Equilibrium collision theory

Flere, we shall concentrate on basic approaches which lie at the foundations of the most widely used models. Simplified collision theories for bimolecular reactions are frequently used for the interpretation of experimental gas-phase kinetic data. The general transition state theory of elementary reactions fomis the starting point of many more elaborate versions of quasi-equilibrium theories of chemical reaction kinetics [27, M, 37 and 38]. 

The collision theory considers the rate to be governed by the number of energetic collisions between the reactants. The transition state theory considers the reaction rate to be governed by the rate of the decomposition of intermediate. Tlie formation rate of tlie intermediate is assumed to be rapid because it is present in equilibrium concentrations. 

If the transition state theory is applied to the reaction of two hard spheres, the result is identical with that of simple collision theory. - pp Because transition state theory is an equilibrium theory, it can be inferred that collision theory is also an equilibrium theory. 

A more interesting possibility, one that has attracted much attention, is that the activation parameters may be temperature dependent. In Chapter 5 we saw that theoiy predicts that the preexponential factor contains the quantity T", where n = 5 according to collision theory, and n = 1 according to the transition state theory. In view of the uncertainty associated with estimation of the preexponential factor, it is not possible to distinguish between these theories on the basis of the observed temperature dependence, yet we have the possibility of a source of curvature. Nevertheless, the exponential term in the Arrhenius equation dominates the temperature behavior. From Eq. (6-4), we may examine this in terms either of or A//. By analogy with equilibrium thermodynamics, we write... 

Although collision theory provides a useful formalism to estimate an upper limit for the rate of reaction, it possesses the great disadvantage that it is not capable of describing the free energy changes of a reaction event, since it only deals with the individual molecules and does not take the ensemble into consideration. As such, the theory is essentially in conflict with thermodynamics. This becomes immediately apparent if we derive equilibrium constants on the basis of collision theory. Consider the equilibrium... 

Applying collision theory to the forward and the reverse reactions, and taking the ratio, we obtain the equilibrium constant ... 

Hence, we find a relation between K and the enthalpy of the reaction, instead of the free energy, and the expression for the equilibrium is in conflict with equilibrium thermodynamics, in particular with Eq. (32) of Chapter 2, since the prefactor can not be related to the change of entropy of the system. Hence, collision theory is not in accordance with thermodynamics. 

Why does the collision theory of reaction rates conflict with equilibrium... 

Absolute Rote Theory(also known as Transition State or Activated Complex Theory). A theory of reaction rates based on the postulate that molecules form, before undergoing reaction, an activated complex which is in equilibrium with the reactants. The rate of reaction is controlled by the concn of the complex present at any instant. In general, the complex is unstable and has a very brief existance(See also Collision Theory of Reaction)... 

The atomic processes that are occurring (under conditions of equilibrium or non equilibrium) may be described by statistical mechanics. Since we are assuming gaseous- or liquid-phase reactions, collision theory applies. In other words, the molecules must collide for a reaction to occur. Hence, the rate of a reaction is proportional to the number of collisions per second. This number, in turn, is proportional to the concentrations of the species combining. Normally, chemical equations, like the one given above, are stoichiometric statements. The coefficients in the equation give the number of moles of reactants and products. However, if (and only if) the chemical equation is also valid in terms of what the molecules are doing, the reaction is said to be an elementary reaction. In this case we can write the rate laws for the forward and reverse reactions as Vf = kf[A]"[B]6 and vr = kr[C]c, respectively, where kj and kr are rate constants and the exponents are equal to the coefficients in the balanced chemical equation. The net reaction rate, r, for an elementary reaction represented by Eq. 2.32 is thus... 

In contrast to the formally analogous van t Hoff equation [10] for the temperature dependence of equilibrium constants, the Arrhenius equation 1.3 is empirical and not exact The pre-exponential factor A is not entirely independent of temperature. Slight deviations from straight-line behavior must therefore be expected. In terms of collision theory, the exponential factor stems from Boltzmann s law and reflects the fact that a collision will only be successful if the energy of the molecules exceeds a critical value. In addition, however, the frequency of collisions, reflected by the pre-exponential factor A, increases in proportion to the square root of temperature (at least in gases). This relatively small contribution to the temperature dependence is not correctly accounted for in eqns 2.2 and 2.3. [For more detail, see general references at end of chapter.]... 

In Sec. 2.18, we looked at the matter of reaction rates from the standpoint of the collision theory. An alternative, more generally useful approach is the transition state (or thermodynamic) theory of reaction rates. An equilibrium is considered to exist between the reactants and the transition state, and this is handled in the same way as true equilibria of reversible reactions (Sec. 18.11). Energy of activation (E act) and probability factor are replaced by, respectively, heat enthalpy) of activation (A/f t) and entropy of activation (A5J), which together make wp free energy of activation (AGt). 

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann energy (or momentum or internal coordinate) distribution of reactants is postulated to persist during a reaction. In the collision theory, mainly due to Hinshelwood,7 the number of energetic, reaction producing collisions is calculated under the assumption that the molecular velocity distribution always remains Maxwellian. In the absolute... 

