Collision frequency

Collision frequency is defined in chemical kinetics and collision theory, in the background of theoretical kinetics. If a system contains n_a type-a molecules per unit volume and n_b type-b molecules in that same unit volume, then the number of a-b collisions per unit time in that volume will be proportional to the product n_an_b.

\nu =z\,n_{a}n_{b}

(a-b collisions per unit time per unit volume)

or, in terms of molar volumes ([x]=n_x /N_A where N_A is the Avogadro constant)

\nu =z\,N_{A}^{2}[a][b]

By kinetic theory it can be shown that:

z=\sigma _{ab}{\sqrt {\frac {8k_{B}T}{\pi \mu _{ab}}}}

where

σ_ab is the collision cross-section (units of area),
k_B is Boltzmann constant,
T is absolute temperature,
μ_ab is the reduced mass of the colliding molecules (μ_ab=m_a m_b/(m_a+m_b))
z has the dimensions of length^3/time

The collision frequency (Z) is usually expressed in units of "moles of collisions" per unit time per unit volume:

Z=zN_{A}=N_{A}\sigma _{ab}{\sqrt {\frac {8k_{B}T}{\pi \mu _{ab}}}}

or equivalently:

Z=\sigma _{ab}{\sqrt {\frac {8RT}{\pi \mu _{ab}}}}

where R is the universal gas constant and μ_ab is measured in atomic mass units.

If each collision resulted in the a and b molecules combining to form a type c molecule (a+b->c), then Z[a][b] would equal the rate of production of c (in moles of type-c particles created per unit time per unit volume). But that's not true; see the article Chemical kinetics which states that the proportion of reactant molecules increases much faster as a function of temperaturethan the number of interactions.

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