Kinetic Theory of Gases

Scenario

Ideal gas

• with pressure $p$, volume $V$, temperature $T$
• with $N$ molecules where each molecule has mass $m$, velocity $v$, kinetic energy $E_k = \frac{1}{2} mv^{2}$
• with density $\rho = \frac{Nm}{V}$

Pressure of Gas

• $p = \frac{1}{3} \rho \overline{v^2}$

Internal Energy of a Molecule

• $U = \overline{E_{k}} = \frac{f}{2} kT$
• degree of freedom $f$ = translational degree of freedom + rotational degree of freedom
• translational degree of freedom is 3
• rotational degree of freedom is 0 for monatomic molecule, is 2 for diatomic molecule, and is 3 for polyatomic molecule
• no potential energy as particles of ideal gas do not interact
• use $k$ for one molecule, $nN_{A}k = nR$ for $n \, \mathrm{mol}$ molecules