Edward A. Desloge
Department of Physics Florida State University Tallahassee, 32306
The Statistical Mechanical Theory of Bimolecular Gas Reactions
In practically every textbook on reaction kinetics, the statistical mechanical treatment of bimolecular reaction rates in gases follows the pioneer activated complex theory of Eyring,' or the equivalent transition state theory of Evans and P ~ l a n y i . ~There is, however, an alternate approach leading to essentially the same results which receives litt,le attention by textbook writers and which is frequently completely ignored. This approach is either implicitly contained in much of the journal literature on bimolecular reactions, or else is an explicit part of attempts a t more comprehensive theories. The purpose of this paper is to present the latter approach in as simple a fashion as possible, in hopes of bringing more teachers and students to a realization of its conceptual and mathematical elegance. The spirit of the presentation follows closely the work of Kecka The details, however, are appreciably different.
Let us assume for fixed T that the probability per unit time that a molecule M I will undergo a reaction is proportional to the probability that at any instant a molecule f i t 2 is within a distance d of it. To determine this probability we consider first a box containing one molecule MI and one molecule P I 2 . If we locate ourselves on the molecule MI then the probability that the molecule Mz falls within a distance d of the molecule sf, is just A V / V where AV 4ada/3. If there were one molecule M I and N2 molecules R'I2 in the box then the probability that at any instant a molecule M2 is within a distance d of the molecule MI is N 2 ( A V / V ) . Finally if there are N 1 molecules M I and N z molecules AiIz then the prohahility that a molecule M I and a molecule At, are within a distance d of one another is N1N2( A V l V ) . It follows for fixed T that R is proportional to N I N ~ / Vand thus r is proportional to N1Nz/V2. We can express this mathematically as follows
--
The-Basic Problem
Let us suppose: (a) we have a rigid container of volume V ; (b) the container encloses a gas consisting initially of N1 molecules of species M I , N 2 molecules of species I t 2 , N I f molecules of species M:, and N2' molecules of species %'; (c) the container is immersed in a heat bath of constant temperature T; and ( d ) the following rcaction is taking place in the gas M,
+ M2
-
MI'
+ M*'
(1)
We futher assumc: (a) the reaction is an elementary homogeneous reaction; and (b) the reaction is taking place sufficiently slowly and the gas is sufficiently dilute that each component in the gas can he treated a t each instaut as an ideal gas in equilibrium at temperature T. Our aim in this paper will be to investigate from a molecular point of view the rate of such a reaction. The Rate Coefficient
We define the reaction rate R for the reaction (1) as the average number of reaction events of type ( 1 ) which take placc in the volume V in unit time, and the specific reaction rate r as the reaction rate per unit volume, that is R / V . I n t,his section we wish to determine the dependence of the specific reaction rate 1. on V , N I , and N?.
I
EYICING, H., J . Chem. I'hlls., 3, 107 (1935). Ev.\Ns, M. G . , .AN,> POI..\NYI, M., Trans. Farnday Soc., 31,
X7R (I!):%). *KI:OK,
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J. C., Arlvanws in Chemical Physics, Journal of Chemical Education
13, 85 (1!167).
where k ( T ) is some function of T. The function k( T ) is called the rate constant or rate coefficient. From eqn. (2) it follows that if N1 = 1, N z = l , and V = 1 then r = k. Thus the rate coefficient k can he interpreted as the probability per unit time that a pair of molecules M I and Mz which is contained in a unit volume will undergo the reaction MI At? -+ MI' M"' -.-'
+
.
+
I n what follows we shall therefore assume that we have a rigid container of unit volume which contains one molecule MI and one molecule M? in thermal equilibrium a t a temperature T . The rate coefficient k will then be simply the probability per unit time that the molecules will undergo the reaction MI Mt -+ MI' M2'.
