Chapter 3 Statistical Thermodynamics

1. Chapter 3Statistical Thermodynamics (Supposed) Student Answers on a Thermo Quiz!

2. Statistical Thermodynamics: The Basics • A slight rewording of the Fundamental Hypothesis of (Equilibrium) Statistical Mechanics: “In a closed system, every (micro-) state is equally likely to be observed”. • A consequence of this is that ALL of Equilibrium Statistical Mechanics & Thermodynamics can be derived!

3. Sections 3.1 & 3.2:Equilibrium Conditions & ConstraintsReversible & Irreversible Processes

4. Let A An isolated macroscopic system initially in internal equilibrium with energy E. The number of accessible states for Ais initially Ωi(E). • Suppose that A contains at least one Internal Constraint • Definition: A ConstraintAny system property that limits the number of accessible states Ωi(E).

5. Example: Consider a gas in a container divided by a partition into 2 equal volumes V. The gas is in equilibrium & confined by the partition to the left side. See figure: Gas V Vacuum V

6. Gas V Vacuum V • Remove the partition:The new situation • is clearlyNOT an equilibrium one. All • accessible states on the right side are • NOT filled. Wait some time. As a result • of collisions between molecules, • Eventually, they distribute themselves uniformly over the entire volume2V.

7. Gas 2V V • When thepartition is removed,& after we wait until a new equilibrium is reached, the • Molecules eventually will be distributed • uniformly over the entire volume2V. • The new equilibrium is as illustrated in the figure. • Compare initial # of accessible states Ωi(E) the molecules had before removing the partition to final # of accessible states Ωf(E) after removing it. • Clearly, Ωf(E) > Ωi(E).

8. In this simple example, after the gas comes to a new equilibrium, Ωf(E)> Ωi(E) • More generally, consider an isolated system A, in internal equilibrium at energy E, subject to one or more constraints. The number of accessible states is Ωi(E). • Remove one or more constraints. Except in very special cases, the new situation is NOT an equilibrium one. Wait some time until a new equilibrium is reached as a result of interactions between particles in A.

9. General Discussion • An isolated system A, in internal equilibrium at energy E, with one or more constraints. Remove one or more constraints. Wait some time until a new equilibrium is reached as a result of interactions between particles in A. • Compare the initial number of accessible states Ωi(E) before removing constraints to the final number of accessible states Ωf(E)after removal. • Generally, it will be true that Ωf(E)> Ωi(E). • In some very special cases, we might find: Ωf(E)= Ωi(E).

10. After system A comes to a new equilibrium, in general we have Ωf(E)Ωi(E) • If the equals holds, Ωf(E)= Ωi(E) & constraint removal is called a Reversible Process. • A Reversible Process is an idealization, which can never happen in real life!

11. After system A comes to a new equilibrium, in general we have Ωf(E)Ωi(E) • If the greater than signholds, Ωf(E)> Ωi(E) & constraint removal is called an Irreversible Process. All Real InteractionProcesses are Irreversible!

12. Section 3.3: Energy Distribution Between Systems in Equilibrium Note! This section is one of the most important sections in the chapter!! The following figure is about the problem of gas molecules in a container.

13. The figures below illustrate the problem of gas • molecules in a container. But, • The following discussion applies to • essentially any thermodynamic system. • For seven molecules in a container, the figures • show 3 possible spatial configurations of • the molecules. The total number of possible • configurations is huge, of course!!

14. Consider the FundamentalHypothesis

15. Each individual microstate is equally probable. Consider the FundamentalHypothesis

16. This microstate is possible,but there aren’t many microstates that give these extreme results. So it is highly improbable! Each individual microstate is equally probable. Consider the FundamentalHypothesis

17. This microstate is possible,but there aren’t many microstates that give these extreme results. So it is highly improbable! Each individual microstate is equally probable. Consider the FundamentalHypothesis If the number of particles is large (>10)the probability functionsfor this type of configuration are sharply peaked. So this type of configuration is highly probable!

18. Does the Fundamental Hypothesis lead to something that isConsistent with Classical Thermodynamics? • Assume systems A1 & A2 are coupled • so they can exchange energy. They • correspond to Reif’s systemsA & A'. E2 = E - E1 E1 Calculation of E1 The most probable energy of system A1 E = E1 + E2 • The probability finding of A1with a particular energy E1 • is proportional to the product of the number of accessible • states of A1 times the number of accessible states of A2 • Consistent with Energy Conservation: E = E1 + E2

19. E2 = E - E1 E = E1 + E2 E1 • Stated another way: • Each configuration is equally probable, • but, the number of states having a particular • energyE1is huge & unknown. We find E1 by • finding the energy where the left side of • the above equation has a maximum.

