19-4 Translational Kinetic Energy#
Prompts
Write \(K_{\text{avg}}\) in terms of \(v_{\text{rms}}\) and in terms of \(T\). How are they related?
At the same temperature, do H\(_2\) and O\(_2\) molecules have the same average translational kinetic energy? The same rms speed?
A gas mixture contains molecules of different masses. Rank them by average kinetic energy. By rms speed.
What does a thermometer measure when you place it in a gas?
Lecture Notes#
Overview#
The average translational kinetic energy per molecule in an ideal gas is directly tied to temperature.
\(K_{\text{avg}} = \frac{3}{2}kT\)—at a given \(T\), all ideal gas molecules have the same \(K_{\text{avg}}\), independent of mass.
Measuring temperature is effectively measuring the average kinetic energy of the molecules.
Average kinetic energy per molecule#
In terms of rms speed (section 19-3):
In terms of temperature:
\(k\): Boltzmann constant. \(T\): temperature in kelvins.
Same \(T\) → same \(K_{\text{avg}}\) for all molecules, regardless of mass.
Heavier molecules have lower \(v_{\text{rms}}\) but the same \(K_{\text{avg}}\).
Total translational kinetic energy#
For \(N\) molecules (or \(n\) moles):
For a monatomic ideal gas, \(E_{\text{int}} = K_{\text{tot}}\) (no rotational or vibrational energy).
Temperature as kinetic energy
A thermometer in a gas measures the average translational kinetic energy of the molecules—temperature and \(K_{\text{avg}}\) are equivalent for an ideal gas.
Poll: Gas mixture
A gas mixture has molecules of types A, B, and C with masses \(m_A > m_B > m_C\). At thermal equilibrium, rank their rms speeds (greatest first).
(A) A, B, C
(B) C, B, A
(C) All equal
Summary#
\(K_{\text{avg}} = \frac{1}{2}m\,v_{\text{rms}}^2 = \frac{3}{2}kT\)—average translational KE per molecule.
\(K_{\text{tot}} = \frac{3}{2}nRT\)—total translational KE.
Same \(T\) → same \(K_{\text{avg}}\); lighter molecules have higher \(v_{\text{rms}}\).