Editing Collision theory (section)

==Rate equations==

The rate for a bimolecular gas-phase reaction, A + B → product, predicted by collision theory is<ref>{{Cite web|url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/Modeling_Reaction_Kinetics/Collision_Theory/The_Collision_Theory|title = 6.1.6: The Collision Theory|date = 2 October 2013}}</ref>

: <math>r(T) = kn_\text{A}n_\text{B}= Z \rho \exp \left( \frac{-E_\text{a}}{RT} \right)</math>

where:
*''k'' is the rate constant in units of (number of molecules)<sup>−1</sup>⋅s<sup>−1</sup>⋅m<sup>3</sup>.
* ''n''<sub>A</sub> is the [[number density]] of A in the gas in units of m<sup>−3</sup>.
* ''n''<sub>B</sub> is the [[number density]] of B in the gas in units of m<sup>−3</sup>. E.g. for a gas mixture with gas A concentration 0.1&nbsp;mol⋅L<sup>−1</sup> and B concentration 0.2&nbsp;mol⋅L<sup>−1</sup>, the number of density of A is 0.1×6.02×10<sup>23</sup>÷10<sup>−3</sup> = 6.02×10<sup>25</sup> m<sup>−3</sup>, the number of density of B is 0.2×6.02×10<sup>23</sup>÷10<sup>−3</sup> = 1.2×10<sup>26</sup> m<sup>−3</sup>
* ''Z'' is the [[collision frequency]] in units of m<sup>−3</sup>⋅s<sup>−1</sup>.
* <math>\rho</math> is the [[steric factor]].<ref name="steric">{{GoldBookRef | file = S05998 | title = steric factor}}</ref>
* ''E''<sub>a</sub> is the [[activation energy]] of the reaction, in units of J⋅mol<sup>−1</sup>.
* ''T'' is the [[temperature]] in units of K.
* ''R'' is the [[gas constant]] in units of J mol<sup>−1</sup>K<sup>−1</sup>.
The unit of ''r''(''T'') can be converted to mol⋅L<sup>−1</sup>⋅s<sup>−1</sup>, after divided by (1000×''N''<sub>A</sub>), where ''N''<sub>A</sub> is the [[Avogadro constant]].

For a reaction between A and B, the [[collision frequency]] calculated with the hard-sphere model with the unit number of collisions per m<sup>3</sup> per second is:  
: <math> Z = n_\text{A} n_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}} = 10^6N_A^2\text{[A][B]} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}</math>

where: 
* ''n''<sub>A</sub> is the [[number density]] of A in the gas in units of m<sup>−3</sup>.
* ''n''<sub>B</sub> is the [[number density]] of B in the gas in units of m<sup>−3</sup>. E.g. for a gas mixture with gas A concentration 0.1&nbsp;mol⋅L<sup>−1</sup> and B concentration 0.2&nbsp;mol⋅L<sup>−1</sup>, the number of density of A is 0.1×6.02×10<sup>23</sup>÷10<sup>−3</sup> = 6.02×10<sup>25</sup> m<sup>−3</sup>, the number of density of B is 0.2×6.02×10<sup>23</sup>÷10<sup>−3</sup> = 1.2×10<sup>26</sup> m<sup>−3</sup>.
*''σ''<sub>AB</sub> is the reaction [[cross section (physics)|cross section]] (unit m<sup>2</sup>), the area when two molecules collide with each other, simplified to <math> \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 </math>, where ''r''<sub>A</sub> the radius of A and ''r''<sub>B</sub> the radius of B in unit m.
* ''k''<sub>B</sub> is the [[Boltzmann constant]] unit J⋅K<sup>−1</sup>.
* ''T'' is the absolute temperature (unit K).
* ''μ<sub>AB</sub>'' is the [[reduced mass]] of the reactants A and B, <math> \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} </math> (unit kg).
* ''N''<sub>A</sub> is the [[Avogadro constant]].
* [A] is molar concentration of A in unit mol⋅L<sup>−1</sup>.
* [B] is molar concentration of B in unit mol⋅L<sup>−1</sup>.
* Z can be converted to mole collision per liter per second dividing by 1000''N''<sub>A</sub>.

If all the units that are related to dimension are converted to dm, i.e. mol⋅dm<sup>−3</sup> for [A] and [B], dm<sup>2</sup> for ''σ''<sub>AB</sub>, dm<sup>2</sup>⋅kg⋅s<sup>−2</sup>⋅K<sup>−1</sup> for the [[Boltzmann constant]], then 
: <math> Z = N_\text{A}^2 \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}[\text{A}][\text{B}] = k [A][B]</math> 
unit mol⋅dm<sup>−3</sup>⋅s<sup>−1</sup>.