Collision frequency

Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of species A and species B is:[1]

Z=N_{\text{A}}N_{\text{B}}\sigma _{\text{AB}}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{\text{AB}}}}},

SI unit of Z is the volumetric collision rate (unit m³⋅s⁻¹).

where:

$N_{\text{A}}$ is the number of A molecules in the gas,
$N_{\text{B}}$ is the number of B molecules in the gas,
$\sigma _{\text{AB}}$ is the collision cross section (unit m²), the area when two molecules collide with each other, simplified to $\sigma _{\text{AB}}=\pi (r_{\text{A}}+r_{\text{B}})^{2}$ , where $r_{\text{A}}$ the radius of A and $r_{\text{B}}$ the radius of B.
$k_{\text{B}}$ is the Boltzmann constant (unit m²⋅kg⋅s⁻²⋅K⁻¹),
$T$ is the temperature (unit K),
$\mu _{\text{AB}}$ is the reduced mass of the reactants A and B, $\mu _{\text{AB}}={\frac {{m_{\text{A}}}{m_{\text{B}}}}{{m_{\text{A}}}+{m_{\text{B}}}}}$ (unit kg)

Collision in diluted solution

Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated from Fick's laws of diffusion.

In the case of equal-size particles at a concentration $n$ in a solution of viscosity $\eta$ , an expression for collision frequency $Z=V\nu$ where $V$ is the volume in question, and $\nu$ is the number of collisions per second, can be written as:[2]

\nu ={\frac {8k_{\text{B}}T}{3\eta }}n,

Where:

$k_{B}$ is the Boltzmann constant
$T$ is the absolute temperature (unit K)
$\eta$ is the viscosity of the solution (pascal seconds)
$n$ is the concentration of particles per cm³

Here the frequency is independent of particle size, a result noted as counter-intuitive. For particles of different size, more elaborate expressions can be derived for estimating $\nu$ .[2]

References

chem.libretexts.org: Collision Frequency
Debye, P. (1942). "Reaction Rates in Ionic Solutions". Transactions of the Electrochemical Society. 82 (1): 265. doi:10.1149/1.3071413. ISSN 0096-4743.

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[1] .libretexts.org: Collision Frequency

[:0-2] Debye, P. (1942). "Reaction Rates in Ionic Solutions". Transactions of the Electrochemical Society. 82 (1): 265. doi:10.1149/1.3071413. ISSN 0096-4743.