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# Summary of ideal gases

Ideal gas equation:

$$pV=nRT=NkT$$
• $N=nN_{\text{A}}$
• $R=kN_{\text{A}}$

For Ideal gas : $U=\frac{3}{2}NkT$

• Mean kinetic energy of a gas particle: $\frac{1}{2}m\langle v^{2}\rangle=\frac{3}{2}kT$

Root mean square speed of a gas particle:

$$v_{\text{rms}}=\sqrt{\langle v^{2}\rangle}=\sqrt{\frac{3kT}{m}}$$

$p=$ pressure; $V=$ volume; $n=$ number of moles; $m=$ mass of a particles of gas; $N=$ number of particles; $R=$ molar gas constant $(R=8.314\text{ m}^{2}\text{ kg}\text{ s}^{-2}\text{ K}^{-1})$; $k=$ Boltzmann constant $(k=1.38\times10^{-23}\text{m}^{2}\text{kg s}^{-2}\text{ K}^{-1})$; $N_{\text{A}}$ Avogadro constant $(N_{A}=6.02\times10^{23})$; $T=$ thermodynamic temperature; $U=$ internal energy.