Ideal gases：練習問題

A sample of an ideal gas contains \(N\) molecules, each of mass \(m\). The mean-square speed of the molecules is \(\langle c^2 \rangle\). What is the correct expression for the pressure \(p\) of the gas in a container of volume \(V\)?

The thermodynamic temperature of an ideal gas is increased from \(100^{\circ}\text{C}\) to \(200^{\circ}\text{C}\). By what factor does the average translational kinetic energy of the molecules increase?

Two different ideal gases, A and B, are kept at the same temperature. The mass of a molecule of gas A is twice the mass of a molecule of gas B (\(m_A = 2m_B\)). What is the ratio of the root-mean-square speed of gas A to the root-mean-square speed of gas B, \(\frac{c_{\text{r.m.s., A}}}{c_{\text{r.m.s., B}}}\)?

Which of the following is a fundamental assumption of the kinetic theory of gases?

A container of fixed volume \(V\) contains an ideal gas at pressure \(p\). The number of molecules in the container is \(N\). If the root-mean-square (r.m.s.) speed of the molecules is doubled while the volume remains constant, what is the new pressure of the gas?

The density of nitrogen gas at a pressure of \(1.0 \times 10^5 \, \text{Pa}\) is \(1.25 \, \text{kg m}^{-3}\). Calculate the root-mean-square (r.m.s.) speed of the nitrogen molecules under these conditions.

Calculate the total internal energy of 2.0 moles of a monatomic ideal gas maintained at a thermodynamic temperature of 300 K.

(a) State two basic assumptions of the kinetic theory of gases.
(b) A sample of neon gas is contained in a vessel of volume \( 0.040 \, \text{m}^3 \) at a pressure of \( 1.5 \times 10^5 \, \text{Pa} \) and a temperature of \( 27 \, ^\circ\text{C} \). Neon behaves as an ideal gas and has a molar mass of \( 20 \, \text{g mol}^{-1} \).
(i) Calculate the number of moles of neon in the vessel.
(ii) Calculate the root-mean-square (r.m.s.) speed of the neon atoms.
(iii) The temperature is increased to \( 127 \, ^\circ\text{C} \). Determine the new r.m.s. speed of the atoms.

A rigid container of volume \( 0.050 \, \text{m}^3 \) contains a mixture of \( 2.0 \, \text{g} \) of Helium (molar mass \( 4.0 \, \text{g mol}^{-1} \)) and \( 14.0 \, \text{g} \) of Nitrogen (molar mass \( 28.0 \, \text{g mol}^{-1} \)) at a temperature of \( 290 \, \text{K} \).
(a) Calculate the total pressure exerted by the gas mixture.
(b) Determine the ratio of the average translational kinetic energy of a Helium atom to that of a Nitrogen molecule.
(c) Calculate the ratio of the root-mean-square speed of Helium atoms to the root-mean-square speed of Nitrogen molecules.