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Kinetic Theory of Gases. Boltzmann's Law and Maxwell's Distribution

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  • Number of collisions exercised by gas molecules on a unit area of the wall surface per unit time: $$\nu = \frac{1}{4} n \langle v \rangle,$$ where $n$ is the concentration of molecules, and $\langle v \rangle$ is their mean velocity.
  • Equation of an ideal gas state: $$ p = nkT.$$
  • Mean energy of molecules: $$\langle\epsilon\rangle = \frac{i}{2} kT,$$ where $i$ is the sum of translational, rotational, and the double number of vibrational degrees of freedom. 
  • Maxwellian distribution: $$dN (v_x) = N \left(\frac{m}{2\pi k T}\right)^{1/2} e^{-m v_x^2/2kT}dv_x,$$ $$dN(v) =  N \left(\frac{m}{2\pi k T}\right)^{3/2} e^{-m v^2/2kT} 4\pi v^2 dv.$$
  • Maxwellian distribution in a reduced form: $$dN(u) = N \frac{4}{\sqrt{\pi}} e^{-u^2} u^2 du,$$ where $!u = v/v_p$, $v_p$ is the most probable velocity.
  • The most probable, the mean, and the root mean square velocities of molecules: $$v_p = \sqrt {2 \frac{kT}{m}}, \; \langle v \rangle = \sqrt{\frac{8}{\pi}\frac{kT}{m}}, \; v_{sq} = \sqrt{3 \frac{kT}{m}}.$$
  • Boltzmann's formula: $$n = n_0 e^{-\left(U-U_0\right)/kT}, $$ where $U$ is the potential energy of a molecules.




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