6qot = 6 c i b = 6EIC,(t,,). The initially perturbed system will have then relaxed to a new statistical equilihrium characterized by a temperature ( T 6T"). The equilibration process takes a finite time (desienated as the statistical relaxation time). Were the enwgylnjected by a short pulse of infrared radiation which is absorbed by molecular vibrations so that Tvibis suddenly increased, the equilibration process would follow a route which is ~reciselvthe inverse of the one described above. When thk temperature is raised from T to T + 6T, there is a change in the population of molecules within the is less than interval E and E A@the ratio NT+~T(E)/N?(E) unitv for values of E less than the average (E), and greater thanunity ior b: > 12; that is, on raising the overall temperature of a system thr populations of the lower states decrease and the populations uf the higher energy state* increase. If wecould observe the energy content ofany one molecule we would find that i t waxes and wanes. A few molecules become energy rich by means of such a random walk (up the enerzv scale) hut thev retain this affluence for a short time only. During that time the unlucky ones may interconvert t o another molecular confieuration. However, i t is important not to forget that these f& which react do so only heEause of the enormous number of molecules of lower energy content which by their collisions boosted them to this state. The nonreactine molecules also play an important role by deexciting th;! nascent (highl$ energized) product species. I quote here an interesting comment by J. Ford (Georgia Institute of Technology). "Over the years, chaos has received a notoriously bad press . . . yet without chaos, there would be no complex systems, no evolution, no life, and no universe as we know it." (4al e an n-atom molecule there are two wells, . S u o ~ o s for designated A and R, of comparable bur not necessarily of poud d e ~ t hfor . two distinct reometries (isomersl. Let these wells he $0 deep that a t the l&el of the potential maximum (El) . ,... of the surface between them. N ( E d is extremely small at the operational temperaturi (for example: cis- and tronsdichloroethvlene at nmm temperature). There are then (3n - 6) nhnm's for each isomer. While the total number of states are the same for both, their moments of inertia and vibrational frequencies differ, so that for any particular E , a,(E) # et(E),and hence NJE) z Nt(E), except by chance. These molecular species remain distinct on mixing (no reaction), hut they do attain the same temperature, that is, they attain the same thermal eauilihrium. (4b) What happens wheh the surface separating the wells is somewhat lowered (or the temverature is raised) so that the magnitudes of NA(E,) and N~(E,) are appreciable, and mutual transfers from A to B can occur, hut not rapidly? How does one count nhnm's and states? Again, the total number of nearly harmonic normal modes does not change, nor does the total number of states; however, the distributions of molecules amona the states does depend on the well depths and these contra the chemical behavior of the system. First, consider only the attainment of chemical equilihrium between A and B. Suppose one had equal volumes of pure A and pure B, a t arbitrary pressures, both in thermal equilibrium a t the same temperature T'. On adiabatic mixing they will react and drift toward chemical and thermal
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equilihrium, possibly a t a somewhat different temperature (T) due to the heat of reaction. The condition for chemical equilibrium is
where AH; is the enthalpy difference between A and B a t 0°K. However, on mixing arbitrary quantities the ratio (@/ Nt)initjalwill equal x K,,(T), where generally K Z 1. Reaction occurs, a t a rate determined by T. An important consequence of the Maxwell-Boltzmann distribution is that at all times on the way to chemical equilihrium the populations of all states which have the same energy E, both for A and B, drift together.
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Then, and only then, is it possible to define a conventional chemical rate constant which depends only on the temperature. However, when all parts of the system are not in statistical equilibrium, a t a single defined temperature, several relaxation processes occur. These can be individually characterized, hut the relaxation times depend on how the system was set up, that is, on the initial system composition as well as on the temperatures of the components. (4c) Now consider (aualitativelv) the svstem dvnamics whenthe two wells areseparated by a rela&ely lo6 barrier (Fia. lh); then the eauilibrium between A and B is rapidly attained. T o perturbthis system one must use a very iapid impulse, and an appropriate fast-response detector is required to measure how quickly equilibrium is re-established. We wish to consider how best to describe this process and to calculate a chemical relaxation time. But first, since there are two chemically recognizable species there are still two sets of (3n - 6) nhnm's and the total number of states is the same as when there is a very high barrier between A and B. However the locations of the A and B states cannot be approximated by a classical model. Quantum mechanically there is a sienificant difference between two wells which are strictly ideitical, as in the two forms of ammonia, and wells which differ even slightly, typical of the structures and relative energies of the syn and anti forms of methyl nitrite. We shall assume (for simplicity) that the quantum mechanical prohlem has been solved and we know the location of these vibrational states. T o formulate the kinetics problem an important abstraction is generally introduced. A "reaction coordinate" is defined to measure the degree of structural change in the conversion from A t o B; i.e., one assumes there f 1 which is a geometric coordinate, ranging from 0 identifies a continuous seauence of structural intermediates. This is an abstraction if i t is represented as being one dimensional. For example, in the acetylacetones, consider the correlated changes in interatomic hond lengths within the ring which must take place to accommodate the motion of the hydrogen atom. Focusing on the movement of the hydrogen atom in a double-well potential as though i t were independent of the rest of the molecule is a gros~oversimplification. Indeed, movement of the more massive, and hence more sluggish carbon and oxygen atoms prohably control the motion uf the hydrogen. Anuther example is the douhle-well notential for the anele of rotation abuut the 0 - N O hond in the akyl nitrites. ~ g entire e molecule must adjust to each confieuration. as demonstrated hv the fact that the fundamental vibrational frequencies differ in the syn conformer from those of the anti. A one-dimensional well potential is a proper representation of the conformation change provided i t is understood that the structure of the entire molecule determines that potential. In contemplating Figure 2 and the specific states that represent the sequence of allowed excitations of the dynamic
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Volume 63 Number 5 May 1986
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content can be specified hy stating that i t is in any one of its g(E) states. During its random walk among the accessible molecular states it may accumulate a total energy equal to or greater than the peak value of the potential surface which separates A from B (Em,Fig. 2). That does not necessarily lead to conversion because for that particular combination of dynamic parameters the energy content in the directly participating mode is insufficient; that is, it is in an (i + a )or in Ci 8) state rather than a y state. One of three courses may follow:
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Figure 2 A slce through a potenf~alenergy surface. rypncal ot the syn-ant, (somersof memyl n trite a ong the dnectly part!c#mnngmode (rotationaoam the 0-N bonal The quantized stator w tnm the wells are represented by la) and (0);those for the t e e rotation above the barrier maximum by (7).The slantlng planes schematically represent the (387 7 ) nearly harmonic supporting mwlss (state iand j, far A and 0 , respectively).
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variable that most directly is involved in the conversion, one must not forget that there are (3n - 7) other dynamic variables that form the basis for characterizing the internal energy content of each of the reactine s~ecies.All must be counted when considering the excited molecules which incorporate sufficient internal energy to make transitions between A and B. T o distinguish them we shall refer to (3n - 7) supporting modes and to one directly participating mode, which in general are linear combinations of nhnm's.
(1) During the next collision this molecule loses energy and remains A (the most likely event). (2) Durine the next collision it either eains sufficient enerw in thedirectly parti~rparin~ modeand is t h s knocked into a 7 &e, or the energy that it already has is redistributed (hy the rollisionj and is thus ronverted toa 2 state. (3) In the interval between rollisionsa rollisiun.free (intromolerular) redistribution of energy may cske place to bring rt into a 3 state. This can occur because the vibrational motions within the molecule are not strictly orthogonal, there are anharmonic coupling terms which allow the transfer of energy between different vibrational modes.
In either case (2) or (3) the y condition is not strictly A or strictly B; i t is geometrically and energetically (with respect to its (3n - 6) fundamentalvihrations) aviable intermediate species, which we call T. I t mav have a lone or a short lifetime; it can either become de-excited duryng the next collision, or, by intramolecular energy redistribution, lose energy from the directly participating mode to the supporting modes. In either case i t reverts to A or becomes B. What determines which way T will transform? For any specific molecule the choice is random; for an assembly of r's the probabilities for transition to A or B need not he equal. Since we are dealing with an enormous number of molecules all that matters for our measurements is the relative probability. The simplest assumption which generally (but not always) applies is
Into the Fored The preceding is prolog to the two questions for which we seek answers during this random stroll:
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(1) What are the detailed dynamics of the interconversions A
B?
Given the shape of the molecular potential surface, what are the rates for intereonversion? (2)
The latter question should be more carefully stated. Whereas there are several theories for calculatine rate constants. these must be compared with experimentil values, but nd one ever measures rate constants. Chemists measure timedependent concentrations, C;(t)'s, under specified conditions, or they record system relaxation times. Molecular beams kineticists measure energy, angle, and time-dependent fluxes of selected species. From the time-dependent changes "rate constants"-are derioed; from scattered beam fluxes cross sections for molecular events are evaluated. I t is necessary not only to measure Ci(t)'s but also to specify the applicable rate laws by comparing integrated expressions for Cj(t) with those observed. Rate constants deduced without demonstration that the postulated rate laws do apply are of questionable significance; indeed, they may be grossly in error. The operational sequence is: to measure system relaxation times or to develop empirical auantitative ex~ressions for Ci(t)'s, then to postdate H set of k~ementarysteps and t o extract rate constants by cornparinn the predicted with the experimental time-depe~den
