Kinetic Theory

Gases are made of a very large number of tiny molecules in constant random motion. The kinetic theory explains the large-scale behaviour of a gas, such as its pressure and temperature, in terms of the motion of these molecules. This chapter begins with the molecular nature of matter and the behaviour of gases, summarised in the ideal gas equation PV = nRT = k_B N T. It also covers Boyle's law, Charles' law and Dalton's law of partial pressures. You will study the assumptions of the kinetic theory of an ideal gas and derive the pressure P = (1/3) n m v^2, where v^2 is the mean of the squared molecular speeds. From this result, the average kinetic energy of a molecule is (3/2)k_B T, giving a kinetic interpretation of temperature and the root mean square speed sqrt(3RT/M). The law of equipartition of energy assigns (1/2)k_B T to each degree of freedom, and is used to find the specific heats of monatomic, diatomic and polyatomic gases and of solids. The chapter ends with the mean free path.

Detailed Worksheet: Kinetic Theory Section A - Definitions (10 marks) 1. State Boyle's law and Charles' law. (2 marks) 2. Write the ideal gas equation in terms of the number of moles and in terms of the number of molecules. (2 marks) 3. State the law of equipartition of energy. (2 marks) 4. Find the ratio of the rms speeds of hydrogen and oxygen molecules at the same temperature. (2 marks) 5. Define the mean free path of a gas molecule. (2 marks) Section B - Calculations and Applications (15 marks) 6. Calculate the rms speed of oxygen molecules at 300 K. Take R = 8.31 J/mol K and M = 0.032 kg/mol. (3 marks) 7. Find the average kinetic energy of a gas molecule at 300 K. Take k_B = 1.38 x 10^-23 J/K. (3 marks) 8. At what temperature will the rms speed of a gas be double its value at 27 degrees Celsius? (3 marks) 9. Estimate the number of molecules in 1 cm^3 of a gas at STP, taking the molar volume as 22.4 litres. (3 marks) 10. Using equipartition, find Cv, Cp and gamma for a monatomic and a diatomic ideal gas. (3 marks) Section C - Diagrams (10 marks) 11. Draw P-V graphs (isotherms) for a fixed mass of ideal gas at two temperatures T1 < T2. (4 marks) 12. Draw a diagram of a molecule colliding elastically with the wall of a cubical container and show the change in its momentum. (3 marks) 13. Draw a diagram showing the path of a molecule between successive collisions and label the mean free path. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. State the assumptions of the kinetic theory of an ideal gas and derive P = (1/3) n m v^2. (5 marks) 15. An oxygen cylinder of volume 30 litres has an initial gauge pressure of 15 atm at 27 degrees Celsius. After some oxygen is withdrawn, the gauge pressure drops to 11 atm at 17 degrees Celsius. Estimate the mass of oxygen withdrawn. Take R = 8.31 J/mol K and treat the given pressures as absolute. (5 marks) 16. An air bubble of volume 1.0 cm^3 rises from the bottom of a lake 40 m deep, where the temperature is 12 degrees Celsius, to the surface, where the temperature is 35 degrees Celsius. Find its volume at the surface. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Take R = 8.31 J/mol K, k_B = 1.38 x 10^-23 J/K, N_A = 6.02 x 10^23 per mol and 1 atm = 1.013 x 10^5 Pa.