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][better source needed]} Z = N A N B σ AB 8 k B T π μ AB , {\displaystyle Z=N_{\text{A}}N_{\text{B}}\sigma _{\text{AB}}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{\text{AB}}}}},} where

N A {\displaystyle N_{\text{A}}} is the number of A particles in the volume,

N B {\displaystyle N_{\text{B}}} is the number of B particles in the volume,

σ AB {\displaystyle \sigma _{\text{AB}}} is the collision cross section, the "effective area" seen by two colliding molecules (for hard spheres, σ AB = π ( r A + r B ) 2 {\displaystyle \sigma _{\text{AB}}=\pi (r_{\text{A}}+r_{\text{B}})^{2}} , where r A {\displaystyle r_{\text{A}}} is the radius of A, and r B {\displaystyle r_{\text{B}}} is the radius of B),

k B {\displaystyle k_{\text{B}}} is the Boltzmann constant,

T {\displaystyle T} is the thermodynamic temperature,

μ AB = m A m B m A + m B {\displaystyle \mu _{\text{AB}}={\frac {m_{\text{A}}m_{\text{B}}}{m_{\text{A}}+m_{\text{B}}}}} is the reduced mass of A and B particles.

In the case of equal-size particles at a concentration n {\displaystyle n} in a solution of viscosity η {\displaystyle \eta } , an expression for collision frequency Z = V ν {\displaystyle Z=V\nu } , where V {\displaystyle V} is the volume in question, and ν {\displaystyle \nu } is the number of collisions per second, can be written as^[2] ν = 8 k B T 3 η n , {\displaystyle \nu ={\frac {8k_{\text{B}}T}{3\eta }}n,} where

k B {\displaystyle k_{B}} is the Boltzmann constant,

T {\displaystyle T} is the absolute temperature,

η {\displaystyle \eta } is the viscosity of the solution,

n {\displaystyle n} is the number density.

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 ν {\displaystyle \nu } .^[2]