Physics 172H Modern Mechanics

Transcription

1 Physics 172H Modern Mechanics Instructor: Dr. Mark Haugan Office: PHYS 282 TAs: Alex Kryzwda John Lorenz Lecture 22: Matter & Interactions, Ch

2 Last Time E = ħ k m = ħω Einstein Model of Solids assume atoms oscillate independently of each other each atom is comprised of 3 independent quantum harmonic oscillators: N oscillators N/3 atoms # microstates (N oscillators, q quanta) Ω = ( q + N ) 1! ( N ) q! 1! Fundamental assumption of statistical mechanics Over time, an isolated system in a given macrostate (total energy) is equally likely to be found in any of its microstates (microscopic distribution of energy).

3 Last time we enumerated the 15 different microstates for 4 energy quanta of distributed among 3 oscillators (1 atom). The Fundamental Assumption of Statistical Mechanics A fundamental assumption of statistical mechanics is that in our state of macroscopic ignorance, each microstate (microscopic distribution of energy) corresponding to a given macrostate (total energy) is equally probable.

4 Q1. Two objects share a total energy E = E 1 +E 2. There are 3 ways to arrange an amount of energy E 1 in the first object and 6 ways to arrange an amount of energy E 2 in the second object. How many different ways are there to arrange the total energy E = E 1 +E 2 so that there is E 1 in the first object and E 2 in the other? A) 3 B) 6 C) 9 D) 18 E) 3 6 =729 6 ways 6 ways object 1 way 1 way 2 object 2 way 3 6 ways 3 x 6 =18 total ways

5 Fundamental Assumption of Statistical Mechanics The fundamental assumption of statistical mechanics is that, over time, an isolated system in a given macrostate is equally likely to be found in any of its microstates. Most likely case: 29% Even distribution of energy 24% 24% 12% 12% atom 1 atom 2 5

6 The chart for THE 4-quantum macrostate of the 2-atom system q 1 q 2 Ω 1 Ω 2 Ω 1 Ω 2 Each line is about a different energy distribution Total = 126 Number of microstates available for the q 1 =2 energy distribution This is the total number of microstates accessible to the entire 2-atom system

7 Two Big Blocks Interacting Add 100 quanta of energy. Most microstates distribute the energy evenly. Ω = ( q + N ) q ( N ) 1!! 1! q 1 q 2 = 100 q 1 Ω 1 Ω 2 Ω 1 Ω E E E E E E E E E E e E E E+88

8 Two Important Observations 1. The most likely circumstance is that the energy is divided in proportion to the # of oscillators in each object: most likely state has 60% of quanta in block with 60% of the oscillators. 2. As the # of oscillators gets larger, the curve gets narrower relative to its height: relatively smaller fluctuations likely. 300,200 3x10 23, 2x10 23 add more oscillators # Microstates 60% q 1

9 Definition of Entropy Entropy: S k ln Ω where Boltzmann s constant is k = J K It is convenient to deal with ln(ω) instead of Ω for several reasons: 1. Ω is typically huge, and ln(ω) is a smaller number to deal with 2. Ω 12 =Ω 1 Ω 2 (multiplies), whereas lnω 12 =lnω 1 + lnω 2 (adds). Double the size, double the entropy. ( Extensive property )

10 Entropy: S k lnω, where k=1.4x10-23 J / K Note smaller numbers: We will see that in equilibrium, the most probable distribution of energy quanta is that which maximizes the total entropy. (Second Law of Thermodynamics).

11 Entropy: S k lnω, where k=1.4x10-23 J / K Why entropy? k ln(ω 1 Ω 2 ) = k ln(ω 1 ) + k ln(ω 2 ) (property of logarithms) When there is more than one object in a system, we can add entropies just like we can add energies: S total =S 1 +S 2, E total =E 1 +E 2. Remember: Ω = ( q + N ) q! ( N 1 )! 1!

12 Approaching Equilibrium energy distribution A energy distribution B Assume system starts in an energy distribution A far away from equilibrium (far from most likely state). What happens as the system evolves? Crucial point: consider our system to contain both blocks. The macrostate of the system is specified by the total # quanta: 100. All microstates compatible with q total =100 are equally likely, but not all energy distributions (values of q 1 ) are equally likely!

13 Approaching Equilibrium Each square represents microstates. near peak far away from peak

14 Entropy Increased! energy distribution A energy distribution B There are many more microstates compatible with 60/40 than with 90/10. Hence, the entropy S = klnω has increased. This behavior is very general...

15 Second Law of Thermodynamics A mathematically sophisticated treatment of these ideas was developed by Boltzmann in the late 1800 s. The result of such considerations, confirmed by all experiments, is THE SECOND LAW OF THERMODYNAMICS If a closed system is not in equilibrium, the most probable consequence is that the entropy of the system will increase. By most probable we don t mean just slightly more probable, but rather so ridiculously probable that essentially no other possibility is ever observed. Hence, even though it is a statistical statement, we call it the 2 nd Law.

16 Irreversibility An isolated system goes easily from a low entropy to high entropy ( ), but never: from high to low. Each square represents microstates. near peak (high S) far away from peak (low S)

17 Order, Disorder, and the 2 nd Law There are many more ways to distribute the perfume molecules throughout the room than there are ways to distribute them in the bottle. The natural tendency is S, meaning the perfume diffuses. Intuitively, this final, higher entropy state is more disordered than the initial state.

18 Leading Q2: This logarithmic plot shows the number of ways 100 quanta can be distributed between box 1 (300 oscillators) and box 2 (200 oscillators). Look at the end where q 1 =0. If you had to guess, which box is hotter? A. Box 1 (300 oscillators, q 1 =0). B. Box 2 (200 oscillators, q 2 =100).

19 Thermal transfer of energy

20 S = S 1 + S 2 Note: highest entropy S is the most likely macrostate (equilibrium). Remember 2nd Law! ds ds ds dq dq dq 1 2 = + = ds dq ds = 0 (at equilibrium) dq ds dq ds = dq (at equilibrium) At equilibrium, these two slopes are equal. What else is equal here?

21 At equilibrium, these two slopes are equal. What else is equal here? The temperature of the two objects that are in contact and have come to equilibrium. This leads to the definition of temperature in statistical mechanics, Temperature: 1 T S E int Remember: the box is hotter when the slope is smaller. We use E int instead of q 1 to be more general.

22 SUMMARY: 1. We learned to count the number of microstates in a certain macrostate, using the Einstein model of a solid. 2. We defined ENTROPY. 3. We observe that the 2nd Law of thermodynamics requires that interacting systems move toward their most likely macrostate (equilibrium reflects statistics!); explains irreversibility. 4. We defined TEMPERATURE to correspond to the observed behavior of systems moving into equilibrium, using entropy as a guide (plus 2nd law, statistics). Entropy: S k lnω Temperature: 1 T S E int 22