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