Chapter 18 Kinetic Theory of Gases
Learning Objectives:
In this chapter you will basically learn:
\(\bullet\) Apply the relationship between the number of moles n, the number of molecules N, and Avogadro’s number \(N_A\).
\(\bullet\) Apply the relationships between the mass m of a sample, the molar mass M of the molecules in the sample, the number of moles n in the sample, and Avogadro’s number NA.
\(\bullet\) Apply either of the two forms of the ideal gas law, written in terms of the number of moles n or the number of molecules N.
\(\bullet\) Relate the ideal gas constant R and the Boltzmann constant k.
\(\bullet\) Calculate the work done by a gas, including the algebraic sign, for an expansion and a contraction along an isotherm.
\(\bullet\) For an isothermal process, identify that the change in internal energy \(\Delta E\) is zero and that the energy Q transferred as heat is equal to the work W done.
\(\bullet\) On a p-V diagram, sketch a constant-volume process and identify the amount of work done in terms of area on the diagram.
\(\bullet\) For the molecules of an ideal gas, relate the root mean- square speed \(v_{rms}\) and the average speed \(v_{avg}\).
\(\bullet\) Relate the pressure of an ideal gas to the rms speed \(v_{rms}\) of the molecules.
\(\bullet\) For an ideal gas, apply the relationship between the gas temperature T and the rms speed \(v_{rms}\) and molar mass Mof the molecules.
\(\bullet\) For an ideal gas, relate the average kinetic energy of the molecules to their rms speed.
\(\bullet\) Apply the relationship between the mean free path, the diameter of the molecules, and the number of molecules per unit volume.
\(\bullet\) Explain how Maxwell’s speed distribution law is used to find the fraction of molecules with speeds in a certain speed range.
\(\bullet\) Sketch a graph of Maxwell’s speed distribution, showing the probability distribution versus speed and indicating the relative positions of the average speed \(v_{avg}\), the most probable speed \(v_P\), and the rms speed \(v_{rms}\).
\(\bullet\) Explain how Maxwell’s speed distribution is used to find the average speed, the rms speed, and the most probable speed.
\(\bullet\) Identify that the internal energy of an ideal monatomic gas is the sum of the translational kinetic energies of its atoms.
\(\bullet\) Apply the relationship between the internal energy \(E_{int}\) of a monatomic ideal gas, the number of moles n, and the gas temperature T.
\(\bullet\) Distinguish between monatomic, diatomic, and polyatomic ideal gases.
\(\bullet\) For monatomic, diatomic, and polyatomic ideal gases, evaluate the molar specific heats for a constant-volume process and a constant-pressure process.
\(\bullet\) Calculate a molar specific heat at constant pressure \(C_p\) by adding R to the molar specific heat at constant volume \(C_V\), and explain why (physically) \(C_p\) is greater.
\(\bullet\) For an ideal gas, apply the relationship between heat Q, number of moles n, and temperature change %T, using the appropriate molar specific heat.
\(\bullet\) Calculate the work done by an ideal gas for a constant pressure process.
\(\bullet\) Identify that a degree of freedom is associated with each way a gas can store energy (translation, rotation, and oscillation).
\(\bullet\) Identify that at low temperatures a diatomic gas has energy in only translational motion, at higher temperatures it also has energy in molecular rotation, and at even higher temperatures it can also have energy in molecular oscillations.
\(\bullet\) Calculate the molar specific heat for monatomic and diatomic ideal gases in a constant-volume process and a constant-pressure process.
\(\bullet\) Identify that in an adiabatic expansion, the gas does work on the environment, decreasing the gas’s internal energy, and that in an adiabatic contraction, work is done on the gas, increasing the internal energy.
\(\bullet\) In an adiabatic expansion or contraction, relate the initial pressure and volume to the final pressure and volume.
\(\bullet\) In an adiabatic expansion or contraction, relate the initial temperature and volume to the final temperature and volume.
\(\bullet\) Calculate the work done in an adiabatic process by integrating the pressure with respect to volume.
\(\bullet\) Identify that a free expansion of a gas into a vacuum is adiabatic but no work is done and thus, by the first law of thermodynamics, the internal energy and temperature of the gas do not change.
18.1 AVOGADRO’S NUMBER
18.2 IDEAL GASES
18.3 PRESSURE, TEMPERATURE, AND RMS SPEED
18.4 TRANSLATIONAL KINETIC ENERGY
18.5 MEAN FREE PATH
18.6 THE DISTRIBUTION OF MOLECULAR SPEEDS
18.7 THE MOLAR SPECIFIC HEATS OF AN IDEAL GAS
18.8 DEGREES OF FREEDOM AND MOLAR SPECIFIC HEATS
18.9 THE ADIABATIC EXPANSION OF AN IDEAL GAS
Solved Problems:
[2] Gold has a molar mass of 197 g/mol. (a) How many moles of gold are in a 2.50 g sample of pure gold? (b) How many atoms are in the sample?
[17] Container A in Fig. 18-22 holds an ideal gas at a pressure of \(5.0 \times 10^5\) Pa and a temperature of 300 K. It is connected by a thin tube (and a closed valve) to container B, with four times the volume of A. Container B holds the same ideal gas at a pressure of \(1.0 \times 10^5\) Pa and a temperature of 400 K.The valve is opened to allow the pressures to equalize, but the temperature of each container is maintained.What then is the pressure?
[27] Water standing in the open at 32.0\(^\circ\)C evaporates because of the escape of some of the surface molecules.The heat of vaporization (539 cal/g) is approximately equal to \(\varepsilon n\), where \(\varepsilon\) is the average energy of the escaping molecules and n is the number of molecules per gram. (a) Find \(\varepsilon\). (b) What is the ratio of \(\varepsilon\) to the average kinetic energy of \(H_2O\) molecules, assuming the latter is related to temperature in the same way as it is for gases?
[49] A container holds a mixture of three nonreacting gases:2.40 mol of gas 1 with \(C_{V1}\) = 12.0 J/mol.K, 1.50 mol of gas 2 with \(C_{V2}\) = 12.8 J/mol.K, and 3.20 mol of gas 3 with \(C_{V3}\) = 20.0 J/mol.K. What is \(C_V\) of the mixture?
[59] Figure 19-26 shows two paths that may be taken by a gas from an initial point i to a final point f. Path 1 consists of an isothermal expansion (work is 50 J in magnitude), an adiabatic expansion (work is 40 J in magnitude), an isothermal compression (work is 30 J in magnitude), and then an adiabatic compression (work is 25 J in magnitude).What is the change in the internal energy of the gas if the gas goes from point i to point f along path 2?