Find the average translational kinetic energy per molecule of an ideal gas at \(300\,\mathrm{K}\). Use \(k_B=1.38\times10^{-23}\,\mathrm{J\,K^{-1}}\).

Find \(v_{\mathrm{rms}}\) for a gas molecule of mass \(5.0\times10^{-26}\,\mathrm{kg}\) at \(320\,\mathrm{K}\).

At the same temperature, gas A has molecular mass four times gas B. Find \(v_{\mathrm{rms},A}/v_{\mathrm{rms},B}\).

Use \(pV=\frac13Nm_0v_{\mathrm{rms}}^2\) to find the pressure of \(1.0\times10^{22}\) molecules of mass \(4.0\times10^{-26}\,\mathrm{kg}\) in volume \(2.0\times10^{-3}\,\mathrm{m^3}\), with \(v_{\mathrm{rms}}=500\,\mathrm{m\,s^{-1}}\).

Find \(v_{\mathrm{rms}}\) for oxygen at \(300\,\mathrm{K}\), using molar mass \(M=3.2\times10^{-2}\,\mathrm{kg\,mol^{-1}}\).

If the Kelvin temperature of an ideal gas is tripled, by what factor does the average translational kinetic energy per molecule change? By what factor does \(v_{\mathrm{rms}}\) change?

A monatomic ideal gas contains \(0.75\,\mathrm{mol}\) at \(310\,\mathrm{K}\). Use the kinetic model to find its total translational kinetic energy.

Starting from \(pV=\frac13Nm_0v_{\mathrm{rms}}^2\) and \(pV=Nk_BT\), derive \(v_{\mathrm{rms}}=\sqrt{3k_BT/m_0}\).

A gas mixture has light molecules and heavy molecules at the same temperature. Explain whether the two species have the same average translational kinetic energy and whether they have the same \(v_{\mathrm{rms}}\).

A gas has the same \(p,V,N\) as an ideal gas prediction, but its molecule-wall collisions are not elastic because some translational energy is lost to internal modes at the wall. Explain which kinetic-model assumption fails and why the simple pressure derivation no longer applies without modification.

A monatomic ideal gas has \(p=1.0\times10^5\,\mathrm{Pa}\) and \(V=2.0\times10^{-2}\,\mathrm{m^3}\). Find the total translational kinetic energy of its molecules without first finding \(n\) or \(T\).

Two ideal gases have the same \(N\), \(V\), and \(T\), but gas B has molecules four times as massive as gas A. Use the kinetic pressure model to show that their pressures are still the same.

A monatomic ideal gas has \(N=3.0\times10^{23}\) molecules at \(T=400\,\mathrm{K}\). Find its total translational kinetic energy.

At fixed \(N\) and \(V\), an ideal gas temperature is doubled. Use the kinetic pressure model to explain why pressure doubles.

A gas has measured pressure lower than the ideal kinetic model predicts at the same \(N,V,T\). Give one molecular reason this can happen.

Derive \(\langle K_{\mathrm{tr}}\rangle=\frac32k_BT\) from the kinetic pressure relation and molecular ideal-gas equation.

Explain why ideal-gas pressure is independent of molecular mass when \(N,V,T\) are fixed.

A gas is in thermal equilibrium but individual molecular speeds vary widely. Explain how one temperature can still describe the gas.

Use the kinetic model to explain why pressure is isotropic in an equilibrium gas.

List the ideal kinetic-model assumptions and explain which two fail first at high density.