20. Some Math Details Start with (1):

21. Some Math Details Start with (1): Take the natural log of both sides:

22. Some Math Details Start with (1): Take the natural log of both sides: Find E1 where this is max by setting the derivative = 0 (2):

23. Some Math Details Start with (1): Take the natural log of both sides: Find E1 where this is max by setting the derivative = 0 (2): Combine (1) & (2):

24. Combine (1) &(2):

25. Combine (1) &(2): Energy is Conserved!  dE1 = -dE2

26. Combine (1) &(2): Energy is Conserved!  dE1 = -dE2 This is the Thermal Equilibrium Condition!!

27. Combine (1) &(2): Energy is Conserved!  dE1 = -dE2 This is the Thermal Equilibrium Condition!! Definition of the parameterβ

28. So, at Equilibrium, it must be true that: The Thermal Equilibrium Condition!! Definition of β • A Fundamental Statistical Definition! • The physical interpretation of βis that it is a measure of • A System’s Temperature!

29. Also, DEFINE the EntropyS of a system as: (kB= 1.380662  10-23 J/K = Boltzmann’s constant) In thermodynamics, we DEFINEthe Kelvin Temperature Scalesuch that: But we just defined: So,   [1/(kBT)]

30. This gives a “Statistical” Definition of Temperature:  Entropy & Temperatureare both related to the number of accessible states Ω(E).  The Fundamental Postulateleads to: 1. An Equilibrium Condition.2 systems are in thermal equilibrium when their temperatures are equal. (Obvious, but we proved it!) 2. A Maximum Entropyfor the coupled systems when they are at equilibrium. As we’ll see, this is related to the The 2nd Law of Thermodynamics. We’ll see all of this in more detail soon. 3. The 3rd Law of Thermodynamics also.

31. Consider 2 macroscopic systems A, A', interacting & in equilibrium. The combined system A0 = A +A', is isolated. AssumeThermal Interactions Only • Recallthat in a pure • Thermal Interaction, • no mechanical work is • done. So the energy • exchange between A • & A' is a pure A' A Heat Exchange!!

32. Back again to the situation which led to the definitions of • Entropy S = kBln & temperature   [1/(kBT)] • Assume systems A1 & A2 are weakly • coupled so they can exchange energy. • They correspond to Reif’s systemsA & A'. E2 = E - E1 E1 Calculation of E1 The most probable energy of system A1 E = E1 + E2 • The probability P(E) finding of A1with energy E1is • proportional to the product of the number of accessible • states of A1 & the number of accessible states of A2 • Consistent with Energy Conservation:E = E1 + E2 P(E) 

33. After much tedious derivation, the probability P(E) of finding • the combined system A0 = A +A'in a state for which the • system A1 has an energy E is approximately a very sharply • peaked Gaussian, which looks like: P(E) ΔE  (10-12)Ē ΔE Ē/(f)½ E Ē For f = 1024, (f)½ = 1012, so ΔE Ē/(f)½ (10-12)Ē

34. Consider 2 macroscopic systems A, A', interacting & in equilibrium. The combined system A0 = A +A', is isolated.Assume • Mechanical Interactions Only • Recallthat in a pure • Mechanical Interaction, • mechanical work is done. So, • the energy exchange between A' A • A, A' occurs becauseThey Do Workon each other! • Recall also that doing mechanical work means that at least • one external parameter (x) of Achanges. Further, this • change in x can be characterized by a generalized force • X  -(E/x)

35. After much tedious derivation, the probability of finding • the combined system A0 = A +A'in a state for which • the system A has a generalized force X looks like: P(X) ΔX <X>/(f)½ ΔX  (10-12)<X> X <X> For f = 1024, (f)½ = 1012, so ΔX <X>/(f)½ (10-12)<X>

36. After much tedious derivation, the probability P(X) of finding • the combined system A0 = A +A' in a state for which the • system A has a generalized force X is approximately • a very sharply peaked Gaussian, which looks like: P(X) ΔX <X>/(f)½ ΔX  (10-12)<X> X <X> For f = 1024, (f)½ = 1012, so ΔX <X>/(f)½ (10-12)<X>

