Théorie cinétique des gaz/Annexe/Démonstration de l'équation d'état du gaz parfait

Homogénéité et isotropie de la distribution des vitesses : $p(v)=4\pi \left({\frac {a}{\pi }}\right)^{3/2}v^{2}e^{-av^{2}}$ avec $a={\frac {m}{2kT}}$

${\rm {d}}^{3}n=d^{*}\,{\rm {d}}^{3}\tau ={\frac {N}{V}}{\rm {d}}^{3}\tau$

On commence le calcul pour les molécules de vitesse v à dv près ${\rm {d}}^{4}n={\frac {N}{V}}{\rm {d}}^{3}\tau \,p(v)\,{\rm {d}}v$

Parmi ces molécules, seules celles qui sont dans le bon angle solide comptent ${\rm {d}}^{6}n={\frac {N}{V}}{\rm {d}}^{3}\tau \,p(v)\,{\rm {d}}v{\frac {{\rm {d}}^{2}\Omega }{4\pi }}$

En sphérique ${\rm {d}}^{3}\tau =r^{2}{\rm {d}}r\,{\rm {d}}\theta \,\sin(\theta )\,{\rm {d}}\varphi$

${\rm {d}}^{2}\Omega ={\frac {\cos(\theta )}{r^{2}}}{\rm {d}}^{2}S$

${\begin{aligned}{\rm {d}}^{4}n&=\int _{\varphi =0}^{\varphi =2\pi }\int _{\theta =0}^{\theta ={\frac {\pi }{2}}}\int _{r=0}^{r=v{\rm {d}}t}{\frac {N}{V}}r^{2}p(v){\rm {d}}v{\frac {\cos(\theta )}{4\pi }}{\frac {{\rm {d}}^{2}S}{r^{2}}}\sin(\theta ){\rm {d}}r\,{\rm {d}}\theta \,{\rm {d}}\varphi \\&={\frac {N}{V}}p(v){\rm {d}}v{\frac {{\rm {d}}^{2}S}{4\pi }}\int _{\varphi =0}^{\varphi =2\pi }\int _{\theta =0}^{\theta ={\frac {\pi }{2}}}\int _{r=0}^{r=v{\rm {d}}t}\cos(\theta )\sin(\theta ){\rm {d}}r\,{\rm {d}}\theta \,{\rm {d}}\varphi \\&={\frac {N}{V}}p(v){\rm {d}}v{\frac {{\rm {d}}^{2}S}{4\pi }}\cdot v{\rm {d}}t\cdot 2\pi \cdot {\frac {1}{2}}\\&={\frac {N}{V}}p(v){\rm {d}}v{\frac {{\rm {d}}^{2}S}{4}}v{\rm {d}}t\end{aligned}}$

On intègre sur les vitesses

${\begin{aligned}d^{3}n&=\int _{v=0}^{v=+\infty }{\frac {N}{V}}p(v)v{\frac {{\rm {d}}^{2}S}{4}}{\rm {d}}t{\rm {d}}v\\&={\frac {N{\rm {d}}^{2}S}{4V}}{\rm {d}}t\int _{v=0}^{v=+\infty }v4\pi \left({\frac {a}{\pi }}\right)^{3/2}v^{2}e^{-av^{2}}{\rm {d}}v\\&={\frac {N{\rm {d}}^{2}S}{V}}{\rm {d}}t\left({\frac {a}{\pi }}\right)^{3/2}\pi \int _{v=0}^{v=+\infty }v^{3}e^{-av^{2}}{\rm {d}}v\\&={\frac {N{\rm {d}}^{2}S}{V}}{\rm {d}}t\left({\frac {a}{\pi }}\right)^{3/2}{\frac {\pi }{a}}\int _{v=0}^{v=+\infty }ve^{-av^{2}}{\rm {d}}v\\&={\frac {N{\rm {d}}^{2}S}{V}}{\rm {d}}t{\frac {\pi }{\pi ^{3/2}}}{\frac {a^{3/2}}{2a^{2}}}\\&={\frac {N}{2V}}{\frac {1}{\sqrt {\pi a}}}{\rm {d}}^{2}S\,{\rm {d}}t\end{aligned}}$