+
+
The System
The atoms which make up the molecules which are participating in a reaction can be combined in a variety of ways to form different sets of molecules. I n order to simplify the discussion of chemical reactiom we shall refer to a particular combination of the atoms into molecules as a chemical state of the atoms. I n particAtz ular we shall refer to the pair of molecules M I as the chemical state M of the atoms which are participating in the reaction (1) and the pair of molecules MI' M?' as the chemical state M' of the same set of atoms. The chemical reaction ( 1 ) can thus he ahhreviated
+
+
M-M'
(3)
According to expression ( 3 ) when a reaction occurs
the group of atoms pass from the chemical state M to the chemical state M'. From a dynamical point of view the system of interest is the atoms which are participating in the reaction. We shall assume: there are s atoms; each of the s atoms in the system can be treated as a point particle; the interaction between the atoms can be described by a potential which is a unique function of the inter-particle distances; and the particles obey the classical laws of motion. It then follows that the configuration of the system can be represented by a , . . . , q3*, in a 3s dimensional consingle point q = q ~ qz, figuration space and the dynamical state of the system by a single point, q, p = a, q2,. . ., qsa, PI, pz, . . ., p3#, in a 6s dimensional phase space. We next assume that there is a region R in configuration space corresponding to the system being in chemical state 14, a region R' corresponding to the system being in chemical state PI', and a surface S which clearly separates the region R from the region R'. If the system undergoes the reaction h'l + M' then the point representing the system in configuration space passes from R to R' through the surface S. Finally we shall assume that there exists a function Q(q) such that if Q < 0 the system is in the region R, if Q = 0 the system is on the surface S, and if Q > 0 the system is in the region R'. Deferminotion of the Rate Coefficient
dPdp+; and the flux through the surface element dq+ on the surface S is [QN+]n=odPdp+dq+. The rate coefficient is equal to the probability per unit time that an arbitrary system in the ensemble will cross the surface S. To obtain this probability we integrate [~N+],-~dPdp+dq+ over all values of q+, P, and p+ for which the flux on the surface S is in the positive Q direction, and then we divide by N. If we do this, and assume that the variables q+ and p+ can be chosen and are chosen in such a way that if P > 0 then the flux across the surface S is in the positive Q direction, we obtain Equation (5) can be expressed in a little prettier mathematical form if we introduce the Dirac delta function, and the unit step function. The Dirac delta function, 6(x), is the distribution function corresponding to a single point located a t the origin. It has the properties that S(x) = 0 whenx f 0,and for an arbitrary function g(x) that
The unit step function, ?(x), is defined as follows: ?(x) = 0 for x < 0, and ?(x) = 1 for x I 0. Malcing use of the Dirac delta function and the unit step function in eqn. (5), we obtain
If the system point is initially in the region R, then the rate coefficient k will be equal to the probability per unit time that the system point will cross the surWe will work with k in the form given by eqn. (6). face S. I n the present section, we will obtain an exIt should be pointed out, however, that the results pression for this number. which we shall obtain do not depend on the introduction We first replace the generalized coordinates q ~ , of the functions 6(Q) and ?(P). The results can be q,, . . . , q,, by a new set of generalized coordinates, one of derived directly from eqn. (5). The functions 6(Q) which is the variable Q(q). We shall designate this and ?(P) are introduced merely to avoid awkward new set by the symbols Q, q ~ + qz+,. , . ., q+3,-1. The set mathematical expressions. of coordinates q+= ql+, qz+, . . . , q,+3,-1 is simply any set We now note from Hamilton's equation of motion, of coordinates which together with the coordinate Q and from the definition of 4, given in eqn. (4),that uniquely specify the configuration of the system. The &+ = (3H/aP)+ = -kT(b+/aP) (7) corresponding generalized momenta will he designated P,,p1+,pz+, . . . , p+3s-1. Substituting eqn. (7) into eqn. (6) we obtain We next consider an ensemble consisting of a large k= -kT f f f f (a6/aP)?(P)6(Q)dPdQdp+dqC(8) number N of macroscopic replicas of the system we are studying, that is each replica consists of a single pair Integrating by parts over the variable P, and making of molecules MI and Mz in a box of unit volume a t a use of the fact that dq(x)/dx = 6(x) we obtain temperature T. The number of systems in the endP, Q between Q semble with P between P and P and Q dQ, p+ between p+ and p+ dp+, and q + Substituting eqn. (4) into eqn. (9) we obtain between q + and q+ dq+ is given, fromstatistical mechanics, by N+dPdQdp+dq+ where
+
+
+
+
and H = H(P,Q,p+,q+) is the Hamiltonian of the system of s atoms. It should be noted that the integration over Q and q+ is restricted to the region R. Let us consider first only those systems in the ensemble for which P lies between P and P d P and p+ lies between p+ and p+ dp+. For such systems: the density in configuration space is N+dPdp+; the Q component of the flux density in configuration space is QN+dPdp+ where Q is the time rate of change of coordinate Q; the Q component of the flux density at the surface S in configuration space is [QN+],-~-
+
Carrying out the integration over P and Q in the numerator, and noting in the denominator that dPdQdp+dq+ = dpdq we obtain
+
where
The Hamiltonian H is just the Hamiltonian of the Volume 47, Number 5, M o y 1970
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systen~in the chemical state hI. The Hamiltonian H + is the Hamiltonian which the system would have if it were constrained in configuration space to remaiu on the surface S. We shall refer to the system in this situation as being in the state hI+. Equation (11) is the desired expression for the rate coefficient k. It is not, however, in its most familiar form. If we define
collision theory. The reason for the agreement is obvious. We are calculating exactly the same quantity in both cases, the only difference being that in the present approach, we are using the techniques of statistical mechanics, whereas, in the collision theory approach, the techniques of kinetic theory are used. This result also agrees with the result one obtairu using transition state theory. Rate of Effusion
where Ho is the minimum value of H; Ho+ is the minimum value of H+; and h is Planck's constant, then eqn. (11) can be rewritten kT Z+ k = e x p h Z where & = HaC - Ho
(17)
The quantity Z is the partition function of the system in the chemical state id. The quantity Z + is the partition function of the system in the chemical state M+. The appearance of H - Ho and H + - Ho+ rather than simply H and H + in the exponentials is due to the fact that in evaluating the Hamiltonian energies in the partition functions it is customary to measure the energy of a particular chemical state with respect to its minimum energy rather than from some common reference energy. The energy Eo is just the energy required to take the system from the lowest energy in chemical state 14 to the lowest energy in chemical state I f + . The Rate of Formation of a Diatomic Molecule
As a simple application of the preceding results, let us consider the reaction A B + AB, where A and B are atoms of masses m* and ma, respectively. We shall assume that the surface separating the chemical state A B from the chemical state AB is simply the surface defined by the equation p = po, where p is the distance between atoms A and B, and po is a constant usually chosen to be the sum of the radii of atoms A and B. It then follows that Z in eqn. (16) is the partition function for two free particles of masses mA and r n ~confined in a box of unit volume, and Z+ is the partition function for a rigid diatomic molecule consisting of two atoms of masses mAand m n separated by a distance po and confined to a box of unit volume. These partition functions can be shown to be
+
+
Substituting these values of Z and Z + in eqn. (16) we obtain k = ooZ [ 8 r k T / [ m ~ m e / ( m ~ me)ll'/~exp(-EolkTj
+
This is the same result which one obtains using simple 380
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Although eqn. (16) was derived for the special case of a chemical reaction, it is possible to apply it to a wide variety of rate problems. As a simple illustration, let us consider the rate of effusion of a gas through a small opening in a container. Suppose we have n identical molecules of mass m in a container of unit volume, then the probability that a particular one of the molecules will hit a unit area on the wall of the container can be obtained from eqn. (16) if we interpret Z as the partition function of a mass m in a box of unit volume, Z + as the partition function of a mass m constained to move on a surface of unit area, and Eo as the energy required to move the mass from rest in the box to rest on the surface. With this interpretation Z = (2mkT/h')"/* Z+ = (2wmkT/ha)
and E.
