A full set of ten original structured questions in the style of Paper 4, covering the whole of Ideal gases: the mole, the equation of state, and the kinetic theory. Several questions deliberately carry one lesson’s idea into the next within the topic. Each is linked to the lessons it draws on; attempt them all, then reveal the worked solutions.
A cylinder contains 0.40 mol of an ideal gas, as shown in Fig. 1.1, at a temperature of 27 °C and a pressure of 1.5 × 10⁵ Pa.
(a) State what is meant by one mole of a substance. [1]
(b) Calculate the number of molecules in the cylinder. [1]
(c) Calculate the volume occupied by the gas. [3]
(d) The gas is heated at constant volume until its temperature is 177 °C. Calculate the new pressure. [3]
(e) Show that the Boltzmann constant k = R / N₊ is about 1.38 × 10⁻²³ J K⁻¹, and state the equation of state written in terms of N and k. [2]
(f) State two assumptions about the molecules of an ideal gas. [2]
Fig. 2.1 represents the molecules of an ideal gas in random motion inside a container.
(a) State three assumptions made in the kinetic theory of an ideal gas. [3]
(b) Kinetic theory gives pV = ⅓Nm⟨c²⟩. Use this together with pV = NkT to show that the mean translational kinetic energy of a molecule is ₃⁄₂ kT. [3]
(c) Calculate the mean translational kinetic energy of a molecule at a temperature of 300 K. [2]
(d) A nitrogen molecule has mass 4.7 × 10⁻²⁶ kg. Calculate its root-mean-square speed at 300 K. [3]
(e) State and explain how the root-mean-square speed changes if the thermodynamic temperature is doubled. [2]
A small bubble of ideal gas rises from the bottom of a lake to the surface, as shown in Fig. 3.1. At the bottom its volume is 1.2 cm³, the pressure is 2.8 × 10⁵ Pa and the temperature is 8 °C. At the surface the pressure is 1.0 × 10⁵ Pa and the temperature is 18 °C.
(a) Write down, in terms of p, V and T, the relationship that holds for a fixed mass of ideal gas between two states. [2]
(b) Calculate the volume of the bubble at the surface. [3]
(c) Explain, in molecular terms, why the pressure of the gas in the bubble is lower at the surface. [3]
(d) Calculate the amount, in mol, of gas in the bubble. [3]
(e) Suggest one reason the real bubble would not behave exactly as predicted. [2]
A container holds 64 g of oxygen gas, which has a molar mass of 32 g mol⁻¹.
(a) Define the mole. [2]
(b) State the relation between the number of molecules N, the amount of substance n and the Avogadro constant. [1]
(c) Calculate the amount of substance, in moles. [2]
(d) Calculate the number of oxygen molecules. [2]
(e) Calculate the number of oxygen atoms. [1]
(f) State the value and unit of the Avogadro constant. [2]
A fixed mass of gas at a pressure of 1.0 × 10⁵ Pa occupies 250 cm³. It is compressed isothermally, as shown in Fig. 5.1.
(a) State Boyle’s law. [2]
(b) Calculate the pressure when the volume is reduced to 100 cm³. [2]
(c) Explain why the temperature must be constant for Boyle’s law to hold. [1]
(d) Describe the shape of the p-V graph for this change. [2]
(e) Calculate the volume at which the pressure would be 4.0 × 10⁵ Pa, at the same temperature. [2]
(f) State one condition on the gas for Boyle’s law to apply. [1]
A fixed mass of gas is kept at constant volume. At 27 °C its pressure is 1.2 × 10⁵ Pa.
(a) Convert 27 °C to kelvin. [1]
(b) State how the pressure depends on the thermodynamic temperature at constant volume. [2]
(c) Calculate the pressure at 127 °C. [2]
(d) Calculate the temperature, in °C, at which the pressure would be 2.0 × 10⁵ Pa. [3]
(e) Explain why the temperature must be in kelvin in these gas laws. [2]
A cylinder contains 0.16 kg of nitrogen gas (molar mass 28 g mol⁻¹) at 300 K and a pressure of 2.0 × 10⁵ Pa. Parts (a) and (b) use the mole; parts (c) and (d) use the equation of state.
(a) Calculate the amount of substance, in moles. [2]
(b) Calculate the number of nitrogen molecules. [2]
(c) Using pV = nRT, calculate the volume occupied by the gas. [3]
(d) The gas is heated at constant pressure to 450 K. Calculate the new volume. [2]
(e) State one assumption of the ideal-gas model used here. [2]
Fig. 8.1 represents the molecules of a gas in rapid random motion.
(a) Write the kinetic theory equation relating pressure to molecular motion. [1]
(b) State three assumptions of the kinetic theory of gases. [3]
(c) A gas has density 1.2 kg m⁻³ at a pressure of 1.0 × 10⁵ Pa. Calculate the root-mean-square speed of its molecules. [3]
(d) State what is meant by the root-mean-square speed. [2]
(e) State and explain how the r.m.s. speed changes if the kelvin temperature is doubled. [2]
This question links the equation of state with the kinetic theory. Part (b) connects pV = NkT with pV = ⅓Nm⟨c²⟩.
(a) Write the expression for the mean translational kinetic energy of a molecule in terms of temperature. [1]
(b) Show that this follows from pV = NkT and pV = ⅓Nm⟨c²⟩. [3]
(c) Calculate the mean translational kinetic energy of a molecule at 300 K. [2]
(d) An oxygen molecule has mass 5.3 × 10⁻²⁶ kg. Calculate its r.m.s. speed at 300 K. [3]
(e) State how the mean translational kinetic energy at a given temperature depends on the type of gas. [2]
A sealed flask of volume 2.0 × 10⁻³ m³ contains helium gas (molar mass 4.0 g mol⁻¹) at 1.5 × 10⁵ Pa and 300 K. This question uses all three lessons of the topic in turn.
(a) Using pV = nRT, calculate the amount of substance. [2]
(b) Calculate the number of helium atoms. [2]
(c) Calculate the mass of helium in the flask. [2]
(d) Calculate the mean translational kinetic energy of a helium atom. [2]
(e) The mass of a helium atom is the molar mass divided by N₊. Calculate the r.m.s. speed of the atoms. [3]
(f) State whether argon atoms (heavier) at the same temperature would have a greater or smaller r.m.s. speed. [1]
Mark this once you have attempted all ten questions and checked your working against the solutions. Revealing the solutions alone does not count.