${\rm {d}}^{6}{\vec {p}}_{\rm {paroi}}={\frac {N}{V}}{\rm {d}}^{3}\tau \,p(v){\rm {d}}v\,(-2mv\cos(\theta ){\vec {u}}_{z}){\frac {{\rm {d}}^{2}\Omega }{4\pi }}$

${\begin{aligned}{\rm {d}}^{4}{\vec {p}}_{\rm {paroi}}&=\int _{\varphi =0}^{\varphi =2\pi }\int _{\theta =0}^{\theta ={\frac {\pi }{2}}}\int _{r=0}^{r=v{\rm {d}}t}{\frac {N}{V}}{\rm {d}}^{3}\tau \,p(v){\rm {d}}v\,(-2mv\cos(\theta )){\frac {{\rm {d}}^{2}\Omega }{4\pi }}{\vec {u}}_{z}\\&=\int _{\varphi =0}^{\varphi =2\pi }\int _{\theta =0}^{\theta ={\frac {\pi }{2}}}\int _{r=0}^{r=v{\rm {d}}t}-{\frac {N}{V}}p(v){\rm {d}}v\,2mv\cos(\theta ){\frac {1}{4\pi }}r^{2}\sin(\theta ){\frac {{\rm {d}}^{2}S\cos(\theta )}{r^{2}}}{\rm {d}}r\,{\rm {d}}\theta \,{\rm {d}}\varphi {\vec {u}}_{z}\\&=-{\frac {Nmv\,p(v)}{2\pi V}}{\rm {d}}v\int _{\varphi =0}^{\varphi =2\pi }\int _{\theta =0}^{\theta ={\frac {\pi }{2}}}\int _{r=0}^{r=v{\rm {d}}t}\cos(\theta )\sin(\theta ){\rm {d}}^{2}S\,{\rm {d}}r\,{\rm {d}}\theta \,{\rm {d}}\varphi \,{\vec {u}}_{z}\\&=-{\frac {Nmv\,p(v)\,{\rm {d}}^{2}S\,{\rm {d}}v}{2\pi V}}2\pi v{\rm {d}}t\,{\frac {1}{3}}{\vec {u}}_{z}\\&=-{\frac {Nm\,{\rm {d}}^{2}S\,{\rm {d}}t}{3V}}v^{2}p(v)\,{\rm {d}}v{\vec {u}}_{z}\end{aligned}}$

On intègre sur les vitesses :

${\begin{aligned}{\rm {d}}^{3}{\vec {p}}_{\rm {paroi}}&=\int _{v=0}^{v=+\infty }-{\frac {Nm\,{\rm {d}}^{2}S\,{\rm {d}}t}{3V}}v^{2}p(v){\rm {d}}v\,{\vec {u}}_{z}\\&=-{\frac {Nm\,{\rm {d}}^{2}S\,{\rm {d}}t}{3V}}\int _{v=0}^{v=+\infty }v^{4}4\pi \left({\frac {a}{\pi }}\right)^{3/2}e^{-av^{2}}{\rm {d}}v\,{\vec {u}}_{z}\\&=-{\frac {Nm\,{\rm {d}}^{2}S\,{\rm {d}}t}{3V}}4\pi \left({\frac {a}{\pi }}\right)^{3/2}{\frac {3}{8a^{2}}}{\sqrt {\frac {\pi }{a}}}{\vec {u}}_{z}\\&=-{\frac {Nm\,{\rm {d}}^{2}S\,{\rm {d}}t}{2V}}{\frac {1}{a}}{\vec {u}}_{z}\\&=-{\frac {Nm\,{\rm {d}}^{2}S\,{\rm {d}}t}{2V}}{\frac {2kT}{m}}{\vec {u}}_{z}\\&=-{\frac {NkT}{V}}{\rm {d}}^{2}S\,{\rm {d}}t\,{\vec {u}}_{z}\\\end{aligned}}$

${\overrightarrow {{\rm {d}}^{2}F_{{\rm {d}}^{2}S}}}=-{\frac {NkT}{V}}{\rm {d}}^{2}S=-p\,{\rm {d}}^{2}S\,{\vec {u}}_{z}$ avec $p={\frac {NkT}{V}}$