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

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.

Collision in diluted solution

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

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 }

.