Find \(v_{\mathrm{rms}}\) for a gas with \(M=2.0\times10^{-2}\,\mathrm{kg\,mol^{-1}}\) at \(300\,\mathrm{K}\).

For a Maxwell-Boltzmann speed distribution, order \(v_{\mathrm{mp}}\), \(\bar v\), and \(v_{\mathrm{rms}}\) from smallest to largest.

If a gas temperature is increased from \(300\,\mathrm{K}\) to \(1200\,\mathrm{K}\), by what factor does \(v_{\mathrm{rms}}\) change?

Find the most probable speed of nitrogen at \(300\,\mathrm{K}\), using \(M=2.8\times10^{-2}\,\mathrm{kg\,mol^{-1}}\).

Find the mean speed of helium at \(300\,\mathrm{K}\), using \(M=4.0\times10^{-3}\,\mathrm{kg\,mol^{-1}}\).

At the same temperature, gas A has molar mass \(M\) and gas B has molar mass \(9M\). Find \(\bar v_A/\bar v_B\).

A gas has molecular radius \(1.5\times10^{-10}\,\mathrm{m}\) and number density \(2.5\times10^{25}\,\mathrm{m^{-3}}\). Estimate its mean free path using \(\lambda=1/[4\sqrt2\pi r^2(N/V)]\).

A molecule has mean free path \(8.0\times10^{-8}\,\mathrm{m}\) and speed \(500\,\mathrm{m\,s^{-1}}\). Estimate its mean free time.

Explain why \(v_{\mathrm{rms}}\) is more directly connected to average kinetic energy than the mean speed \(\bar v\).

A gas is compressed isothermally so its number density doubles. In the hard-sphere model, derive how the mean free path and mean free time change, assuming the characteristic molecular speed is unchanged.

At what temperature would helium \((M=4.0\times10^{-3}\,\mathrm{kg\,mol^{-1}})\) have the same \(v_{\mathrm{rms}}\) as nitrogen \((M=2.8\times10^{-2}\,\mathrm{kg\,mol^{-1}})\) at \(300\,\mathrm{K}\)?

For nitrogen at \(300\,\mathrm K\), compare \(v_{\mathrm{mp}}\), \(\bar v\), and \(v_{\mathrm{rms}}\) using their formula factors.

A gas has mean free path \(1.2\times10^{-7}\,\mathrm m\) and mean speed \(480\,\mathrm{m\,s^{-1}}\). Estimate the collision frequency.

A gas expands isothermally to four times its original volume with the same number of molecules. How do mean free path and mean free time change?

A gas has molecular radius doubled while number density and temperature stay fixed. In the hard-sphere model, how does mean free path change?

Derive \(v_{\mathrm{rms}}=\sqrt{3RT/M}\) from \(\langle K_{\mathrm{tr}}\rangle=3k_BT/2\).

Explain why the Maxwell-Boltzmann distribution broadens and shifts right as temperature increases.

A gas mixture contains helium and argon at the same temperature. Explain why their speed distributions are different but their average translational kinetic energies are the same.

Derive the dependence of mean free path on pressure for an ideal gas at fixed temperature and molecular radius.

Why is \(v_{\mathrm{rms}}\) not the same as the speed of every molecule, and why is that distinction important?