1. PHYS 172: Modern Mechanics Lecture 25 - Kinetic theory of gases; pressure Read 12.8 Summer 2012

2. Counting states Entropy Temperature Boltzmann distribution Heat capacity Summary of Statistical Mechanics (Ch 12)

3. Boltzmann Distribution Blast from the Past

4. Distribution of Speeds in a Gas X- ComponentSpeed Distribution

5. Use this “distribution function” to calculate averages: This is determined by assuming that all of a molecule’s energy is kinetic energy, and then averaging over all directions (see page 399-400). The Maxwell-Boltzmann Speed Distribution:

6. Average K trans in a Gas Boltzmann Probability of having K trans Actual probability: 100% chance of having some velocity, so: (buncha math) Average K trans over the probability: (buncha math) Average K trans Related to T Important Result

7. Average K trans in a Gas Boltzmann Probability of having K trans Average K trans Related to T Important Result In HOTTER gases, molecules move FASTER

8. What’s the Pressure? Big Box of Gas

9. The Kinetic Theory of Gases PRESSURE: P = F/A force per area On average, K trans is conserved:

10. What’s the Pressure? PRESSURE: P = F/A L A vxvx V = LA Volume N right = n V / 2 Half of the atoms are going to the right:

11. What’s the Pressure? PRESSURE: P = F/A

12. What’s the Pressure? PRESSURE: P = F/A We really mean the average of

13. IDEAL GAS LAW! What’s the Pressure? PRESSURE: P = F/A We really mean the average of Same

14. Boltzmann Distribution IDEAL GAS LAW! Comes from Boltzmann Distribution

15. A.The tank weighed less after the nitrogen escaped. B.The weight of the tank did not change, because the nitrogen gas molecules don’t rest on the bottom of the tank. C.The weight of the tank did not change, because any nitrogen gas molecules striking the bottom of the tank are balanced by nitrogen gas molecules striking the top of the tank. CLICKER QUESTION A large, standard gas cylinder (tank) has a volume of about 100 liters and is filled with nitrogen gas to a pressure of 2500 psi. Somebody opens the valve and all the nitrogen escapes. Choose the correct statement:

16. What Is The Weight of a Box of Gas? x y h F top,y =P top A F bottom,y =P bottom A F net,y = P top A – P bottom A = AΔP = ? How do we evaluate ΔP? The only possible source of a force on the scale pan would be collisions of the molecules with the box walls – i.e., pressure forces.

17. The density n is also governed by the Boltzmann distribution: How Does Pressure Change With Height? ≈1

18. For any container of height h, there is always a difference in pressure between the top and the bottom such that the force on the top is less than the force on the bottom by an amount exactly equal to the weight of the enclosed gas molecules. What Is The Weight of a Box of Gas?

19. Ice water steam (gas) A.Ice system weighs most - all the force is directed down. B.Water system weighs less than ice - some of the force is directed to the side walls. C.Steam system weighs least - most molecules don’t touch the bottom. D.They have equal weights. Three identical rigid containers rest on a table. One contains a block of ice, one contains water, and one has just steam (water vapor). All three contain the same number of water molecules. Which is the correct statement? CLICKER QUESTION