Chemistry: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:

[math]\displaystyle{ Z = N_\text{A} N_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}, }[/math]

which has units of [volume][time]⁻¹.

Here,

• [math]\displaystyle{ N_\text{A} }[/math] is the number of A molecules in the gas,
• [math]\displaystyle{ N_\text{B} }[/math] is the number of B molecules in the gas,
• [math]\displaystyle{ \sigma_\text{AB} }[/math] is the collision cross section, the "effective area" seen by two colliding molecules, simplified to [math]\displaystyle{ \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 }[/math], where [math]\displaystyle{ r_\text{A} }[/math] the radius of A and [math]\displaystyle{ r_\text{B} }[/math] the radius of B.
• [math]\displaystyle{ k_\text{B} }[/math] is the Boltzmann constant,
• [math]\displaystyle{ T }[/math] is the temperature,
• [math]\displaystyle{ \mu_\text{AB} }[/math] is the reduced mass of the reactants A and B, [math]\displaystyle{ \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} }[/math]

Collision in diluted solution

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

[math]\displaystyle{ \nu = \frac{8 k_\text{B} T}{3 \eta} n, }[/math]

Where:

• [math]\displaystyle{ k_B }[/math] is the Boltzmann constant
• [math]\displaystyle{ T }[/math] is the absolute temperature (unit K)
• [math]\displaystyle{ \eta }[/math] is the viscosity of the solution (pascal seconds)
• [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ \nu }[/math].