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The role of diffusion

The role of the transition state in determining rate of reaction

The Arrhenius equation is based on the empirical observation that rates of reactions increase with temperature. This is accounted for by an increase in rate consant with temperature, which led to the concept of an activation barrier in the reaction pathway. The height of barrier is given by the activation energy, E_act.

where k_¥ is the rate constant when Eact is zero, often called the pre-exponential factor. In collisional ractions, k_¥ is the diffusion limited rate constant (see diffusion note above).

where r_m and r_n are the radii of reacting species (in cm), D_m and D_n are diffusion coefficients (units of cm².s^-1) , N_o is the Avogadro number, and the division by 1000 cm³ gives k in M^-1 s^-1.

Eyring, Randall-Wilkins theory:
Consider the reaction below, which proceeds through an activation barrier populated by the species MN^#:

                                    k₁             k₂
M + N <==> M--N <==> MN^# ----> P
                                   k_-1

The probability for forming MN^#, the intermediate at the top of the activation barrier, is determined by the equilibrium constant for formation (K^#), and hence the free-energy:

The rate of formation of products is given by:

The rate constant k₂ is considered to occur in the first vibrational transition, and decay of MN^# is assumed to go with equal probability forward or back, so that:

where k_B is the Boltzmann constant, h is the Planck constant, and k_BT/h has a value of ~10¹³ s^-1. Substituting:

Since we can also express the rate in the conventional form:

and since K^# = exp[-DG^act/RT]

We introduce a second factor, k, to account for differences between adiabatic and non-adiabatic processes (see Lecture 19), and convert to molecular units by substituting k_B for R, to get:

More generally, we can consider that the limiting step of a reaction may be the collisional process, in which case the overall rate will be determined by the rate of formation of M--N through the second order process above. Then:

where B is either collision frequency (the diffusion limited rate constant, also called A above, 10¹¹ M^-1.s^-1) if liquid phase collisions are involved, or B is vibrational frequency (k_BT / h (~10¹³ s^-1)) if the reaction is intramolecular or monomolecular (first order), or some mixed term if a collisional process proceeds via an intramolecular complex.

If we equate E^act of the Arrhenius equation with DH^act, then: