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Collision frequency

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Main page: Chemistry:Collision theory

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]

[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:[2]

[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].[2]

References

  1. chem.libretexts.org: Collision Frequency
  2. 2.0 2.1 Debye, P. (1942). "Reaction Rates in Ionic Solutions" (in en). Transactions of the Electrochemical Society 82 (1): 265. doi:10.1149/1.3071413. ISSN 0096-4743. https://doi.org/10.1149/1.3071413. 




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