1. Dr. Jie ZouPHY 13611 Chapter 21 The Kinetic Theory of Gases

2. Dr. Jie ZouPHY 13612 Outline Molecular model of an ideal gas (21.1) Kinetic theory Application of the kinetic theory (21.1) Derivation of an expression for the pressure Molecular interpretation of temperature Theorem of equipartition of energy (21.1)

3. Dr. Jie ZouPHY 13613 Molecular model of an ideal gas A microscopic model of an ideal gas. Assumptions: 1. The number of molecules in the gas is large, and the average separation between them is large compared with their dimensions. 2. The molecules obey Newton’s laws of motion, but as a whole they move randomly. 3. The molecules interact only by short-range forces during elastic collisions. 4. The molecules make elastic collisions with the walls. 5. The gas under consideration is a pure substance; that is, all molecules are identical. Kinetic theory: Gas molecules move about in a random fashion, colliding with the walls of their container and with each other – a physical basis for our understanding of the concept of temperature.

4. Dr. Jie ZouPHY 13614 Application of kinetic theory: derivation of an expression for the pressure Apply kinetic theory and derive an expression for the pressure of N molecules of an ideal gas in a container of volume V in terms of microscopic quantities: Results (derivation or proof): Total pressure exerted on the wall: Physical meaning: The pressure of a gas is proportional to the number of molecules per unit volume (N/V) and to the average translational kinetic energy of the molecules,. Significance: A key link is established between the molecular world and the large-scale world. Everyday observations: Increase the pressure inside of tire by adding air to a tire. Increase the pressure inside of tire by increasing the temperature.

5. Dr. Jie ZouPHY 13615 Molecular interpretation of temperature From the kinetic theory (see previous slide): From the ideal gas law (based on experimental facts): PV = Nk B T k B = R/N A = 1.38  10 -23 J/K is the Boltzmann’s constant. A comparison gives: Physical meaning: Temperature is a direct measure of average molecular kinetic energy. Or, temperature is proportional to average kinetic energy.

6. Dr. Jie ZouPHY 13616 Theorem of equipartition of energy Average translational kinetic energy per molecule: Since, it follows that Physical meaning: Each translational degree of freedom contributes an equal amount of energy, (1/2) k B T, to the gas. Total translational kinetic energy of N molecules: Theorem of equipartition of energy (A generalization): Each degree of freedom contributes (1/2) k B T to the energy of a system, where possible degrees of freedom in addition to those associated with translation arise from rotation and vibration of molecules. The internal energy of an ideal gas depends only on the temperature. Root-mean-square (rms) speed:

7. Dr. Jie ZouPHY 13617 Example 21.2 A tank of Helium A tank used for filling helium balloons has a volume of 0.300 m 3 and contains 2.00 mol of helium gas at 20.0°C. Assume that the helium behaves like an ideal gas. (A) What is the total translational kinetic energy of the gas molecules? (B) What is the average kinetic energy per molecule?

8. Dr. Jie ZouPHY 13618 Homework Ch. 21, P. 662, Problems: #2, 4, 6, 8, 10.