37. Example:If the external parameter for a gas is the • volume, x = V, then we’ve already seen that the resulting • generalized force is the gas pressure p, X = p. P(p) Δp  (10-12)<p> Δp <p>/(f)½ p <p> For f = 1024, (f)½ = 1012, so Δp <p>/(f)½ (10-12)<p>

38. Consider 2 macroscopic systems A, A', interacting & in equilibrium. The combined system A0 = A +A', is isolated. Assume that Both Thermal & Mechanical Interactions Occur at the Same Time A A'

39. The probability P(E) of finding the combined system • A0 = A +A'being in a state for which the system A has • energy E is a very sharply peaked Gaussian, as before. So, P(E) ΔE  (10-12)Ē ΔE Ē/(f)½ E Ē For f = 1024, (f)½ = 1012, so ΔE Ē/(f)½ (10-12)Ē

40. The probability P(X) of finding the combined system A0 = A +A' • in a state for which the system A has a generalized force • X is a very sharply peaked Gaussian, as before P(X) ΔX <X>/(f)½ ΔX  (10-12)<X> X <X> For f = 1024, (f)½ = 1012, so ΔX <X>/(f)½ (10-12)<X>

41. Section 3.4:The Approach to Thermal Equilibrium:Entropy & the 2nd Law ofThermodynamics

42. As we’ve seen, Entropyis DEFINED as Boltzmann’s constant times the natural log of the number of accessible states S kBlnΩ(E)

43. EntropyDEFINITION S kBlnΩ(E) • As already discussed, for two interacting systems, A & A', the number of accessible states can be written: Ω0(E,E') = Ω(E)Ω'(E') • So the entropy of the combined system can be written: S0kBln[Ω0(E,E')] = kBln[Ω(E)Ω'(E')] = kBln[Ω(E)] + kBln[Ω'(E')] or S0 = S(E) + S'(E')

44. The 2nd Law of Thermodynamics • Consider 2 macroscopic systems A, A', each initially isolated & not interacting with each other. • Define:Ωi(E) ≡Initial number of accessible states at energy E for A. Ω'i(E')  Initial number of accessible states at energy E'for A'. • Now, bring A &A' into contact, let them interact & wait long enough that they come to equilibrium. • The combined system A0 = A +A', is isolated.

45. Combined system A0 = A +A', is isolated. • After they come to equilibrium: Define: Ωf(E) Final number of accessible states at energy E for A. Ω'f(E') Final number of accessible states at energy E'for A'. • As we’ve also seen, it will be true in general for A &A' that Ωf(E)Ωi(E)&Ω'f(E')Ω'i(E')

46. After systems A &A' come to equilibrium, Ωf(E)Ωi(E)&Ω'f(E')Ω'i(E') • If the equals signs hold, Ωf(E)= Ωi(E) & Ω'f(E')= Ω'i(E') & the interaction of A &A' is called a Reversible Process. A Reversible Process is an idealization!

47. After systems A &A' come to equilibrium, Ωf(E)Ωi(E)&Ω'f(E')Ω'i(E') • If the greater than signs hold, Ωf(E)> Ωi(E) & Ω'f(E')> Ω'i(E'), the interaction of A &A' is called an Irreversible Process. All Real InteractionProcesses are Irreversible!

48. In terms of Entropies, S = kBln(Ω), this means that after systemsA &A'come to equilibrium: Sf(E)Si(E)&S'f(E')S'i(E')(1) • If the equals signs hold, Sf(E)=Si(E)& S'f(E')=S'i(E') & the interaction of A &A' is called a Reversible Process. A Reversible Process is an idealization!

49. In terms of Entropies, S = kBln(Ω), this means that after systemsA &A'come to equilibrium: Sf(E)Si(E)&S'f(E')S'i(E') (1) • If the greater than signs hold, Sf(E)>Si(E)&S'f(E')>S'i(E') the interaction of A &A' is called an Irreversible Process. All Real InteractionProcesses are Irreversible!

50. Further, as we’ve seen, the probability distribution P(E) is extremely sharply peaked about its maximum at Ē. So, as A &A' interact, it is overwhelmingly probable that combined system approaches a new equilibrium where the energy of A is Ē. & P(E) is maximized about Ē.