Editing Kinetic theory of gases (section)

=== Collisions with container wall ===
For an ideal gas in equilibrium, the rate of collisions with the container wall and velocity distribution of particles hitting the container wall can be calculated<ref name="OCW">{{cite web |title=5.62 Physical Chemistry II |url=https://ocw.mit.edu/courses/chemistry/5-62-physical-chemistry-ii-spring-2008/lecture-notes/29_562ln08.pdf |website=MIT OpenCourseWare}}</ref> based on naive kinetic theory, and the results can be used for analyzing [[Effusion#Physics in Effusion|effusive flow rate]]s, which is useful in applications such as the [[Gaseous diffusion#Technology|gaseous diffusion]] method for [[Isotope separation#Diffusion|isotope separation]].

Assume that in the container, the number density (number per unit volume) is <math>n = N/V</math> and that the particles obey [[Maxwell-Boltzmann distribution|Maxwell's velocity distribution]]:
<math display="block">f_\text{Maxwell}(v_x,v_y,v_z) \, dv_x \, dv_y \, dv_z = \left(\frac{m}{2 \pi k_\text{B} T}\right)^{3/2} e^{- \frac{mv^2}{2k_\text{B}T}} \, dv_x \, dv_y \, dv_z</math>

Then for a small area <math>dA</math> on the container wall, a particle with speed <math>v</math> at angle <math>\theta</math> from the normal of the area <math>dA</math>, will collide with the area within time interval <math>dt</math>, if it is within the distance <math>v\,dt</math> from the area <math>dA</math>. Therefore, all the particles with speed <math>v</math> at angle <math>\theta</math> from the normal that can reach area <math>dA</math> within time interval <math>dt</math> are contained in the tilted pipe with a height of <math>v\cos (\theta) dt</math> and a volume of <math>v\cos (\theta) \,dA\,dt</math>.

The total number of particles that reach area <math>dA</math> within time interval <math>dt</math> also depends on the velocity distribution; All in all, it calculates to be:<math display="block">n v \cos(\theta) \, dA\, dt \times\left(\frac{m}{2 \pi k_\text{B}T}\right)^{3/2} e^{- \frac{mv^2}{2k_\text{B}T}} \left( v^2 \sin(\theta) \, dv \, d\theta \, d\phi \right).</math>

Integrating this over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math> yields the number of atomic or molecular collisions with a wall of a container per unit area per unit time:
<math display="block">J_\text{collision} = \frac{\displaystyle\int_0^{\pi/2} \cos(\theta) \sin(\theta) \, d\theta}{\displaystyle\int_0^\pi \sin(\theta) \, d\theta}\times n \bar v
= \frac{1}{4} n \bar v
= \frac{n}{4} \sqrt{\frac{8 k_\mathrm{B} T}{\pi m}}. </math>

This quantity is also known as the "impingement rate" in vacuum physics. Note that to calculate the average speed <math>\bar{v}</math> of the Maxwell's velocity distribution, one has to integrate over <math>v > 0 </math>, <math>0 < \theta < \pi </math>, <math>0 < \phi < 2\pi</math>.

The momentum transfer to the container wall from particles hitting the area <math>dA</math> with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is:
<math display="block">[2mv \cos(\theta)]\times n v \cos(\theta) \, dA\, dt \times\left(\frac{m}{2 \pi k_\text{B}T}\right)^{3/2} e^{- \frac{mv^2}{2k_\text{B}T}} \left( v^2 \sin(\theta) \, dv \, d\theta \, d\phi \right).</math>
Integrating this over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math> yields the [[pressure]] (consistent with [[Ideal gas law]]):
<math display="block">P = \frac{\displaystyle 2\int_0^{\pi/2} \cos^2(\theta) \sin(\theta) \, d\theta}{\displaystyle \int_0^\pi \sin(\theta) \, d\theta}\times n mv_\text{rms}^2
= \frac{1}{3} n mv_\text{rms}^2
= \frac{2}{3} n\langle E_\text{kin}\rangle
= n k_\mathrm{B} T </math>
If this small area <math>A</math> is punched to become a small hole, the [[Effusion#Physics in Effusion|effusive flow rate]] will be:
<math display="block">\Phi_\text{effusion} = J_\text{collision} A= n A \sqrt{\frac{k_\mathrm{B} T}{2 \pi m}}. </math>

Combined with the [[ideal gas law]], this yields
<math display="block">\Phi_\text{effusion} = \frac{P A}{\sqrt{2 \pi m k_\mathrm{B} T}}. </math>

The above expression is consistent with [[Graham's law]].

To calculate the velocity distribution of particles hitting this small area, we must take into account that all the particles with <math>(v,\theta,\phi)</math> that hit the area <math>dA</math> within the time interval <math>dt</math> are contained in the tilted pipe with a height of <math>v\cos (\theta) \, dt</math> and a volume of <math>v\cos (\theta) \, dA \, dt</math>; Therefore, compared to the Maxwell distribution, the velocity distribution will have an extra factor of <math>v\cos \theta</math>:
<math display="block">\begin{align}
f(v,\theta,\phi) \, dv \, d\theta \, d\phi
&= \lambda v\cos{\theta} \left(\frac{m}{2 \pi k T}\right)^{3/2} e^{- \frac{mv^2}{2k_\mathrm{B} T}}(v^2\sin{\theta} \, dv \, d\theta \, d\phi)
\end{align}</math>
with the constraint <math display="inline">v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2}</math>, <math>0 < \phi < 2\pi</math>. The constant <math>\lambda</math> can be determined by the normalization condition <math display="inline">\int f(v,\theta,\phi) \, dv \, d\theta \, d\phi=1</math> to be <math display="inline">4/\bar{v} </math>, and overall:
<math display="block">\begin{align}
f(v,\theta,\phi) \, dv \, d\theta \, d\phi
&= \frac{1}{2\pi} \left(\frac{m}{k_\mathrm{B} T}\right)^2e^{- \frac{mv^2}{2k_\mathrm{B} T}}
(v^3\sin{\theta}\cos{\theta} \, dv \, d\theta \, d\phi) \\
\end{align};\quad v>0,\, 0<\theta<\frac \pi 2,\, 0<\phi<2\pi</math>