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:


 * $$ Z = N_\text{A} N_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}},$$

which has units of [volume][time]−1.

Here,
 * $$N_\text{A}$$ 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, the "effective area" seen by two colliding molecules, 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,
 * $$T$$ is the temperature,
 * $$\mu_\text{AB}$$ is the reduced mass of the reactants A and B, $$ \mu_\text{AB} = \frac{{m_\text{A}} + {m_\text{B}}} $$

Collision in diluted solution
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:


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

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 $$\nu$$.