Entropy, probability and disorder

1. Entropy, probability and disorder

2. Thermal equilibrium • Experience tells us: two objects in thermal contact will attain the same temperature and keep that temperature • Why? More than just energy conservation! • Involves concept of entropy

3. Entropy and disorder • It is often said that entropy is a measure of disorder, and hence every system in isolation evolves to the state with “most disorder” • Consider a box sliding on a floor: • internal energy due to disorderly motion of the molecules • kinetic energy (of the box) due to the collective, orderly, motion of all the molecules

4. Entropy and disorder II • Now the box comes to rest due to friction • Temperature rise in both floor and box so the internal energy increases • No more collective motion: all K.E. has been transferred into internal energy • More disorder, so entropy has increased

5. A vessel of two halves • Large number of identical molecules – distribution? • About 50% in left half, 50% in right half • Why?

6. Definitions • Microstate: position and momentum of each molecule accurately specified • Macrostate: only overall features specified • Multiplicity: the number of microstates corresponding to the same macrostate

7. Fundamental assumption • Statistical Mechanics is built around this one central assumption: Every microstate is equally likely to occur • This is just like throwing dice:

8. A throw of the dice • Roll one die: 1/2/3/4/5/6 all equally likely • Roll a pair of dice: • for each 1/2/3/4/5/6 equally likely • the sum 7 is most likely, then 6 and 8, etc. • Why? 6 combinations (microstates) give 7 (the macrostate): 1+6, 2+5, 3+4, 4+3, 5+2, 6+1. There are 5 combinations that give 6 or 8, etc.

9. Four identical molecules • 4 molecules ABCD • 5 macrostates:

10. Four identical molecules (2) • left: right: • A&B&C&D - • A&B&C D • A&B&D C • A&C&D B • B&C&D A multiplicity: 1 multiplicity: 4

11. Four identical molecules (3) • left: right: • A&B C&D • A&C B&D • A&D B&C • B&C A&D • B&D A&C • C&D A&B multiplicity: 6

12. Four identical molecules (4) • #left #right multiplicity probability • 4 0 1 1/16 • 3 1 4 4/16 • 2 2 6 6/16 • 1 3 4 4/16 • 0 4 1 1/16 16

13. Ten identical molecules • Multiplicity to find 10 –...– 0 molecules on left • 1–10–45–120–210–252–210–120–45–10–1 • Probability of finding #left = 4, 5 or 6: • For large N: extremely likely that #left is very close to N/2

14. Generalisation • Look at a gas of N molecules in a vessel with two “halves”. • The total number of microstates is 2N: • two possible locations for each molecule • we’ve just seen the N=4 example

15. Binomial distribution I • A gas contains N molecules, N1 in the left half (“state 1”) and N2 = N – N1 in state 2 (the right half). How many microstates correspond to this situation? N1 N2

16. Binomial distribution II • Pick the molecules one by one and place them in the left hand side: • choose from N molecules for the first molecule • choose from N – 1 for the second • choose from N – 2 for the third, … • choose from N – N1 + 1 for the N1-th molecule

17. Binomial distribution III • Number of ways of getting N1 molecules into left half: • The macrostate doesn’t depend on the order of picking these molecules; there are N1! ways of picking them. Multiplicity W is mathematical “combination”:

18. Verification • Look at a gas with molecules A,B,C,D,E. • Look at the number of ways of putting 2 molecules into the left half of the vessel. • So: N = 5, N1 = 2, N – N1 = 3

19. Verification II • The first molecule is A, B, C, D, or E. • Pick the second molecule. If I first picked A then I can now pick B, C, D or E, etc: AB BA CA DA EA AC BC CB DB EB AD BD CD DC EC AE BE CE DE ED • That is possibilities

20. Verification III • In the end I don’t care which molecule went in first. So all pairs AB and BA, AC and CA, etc, really correspond to the same situation. We must divide by 2!=2. A B = B A

21. Binomial distribution plotted • Look at N=4, 10, 1000:

22. Probability and equilibrium • As time elapses, the molecules will wander all over the vessel • After a certain length of time any molecule could be in either half with equal probability • Given this situation it is overwhelmingly probable that very nearly half of them are in the left half of the vessel

23. Second Law of Thermodynamics • Microscopic version: If a system with many molecules is permitted to change in isolation, the system will evolve to the macrostate with largest multiplicity and will then remain in that macrostate • Spot the “arrow of time”!

