21. Kinetic Theory of Ideal Gases

In a sample of gas, you pick a particle at random. The mass of the particle is 1.67 × 10^-27 kg and you measure its speed to be 1600 m/s. If that particle's kinetic energy is equal to the average kinetic energy of the gas particles, what is the temperature of the sample of gas?

Oxygen (O₂) has a molar mass of $32.0$ g/mol. What is the average translational kinetic energy of an oxygen molecule at a temperature of $300$ K?

A 6.0 m ✕ 8.0 m ✕ 3.0 m room contains air at 20℃. What is the room's thermal energy?

The rms speed of the atoms in a 2.0 g sample of helium gas is 700 m/s. What is the thermal energy of the gas?

Liquid helium boils at 4.2 K. In a flask, the helium gas above the boiling liquid is at the same temperature. What are (a) the mean free path in the gas, (b) the rms speed of the atoms, and (c) the average energy per atom?

1.0 mol of argon has 3100 J of thermal energy. What is the gas temperature in °C?

The molecules in a six-particle gas have velocities:

$\begin{aligned}\vec{v}_1 &= (20\hat{i} - 30\hat{j}) \text{ m/s} \\\vec{v}_2 &= (40\hat{i} + 70\hat{j}) \text{ m/s} \\\vec{v}_3 &= (-80\hat{i} + 20\hat{j}) \text{ m/s} \\\vec{v}_4 &= 30\hat{i} \text{ m/s} \\\vec{v}_5 &= (40\hat{i} - 40\hat{j}) \text{ m/s} \\\vec{v}_6 &= (-50\hat{i} - 20\hat{j}) \text{ m/s}\end{aligned}$

Calculate (a) ${\vec{v}}_{\text{avg}}$, (b) $v_{\text{avg}}$, and (c) $v_{\text{rms}}$.

At what temperature would the average kinetic energy (Chapter 18) of a molecule of hydrogen gas (H₂) be sufficient to excite a hydrogen atom out of the ground state?

A rubidium atom (m = 85 u) is at rest with one electron in an excited energy level. When the electron jumps to the ground state, the atom emits a photon of wavelength ⋋ = 780 nm. Determine the resulting (nonrelativistic) recoil speed v of the atom.

A rubidium atom (m = 85 u) is at rest with one electron in an excited energy level. When the electron jumps to the ground state, the atom emits a photon of wavelength ⋋ = 780 nm. The recoil speed sets the lower limit on the temperature to which an ideal gas of rubidium atoms can be cooled in a laser-based atom trap. Using the kinetic theory of gases (Chapter 18), estimate this “lowest achievable” temperature.