Second Law -Probability

1. Second Law -Probability In a nutshell… Natural irreversible processes are

2. Second Law -Probability In a nutshell… Natural irreversible processes are overwhelmingly probable.

3. Second Law -Probability In a nutshell… Natural irreversible processes are overwhelmingly probable. We want to answer: what is T ?

4. Second Law -Probability In a nutshell… Natural irreversible processes are overwhelmingly probable. We want to answer: what is T ? why does heat flow from hot cold

5. Second Law -Probability In a nutshell… Natural irreversible processes are overwhelmingly probable. We want to answer: what is T ? why does heat flow from hot cold We will use combinatorics

6. Two State Systems • There are many two state systems magnetism, excitation – deexcitation

7. Two State Systems • There are many two state systems magnetism, excitation – deexcitation • This is easily modeled possible outcomes

8. Two State Systems • There are many two state systems magnetism, excitation – deexcitation • This is easily modeled possible outcomes • Terminology • MICROSTATE every possible outcome

9. Two State Systems • There are many two state systems magnetism, excitation – deexcitation • This is easily modeled possible outcomes • Terminology • MICROSTATE every possible outcome • MACROSTATE equivalent outcomes

10. Two State Systems • There are many two state systems magnetism, excitation – deexcitation • This is easily modeled possible outcomes • Terminology • MICROSTATE every possible outcome • MACROSTATE equivalent outcomes • MULTIPLICITY number of microstates in a macrostate Ω(n) if there are n microstates

11. Probability • For a two state system (heads and tails)

12. Probability • For a two state system (heads and tails) • Probability of n heads = Ω(n)/Ω(all)

13. Probability • For a two state system (heads and tails) • Probability of n heads = Ω(n)/Ω(all) • The multiplicity can be found by Binomial • Coefficient

14. Probability • For a two state system (heads and tails) • Probability of n heads = Ω(n)/Ω(all) • The multiplicity can be found by Binomial • Coefficient ( Nn ) = N! • n!(N-n)!

15. Probability • For a two state system (heads and tails) • Probability of n heads = Ω(n)/Ω(all) • The multiplicity can be found by Binomial • Coefficient ( Nn ) = N! • n!(N-n)! • Lets apply this to a two-state paramagnet

16. PARAMAGNETISM • A two-state paramagnet showing spin up and spin down dipoles.

17. Probability The multiplicity of an up state is: • Ω (Nup) = N! Nup! Ndn! N = Nup + Ndn The probability for an up state is Probability = Ω (Nup) = 1/ 2Nup! Ndn! ΣΩ(N)

18. Einstein Model of a Solid • Consider a system composed of a collection of microscopic systems that can store any number of energy units.

19. Einstein Model of a Solid • Consider a system composed of a collection of microscopic systems that can store any number of energy units. • PE = ½ ks x2

20. Einstein Model of a Solid • Consider a system composed of a collection of microscopic systems that can store any number of energy units. • PE = ½ ks x2 Size of E = hf

21. Einstein Model of a Solid • Consider a system composed of a collection of microscopic systems that can store any number of energy units. • PE = ½ ks x2 Separation of E = hf • f = 1 / (2π) √(ks/m)

22. Einstein Model of a Solid • Consider a system composed of a collection of microscopic systems that can store any number of energy units. • PE = ½ ks x2 Separation of E = hf • f = 1 / (2π) √(ks/m) This model represents vibrational modes of diatomic and polyatomic gas molecules and also a solid.

23. Einstein Model of a Solid • Consider N classical oscillators having frequency f, spaced in energy E = hf

24. Einstein Model of a Solid • As an example let’s look at Einstein solid containing three oscillators N = 3

25. Einstein Model of a Solid • As an example let’s look at Einstein solid containing three oscillators N = 3 • Also there are three units of energy q = 3

26. Einstein Model of a Solid • As an example let’s look at Einstein solid containing three oscillators N = 3 • Also there are three units of energy q = 3 What is the multiplicity of Ω (0) = ?

27. Einstein Model of a Solid • As an example let’s look at Einstein solid containing three oscillators N = 3 • Also there are three units of energy q = 3 What is the multiplicity of Ω (0) = ? Here N=3, q=0

28. Einstein Model of a Solid • As an example let’s look at Einstein solid containing three oscillators N = 3 • Also there are three units of energy q = 3 What is the multiplicity of Ω (0) = ? Here N=3, q=0 Ω (N’ ,q) = (q + N -1 q ) = Ω(3,0)

29. Einstein Model of a Solid • As an example let’s look at Einstein solid containing three oscillators N = 3 • Also there are three units of energy q = 3 What is the multiplicity of Ω (0) = ? Here N=3, q=0 Ω (N’ ,q) = (q + N -1 q ) = Ω(3,0) = (3-1)! / (0!2!) = 1

30. Einstein Model of a Solid • As an example let’s look at Einstein solid containing three oscillators N = 3 • Also there are three units of energy q = 3 What is the multiplicity of Ω (0) = ? Here N=3, q=0 N’ = q+N-1 Ω (N’ ,q) = (q + N -1 q ) = Ω(3,0) = (3-1)! / (0!2!) = 1 Argument to show N’ = q+N-1 is in text p 55

31. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2

32. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) =

33. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!)

34. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6

35. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3

36. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3 Ω(q) = Ω (3) = 10

37. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3 Ω(q) = Ω (3) = 10 Ω (0) = 1,

38. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3 Ω(q) = Ω (3) = 10 Ω (0) = 1, Ω (1) = 3,

39. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3 Ω(q) = Ω (3) = 10 Ω (0) = 1, Ω (1) = 3, Ω (2) = 6,

40. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3 Ω(q) = Ω (3) = 10 Ω (0) = 1, Ω (1) = 3, Ω (2) = 6, Ω (3) = 10

41. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3 Ω(q) = Ω (3) = 10 Ω (0) = 1, Ω (1) = 3, Ω (2) = 6, Ω (3) = 10, Ω(Total) =

42. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3 Ω(q) = Ω (3) = 10 Ω (0) = 1, Ω (1) = 3, Ω (2) = 6, Ω (3) = 10, Ω(Total) = ΣΩ(q)

43. Einstein Model of a Solid Multiplicity of Ω (2) = ? N=3, q=2 Ω (4,2) = (q + N -1 q ) = (2+3-1)! / (2!(3-1)!) = 4!/(2!2!) = 6 Going through similar treatment for q = 3 Ω(q) = Ω (3) = 10 Ω (0) = 1, Ω (1) = 3, Ω (2) = 6, Ω (3) = 10, Ω(Total) = ΣΩ(q) =20