THEORIES OF UNIMOLECULAR REACTION RATES
1. LINDEMANN / LINDEMANN-HINSHELWOOD THEORY
This is the simplest theory of unimolecular reaction rates, and was the first to successfully explain the observed first-order kinetics of many unimolecular reactions. The proposed mechanism actually consists of a second-order bimolecular collisional activation step, followed by a rate-determining unimolecular step. k1 A + M Ë A* + M k-1 k2 A* → P Applying the steady-state approximation to the concentration of A* gives [A*] = so that the overall rate is k1 [A][M] k-1 [M] + k2
d[P] k1 k2[A][M] = k2[A*] = dT k-1 [M] + k2
This is often written as d[P] = keff[A] dT k1 k2[M] is an effective first-order rate constant. keff is, of course, a function of pressure. At k-1 [M] + k2 high pressures, collisional deactivation of A* is more likely than unimolecular reaction, keff reduces to k1 k2/k-1 and the reaction is truly first order in A. At low pressures, bimolecular excitation is the rate determining step; once formed A* is more likely to react than be collisionally deactivated. The rate constant reduces to keff=k1 [M] and the reaction is second order. where keff = Lindemann theory breaks down for two main reasons: i) The bimolecular step takes no account of the energy dependence of activation; the internal degrees of freedom of the molecule are completely neglected, and the theory consequently underestimates the rate of activation. ii) The unimolecular step fails to take into account that a unimolecular reaction specifically involves one particular form of molecular motion (e.g. rotation around a double bond for cis-trans isomerization). Subsequent theories of unimolecular reactions have attempted to address these problems. offers a solution to problem i). Hinshelwood theory
2. HINSHELWOOD THEORY
Hinshelwood modelled the internal modes of A by a hypothetical molecule having s equivalent simple harmonic oscillators of frequency ν and using statistical methods to determine the probabality of the molecule being collisionally activated to a reactive state.
The number of ways of distributing a given number of quanta, v, among the s oscillators (i.e. the number of degenerate states of the system at an energy (v+½)hν) is (v+s-1)! gv = v!(s-1)! (a handwavy explanation of where this comes from is that (v+s-1)! is the number of permutations of all the quanta and all the harmonic oscillators. This has to be divided by the number of ways in which the quanta can be permuted amongst themselves, v!, and the number of ways the oscillators can be permuted amongst themselves, (s-1)!) The fraction of molecules in state v is given by the Boltzmann distribution nv gve-vhν/kT = N q 1 q = -hν/kT 1-e 3
Hinshelwood now made the strong collision assumption. He assumed that the probability of deactivation of A* in any given collision is unity, so that the rate constant k of the Lindemann mechanism is equal to the collision -1 frequency Z. Because the collisions promote equilibrium, the probability of forming a state v in a collision is given by the Boltzmann distribution. The rate constant for activation to state v is therefore given by k1 v = Z gv e-vhν/kT q
The overall rate of activation (i.e. rate of formation of collisionally excited A* with enough energy to react) is found by summing the k1 v over all the energy levels which can dissociate i.e. all levels with an energy greater than the critical energy E0 which the molecule needs to react. If the vibrational quantum number of the state with energy E0 is m, we have ∞ gv e-vhν/kT k1 = Σ Z q m The energies involved are usually large, with E0 >> hν. Hinshelwood developed equations for the case in which the energy levels can be assumed to be continuous (kT >> hν). The expression then becomes dk1 = Z N(E) e-E/kT dE q
where N(E) is the density of states; N(E)dE is therefore the number of energy levels with energy between E and E+dE, and dk 1 is the rate constant for activation into this...
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