Theoretical chemistry is the discipline that uses quantum mechanics, classical mechanics, and statistical mechanics to explain the structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of (1) electronic structure, potential energy surfaces, and force fields (2) vibrational-rotational motion and (3) equilibrium properties of condensed-phase systems and macromolecules. Chemical dynamics includes (1) bimolecular kinetics and the collision theory of reactions and energy transfer (2) unimolecular rate theory and metastable states and (3) condensed-phase and macromolecular aspects of dynamics. 

Changes in concentration Adjusting the concentrations of either the reactants or the products puts a stress on the equilibrium. In Chapter 16, you read about collision theory, which states that particles must collide in order to react. The number of collisions between reacting particles depends on the concentration of the particles, so perhaps the chemist can change the equilibrium by changing concentrations. 

Now, let us find a little flaw in the theory equation (2-86) predicts only first-order behavior for the unimolecular reaction, something we know in fact is not true at low pressures. The reason for this failure in TST is the assumption of universal equilibrium between reactants and the transition state complex. At low pressures the collisional deactivation process becomes very slow, since collisions are infrequent, and the rate of decomposition becomes large compared to deactivation. In such an event, equilibrium cannot be established nearly every molecule which is activated will decompose to product. However, the magnitude of the rate of decomposition of the transition complex is much larger than the decomposition of the activated molecule in the collision theory scheme, so one must resist the temptation to equate the two. Since the transition state complex represents a configuration of the reacting molecule on the way from reactants to products, the activated molecule must be a precursor of the transition state complex. 

The interpretation of the shock tube results is that during the induction period the branching chain reaction predominates until significant amounts of H2 and O2 have reacted and back reactions have become important. After the induction period, the concentration of free radical propagators go through a maximum and then slowly approach equilibrium values until the higher-order termination steps limit chain propagation, and the back reactions become important. All of the individual rate constants in the mechanism could be evaluated and are consistent with simple collision theory. 

On the basis of the preceding assumptions, Hinshelwood s theory proceeds as follows. Let c be the concentration of reactant, W the equilibrium fraction of activated molecules, i.e., that at high pressures, where the rate of reaction is extremely small relative to the rate of deactivation, and c the concentration of activated molecules at any pressure. The formal expression for the number of deactivating collisions at high pressures, namely, accW, where a was found from elastic-hard-sphere collision theory to be equal to 2nd kT/ (= Z jc represents the rate of activation at all pressures. Since at the steady state at any pressure this rate of activation will be equal to the rate of deactivation plus the rate of reaction,... 

To be of value the treatment must explain the enormous enhancement of rate in the Finkelstein reaction above, and it is clear that no single bulk-solvent property is adequate. If the transition state is in chemical equilibrium with reactants or, in collision theory parlance, if the stability of the encounter complex is directly reflected in the rate expression, then the change in rate with medium must reflect the free energy difference between the reactants and the transition state or encounter complex. This free energy difference must itself reflect the solvation free energies of the reactants and the transition state or encounter complex, as long as these latter two can be considered to retain their identity upon solvent change. 

If the potential energy surface has a col, then a separation of the reaction coordinate is possible also in the vicinity of the saddle-point. Supposing that the system remains there sufficiently long time for a stationary state to be established, we may assume the existence of a quasi-equilibrium energy distribution in that transition state, too. Then, we can apply the above statistical treatment of reactants also to the transition state of the reacting system. However, we will not introduce at this stage the above restrictive assumptions, since our aim is to derive a collision theory rate expression of a possibly general validity. 

The collision theory expression (51 HI) represents the most general formulation of the rate constant based on the unique assumption that the reactants are in thermal equilibrium. It does not involve any hypothesis concerning the intermediate stages of reaction and applies to any reaction regardless of the shape of the potential energy surface. 

However, later work showed that rather large deviations from experiment are obtained for reactions in which the reacting molecules are more complicated. This collision theory is evidently too simple and unlikely to be generally reliable. One weakness is the assumption that molecules are hard spheres, which implies that any collision with sufficient energy will lead to reaction if the molecules are more complicated, this is not the case. A more fundamental objection to the treatment is that when applied to forward and reverse reactions it cannot lead to an expression for the equilibrium constant that involves the correct thermodynamic parameters. More recent work has involved a similar approach but has treated molecular collisions in a more realistic and detailed way. 

The interpretation of these reactions was a considerable triumph for conventional transition-state theory. Simple collision theory proved unsatisfactory for trimolecular reactions, owing to the difficulty of defining a collision between three molecules, and usually led to very serious overestimations (by several powers of ten) of the rate constants. Similar difficulties are encountered with dynamical treatments, and these have still not been satisfactorily resolved. Conventional transition-state theory, by regarding the activated complex as being in equilibrium with the reactants, leads to a very simple formulation of the rate constant and to values in good agreement with experiment. It also very neatly explains the rather marked negative temperature dependence of the pre-exponential factors for these reactions. 

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