1. Chapter 2 The Second Law

2. Why does Q (heat energy) go from high temperature to low temperature? cold hot Q flow Thermodynamics explains the direction of time. The Big Bang Hot objects are faster so they are more quick to move to the cold side. BUT in a solid objects aren’t actually moving from one side to the other. ?. ?. ?

3. Why does Q (heat energy) go from high temperature to low temperature? cold hot Q flow Let’s look at how probability tells us which way energy should flow. How many ways can energy be arranged? Which arrangements are most likely? E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E ENTROPY

4. Simple Probability In the mid-1960s, the Adams gum company acquired American Chicle and introduced a new slogan for Trident: "4 out of 5 Dentists surveyed would recommend sugarless gum to their patients who chew gum." The phrase became strongly associated with the Trident brand.

5. Some New Terms Mutually Exclusive –Outcomes of events have a single possibility, others are excluded A coin flip is heads or tails, not both Collectively Exhaustive –The full set of propositions or outcomes Heads and tails are all possible outcomes Independent –An event or outcome does not depend of previous or future events or outcomes A previous heads does not determine the next coin flip Multiplicity –The number of ways to get a particular outcome W or  is typically used as the variable Conditional Probability –An outcome depends on a previous event Drawing colored balls from a bag

6. Distributions Continued Discrete –Coin flips List the macrostates, probability, microstates, & multiplicity One coin Two coins Three coins

7. Distributions Continued Discrete –Four coin flip How can we predict what will happen? How can we talk about outcomes in percentages?

8. Distributions Continued Discrete –Four coin flip

9. Pascal’s Triangle

10. Paramagnets B = 0 B ≠ 0

11. Cards

12. Harmonic Oscillators – Einstein Solid Schrodinger Eqn. solutions for energy are n = 0 n = 1 n = 2 n = 3 n = 4 n = 5

13. Einstein Solid Total energy 0 1 Oscillator#1#2#3 

14. Einstein Solid Total energy 0 1 2 000 Oscillator#1#2#3  1 0 0 0 1 0 0 0 1

15. Einstein Solid Total energy 3 Oscillator#1#2#3 

16. Einstein Solid

17. Einstein Solids in Thermal Equilibrium

20. 3059 P 70/3059 = 0.023 350/3059 = 0.114 825/3059 = 0.270 1100/3059 = 0.360 714/3059 = 0.233 1.000 sum

21. Einstein Solids in Thermal Equilibrium Normalized probability for an energy arrangement For N A = N B = 100 oscillators P(n A =30) = P(n A =70) = 0.004 = 0.4%

22. Einstein Solids in Thermal Equilibrium Normalized probability for an energy arrangement For N A = N B = 100 oscillators P(n A =30) = P(n A =70) = 0.004 = 0.4%

23. Numbers Small –1, 5, 10, 235, etc. Large –10 10, 10 23, 10 114, etc. Very Large The Universe: 10 18 s old (Big Bang) ~10 80 atoms

24. Computer Numbers and the 2 nd Law 64 bit floating point numbers –52 bit mantissa –11 bit exponent –1 bit sign

25. The Second Law The second law of thermodynamics: "You can't even break even, except on a very cold day." Energy will "flow" until the state of maximum multiplicity is obtained. S = k B ln (  )

26. Very Large Numbers Stirling’s Approximation

27. See appendix B A more rigorous derivation comes from

28. Multiplicity of a Large Einstein Solid

30. High temperature limit: Assume n >> N

31. Multiplicity of a Large Einstein Solid High temperature limit: Assume n >> N This can’t be plotted for very large systems Neither can this. But this can.

32. Multiplicity of a Large Einstein Solid High temperature limit: Assume n >> N

33. Multiplicity of a Large Einstein Solid High temperature limit: Assume n >> N

34. Multiplicity of a Large Einstein Solid High temperature limit: Assume n >> N

35.  is a Gaussian Function N A =N B =100 n=500 22

36.  is a Gaussian Function

37. Multiplicity of a Monatomic Ideal Gas z y x A container of monatomic gas. How can we describe the ways to arrange the atoms and the energy they contain?

38. Multiplicity of a Monatomic Ideal Gas z y x In 2D momentum space, constant energy is defined by a circle.

39. Multiplicity of a Monatomic Ideal Gas z y x

40. z y x

41. z y x

42. z y x

43. Multiplicity of an Ideal Gas What is the relative probability of a gas taking the full volume of its container to the probability of taking half the volume of its container?

44. Multiplicity of an Ideal Gas

45. Exchanges possible N A N B : Diffusive Equilibrium V A V B : Pressure Equilibrium U A U B : Thermal Equilibrium

46. Entropy

47. Entropy – Ideal Gas

48. Creating Entropy Free Expansion W = 0 because there is nothing to push against in a vacuum. Q = 0 because this is an adiabatic process, and insulated from the surroundings.  U = Q + W = 0. BUT there is a volume change!

49. Entropy of Mixing

51. Entropy – Einstein Solid

52. Entropy

54.  vs. ln(  ) N A =N B =1000 n=1000 N A =N B =10 23 n=10 24

55. Creating Entropy Free Expansion

56. Entropy

57. Experiment Flip n=10 coins Count and record number of heads, n H Repeat N=1000 times Create a histogram from 0 to 10 of n H Computer Simulation Generate n=10 random 1 or 0 (heads or tails) 1 1 1 0 0 1 1 1 0 1 Count and record number of heads, n H Repeat N=1000 times Create a histogram from 0 to 10 of n H Fit a gaussian to the data My results x 0 = 4.93 ± 0.04  = 2.27 ± 0.09