Kinetic Model of Gases: Simplified Mechanics & Ideal Gas Theory

1. 13. Matter very simple 13.2 The kinetic model

2. What’s the relationship

3. What’s the relationship

4. What temperature does water boil?

9. Using the ideal gas relationship

10. Answers • 0.103 m3 • 0.11 m3 • 7.14 litres • If the lab is roughly 3 m high x 10 m x 10 m, volume = 300 m3, then about 12 kmol (with a mass of 360 kg) • 2.38 kg m–3 • 0.51 kg m–3

12. 13. 2 The kinetic model • Modelling of the real world using simplifications • A gas is simply a collection of fast-moving, colliding particles, obeying the laws of mechanics

13. mv

14. Ft +mv mv -2mv +2mv -mv

16. F Δt t

17. p y z x

18. Calculating pressure

19. Improve model: add particles mv mv mv mv

20. F t

21. Improve model: random directions mv mv mv mv

22. 1/3 as many collision F t

24. Improve model: random speeds mv mv mv mv

25. 1/3 as many collision F t

30. The ideal gas • For the model to work • Sample of gas must be large enough • Gas molecules are moving randomly in all directions with variable speeds • Collisions are elastic • Makes better predictions when: • Density of gas is low (space molecules take up is zero) • Energy of particles is large (no interactions)

31. The end of the clockwork Universe • Maxwell and Boltzmann were using Newtonian mechanics but in a new way Statistics

32. Temperature and Energy

33. Temperature and Energy

34. Energy in a gas We can determine the total internal energy of a monatomic gas just by knowing the pressure and volume, which are easy to measure!

35. Speed of a molecule • If you know the average kinetic energy then you can determine the average speed

38. Seeing primordial living motion

39. Let’s flip a coin

40. How far will a molecule go after N steps?

41. Worked examples • If a particles follows a random walk of 10-7 m, and made 1010 steps. • What is the distance it has travelled? • What is the displacement?

42. The real stuff • Treating real gases as ideal is a simplification but a very useful one • This is what physics is, useful simplification!

43. Quick Check 1 • A ball of mass 0.2 kg travels at 2 ms-1 towards a ball and bounces back at the same speed. Show that it gives a momentum of 0.8 kg ms-1 to the wall

44. Quick Check 2 • Show that 20 mol of ideal gas contains 12 x 10^24 molecules.

45. Quick Check 3 • A mole of hydrogen occupies a volume of 0.024 m3 at a pressure of 105 Pa and a temperature of 300 K. Show that the mean kinetic energy of a hydrogen molecules is about 10-20 J.

46. Quick Check 4 • Show that the root mean square speed of H2 molecules (m = 3x10-27 kg) is about 2.6 kms -1. This is less than the escape velocity from the Earth, which is about 11 ms-1. Explain why hydrogen molecules do nethertheless escape from the atmosphere.

47. Quick Check 5 • Show that a typical molecule travelling at about 500 ms-1 will take only 20 ms to cross a room 10 m wide. Explain why a given molecule will actually take very much longer than this to travel only a few cm.

48. Quick check 6 • Show that in a mixture of oxygen (Mr = 32) and hydrogen (2) molecules the root mean square speeds of the two kinds of molecules will differ by a factor of four. Which is the fastest?