24. Boltzmann’s Epitaph: S = k logW • Boltzmann linked heat, temperature, and multiplicity (!) • Entropy defined by S = k ln W • W: multiplicity; k: Boltzmann’s constant • s = “dimensionless entropy” = ln W

25. Second Law of Thermodynamics • Macroscopic version: A system evolves to attain the state with maximum entropy • Spot the “arrow of time”!

26. Question 1 • Is entropy a state variable? a) Yes b) No c) Depends on the system

27. Question 2 • The total entropy of two systems, with respective entropies S1 and S2, is given by a) S = S1 + S2 b) S = S1 · S2 c) S = S1 – S2 d) S = S1 / S2

28. Entropy and multiplicity • Motion of each molecule of a gas in a vessel can be specified by location and velocity  multiplicity due to location and velocity • Ignore the velocity part for the time being and look at the multiplicity due to location only

29. Multiplicity due to location I • Divide the available space up into c small cells. Put N particles inside the space: W=cN. • For c=3, N=2: W=32=9 AB B A B A A B AB B A A B A B AB

30. Multiplicity due to location II • Increasing the available space is equivalent to increasing the number of cells c. • The volume is proportional to the number of cells c • Hence WVN

31. “Slow” and “fast” processes • Slow processes are reversible: we’re always very close to equilibrium so we can run things backwards • Fast processes are irreversible: we really upset the system, get it out of equilibrium so we cannot run things backwards (without expending extra energy)

32. Slow isothermal expansion of ideal gas; small volume change “velocity part” of multiplicity doesn’t change since T is constant Slow isothermal expansion D V V

33. Use the First Law: Big numbers  take logarithm Slow isothermal expansion (2) D V V

34. Manipulation: or Slow isothermal expansion (3) D V V

35. Use definition of entropy: valid for slow isothermal expansion Slow isothermal expansion (4) D V V

36. Example • To melt an ice cube of 20 g at 0 °C we slowly add 6700 J of heat. What is the change in entropy? In multiplicity? • 24.5 J K-1;

37. Expand very rapidly into same volume V+DV which is now empty Isothermal: same #collisions, #molecules, etc. Entropy change: NO! Entropy is a state variable and therefore DS = same as for slow isothermal expansion Very fast adiabatic expansion

38. Slow adiabatic expansion • Same volume change, but need to push air out of the way so temperature drops • Again we ask: • YES! • The “location part” of multiplicity increases as with slow isothermal expansion • The “velocity part” decreases as temperature drops • The two exactly cancel

39. Constant volume process • Heat is added to any (ideal or non-ideal) gas whose volume is kept constant. What is the change in entropy? • Integrate (assuming CV is constant)

40. Constant pressure processes • Heat is added to an ideal gas under constant pressure. What is the change in entropy? a) b) c) d) 0

41. Entropy and isothermal processes • An ideal gas expands isothermally. What is the change in entropy? • Constant temperature so • First Law: (done previously) • Therefore

42. Entropy and equilibrium • We have established a link between multiplicity and thermodynamic properties such as heat and temperature • Now we see how maximum entropy corresponds to maximum probability and hence to equilibrium

43. Equilibrium volume • In general the number of microstates depends on both the volume available and the momentum (velocity) of the molecules • Let’s ignore the momentum part and look at the spatial microstates only.

44. Equilibrium volume II • Say we have 3 molecules in a vessel which we split up into 6 equal parts. A partition can be placed anywhere between the cells. One molecule is on the left-hand side, the other two on the right-hand side. What is the equilibrium volume? • Look for maximum entropy!

45. Equilibrium volume III • Number of cells on the left c1, on the right c2. • We’ll look at c1=4, c2=2: A BC C B A B C BC A A

46. Equilibrium volume IV • Left: W1= c1=4. • Right: W2= (c2)2=4. • s = ln 4 + ln 4 = ln 16 = 2.77 A BC C B A B C BC A A

47. Question • The dimensionless entropy of this system of 6 cells and one partition dividing it into c1 and c2 cells is a) s = ln (c1+ c2) b) s = ln (c1+ c22) c) s = ln c1+ln 2c2 d) s = ln c1+2·ln c2

48. Equilibrium volume V c1Ws P(W) 1 25 3.22 0.25 2 32 3.47 0.31 3 24 3.18 0.24 4 16 2.77 0.16 5 5 1.61 0.05 total 102 1

49. Maximum entropy and probability

50. Maximum probability • Probability maximum coincides with entropy maximum • Volume V1= c1·dV where dV is the cell size • Most likely situation when • Same density on both sides: