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تسجيل

State-to-state dynamics

المؤلف:  Peter Atkins، Julio de Paula

المصدر:  ATKINS PHYSICAL CHEMISTRY

الجزء والصفحة:  ص886-887

2025-12-29

439

State-to-state dynamics

The concept of collision cross-section was introduced in connection with collision theory in Section 24.1, where we saw that the second-order rate constant, k₂, can be expressed as a Boltzmann-weighted average of the reactive collision cross-section and the relative speed of approach. We shall write eqn 24.12 as

k₂ = σv_rel NA

where the angle brackets denote a Boltzmann average. Molecular beam studies provide a more sophisticated version of this quantity, for they provide the state-to-state cross-section, σnn′, and hence the state-to-state rate constant, knn′:

k_nn′ = σ_nn v_rel NA

The rate constant k₂ is the sum of the state-to-state rate constant over all final states (because a reaction is successful whatever the final state of the products) and over a Boltzmann-weighted sum of initial states (because the reactants are initially present with a characteristic distribution of populations at a temperature T):

k2 = ∑_{n, n′} k_nn′(T)f_n(T)

where f_n(T) is the Boltzmann factor at a temperature T. It follows that, if we can determine or calculate the state-to-state cross-sections for a wide range of approach speeds and initial and final states, then we have a route to the calculation of the rate constant for the reaction.