= 0
Substituting these values in eqn. (16) we obtain k = (kT/2m)'h = G14 where e is the average speed of the mass m. I t follows that the rate of effusionthrough a hole of area A in the container will he nAa/4. This agrees with the result obtained using more conventional kinetic theory arguments. The Equilibrium Constant
When the gas described a t the beginning of this paper is in equilibrium the rate of the reaction 14 + h4' will be equal to the rate of the reaction PI'+ 14. It follows from eqns. (2), (16), and (17) that
where Z' is the partition function of the atoms in the chemical state M', and Ha' is the minimum value of the Hamiltonian energy H i of the atoms in the chemical state M'. Equation (18) can be rewritten
The quantity K , is called the equilibrium constant for the reaction M + M'. The quantity H,' - H. is the energy required to take the system from its lowest energy state in chemical state M to its lowest energy state in chemical state $4'. The relationship (19) which we have derived is the usual starting point in transition state theory, from which one ultimately derives eqn. (16).
Conclusion
The preceding derivation of the rate coefficient for a reaction in a gas avoids a number of the sticky points involved in the usual textbook derivation. I n the usual derivation, one assumes that the system moves from the chemical state M through an intermediate chemical state h l f , called the transition state, to the chemical state M'. The state Mi corresponds to a volume in configuration space, and is thus not the same as the state M+ which corresponds to a surface. The resulting calculations then involve the necessity of singling out a particular degree of freedom in the transition state corresponding to motion along the reaction path. The motion associated with this degree of freedom is then assumed to be describable as the motion of a free particle of unknown mass m, or as the motion of a simple harmonic oscillator of unknown frequency V . The mass m and frequency v convenientIy cancel out in the ensuing calculations, and the final expression for the rate coefficient is formally the same as eqn. (16). However, the interpretation of Z C is not quite the same. I n the approach taken in this paper Z+ is the partition function which the system would have if it were constrained in configuration space to remain on the surface Q(q) = 0. I n the transition state approach, Z+ is the partition function of the system in the transition state allowing for all degrees of freedom except that corresponding to the reaction coordinate. Since the degree of freedom corresponding to the reaction coordinate is somewhat vague in nature, the corresponding interpretation of Z+ is also a little vague. I n practical calculations, however, both interpretations usually converge, as was seen when we considered the reaction A C B + AB.
Another aspect of the present derivation is that i t clearly brings out the relation between the reaction rater for a gas reaction and other rate processes. This relation can be useful not only because it provides us with a means for calculating the rates of a variety of other processes, but also because it provides us insight into the behavior of chemical reactions. For example, the expression we obtained for the rate of effusion of a gas is invalid if the hole through which the gas is effusing is too large, since the resulting rapid flow of gas out the hole will destroy the equilibrium of the gas. An analogous situation occurs in the case of chemical reactions; if the reaction rate is too large, equilibrium will he destroyed, and eqn. (1G) will be invalid. It is also interesting to compare the starting points of the two derivations. The transition state theory starts with the assumption that the student knows the expression in eqn. (19) for the equilibrium constant of a gas reaction in terms of the partition functions of the reactants and the products. The usual derivation of this result is not trivial. I n fact in many elementary textbooks on reaction kinetics the student is simply required to accept this as given. I n the present approach this result is not needed, and can even he derived as a consequence of the theory. However, the student using the present approach is expected to be familiar with the distribution in eqn. (4), and with Hamilton's equations of motion. Which approach is more appealing to a student will undoubtedly depend on his background. I n conclusion, the author hopes that the present paper will help others avoid some of the difficulties he encountered when he recently became interested in the subject of reaction kinetics.
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