PHYS 172: Modern Mechanics Lecture 22 – Entropy and Temperature Read Summer 2012

Clicker Question: Consider the energy principle, ΔE system =W+Q. Choose the correct statement: A.If Q is added to the system, the system temperature must increase. B.If Q is added, the temperature must increase if W=0. C.If Q is added, the temperature might not change at all. D.None of the above.

First, an important reminder about multiplying probabilities: A deck of 52 cards has 4 aces. Starting from a randomly- shuffled deck, what is the probability that an ace is the first card dealt? Once this happens, what is the probability that the second card is an ace? So, what is the probability that the first two cards are aces?

Last Time Einstein Model of Solids # microstates (N oscillators, q quanta) 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).

These show the 15 different microstates where 4 quanta of energy are 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.

q1q1 q2q2 Ω1Ω1 Ω2Ω2 Ω 1 Ω atom 1 (3 oscillators) atom 2 (3 oscillators) Macrostate = 4 quanta of energy in 2 atoms “Microstates” = q 1 quanta in atom 1, q 2 quanta in atom 2

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 it’s microstates. Thus, our system of 2 atoms is most likely to be in a microstate where energy is split up 50/50. atom 1atom 2 29% 12% 24% 12% 24%

Interacting Blocks q1q1 q 2 = 100–q 1 Ω1Ω1 Ω2Ω2 Ω 1 Ω E E E E E E E E E E e E E E+88 …………… As quanta are transferred from one block to another, the system transitions from one macrostate to another. Each of these macrostates has a different number of microstates Ω 1 Ω 2.

more oscillators Two Important Observations 1.the most likely circumstance is that the energy divided in proportion to the # of oscillators in each object: 2.as the # of oscillators get large, only one such circumstance is observed: the curve gets narrower: most likely state has 60% of quanta in block with 60% of the oscillators. more oscillators

Probability of macrostates becomes extremely narrow as the numbers increase! Thermal equilibrium of two blocks in contact: quanta

Definition of Entropy It is convenient to deal with lnΩ instead of Ω for several reasons: 1.Ω is typically huge, and ln(Ω) is a smaller number to deal with 2.dΩ/dE is usually huge, and is easier to deal with. 3.Ω 12 =Ω 1 Ω 2 (multiplies), whereas lnΩ 12 =lnΩ 1 + lnΩ 2 (adds). This will make the connection between entropy and temperature easier to see later…. where Boltzmann’s constant is entropy:

We will see that in equilibrium, the most probable distribution of energy quanta is that which maximizes the total entropy. (Second Law of Thermodynamics). Note smaller numbers:

Why entropy? (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:

Approaching Equilibrium assume system starts in a macrostate A far away from equilibrium (most likely state). Ω A is “small” and hence S A = k·lnΩ A is small. what happens as the system evolves? macrostate Amacrostate B

If the initial state is not the most probable, energy is exchanged until the most probable distribution is reached. NOTE: This is a random, statistical process, consistent with the fundamental assumption of statistical mechanics that all microstates are equally likely.

Approaching Equilibrium assume system starts in a macrostate A far away from equillibrium. Ω A is “small” and hence S A = k·lnΩ A is small. as the system evolves (moves from one microstate to the next), it is MUCH more likely to end up in macrostate B, where Ω B is “large” and hence S B = k·lnΩ B is large. from a purely statistical view, we expect that the entropy of the system will increase. macrostate Amacrostate B Note: The number of microstates for is MUCH larger than the number of microstates for

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” and not the “2 nd Likelihood”.

Lecture question: 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).

Note: highest entropy S is the most likely macrostate (equilibrium). Remember 2nd Law! 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 a natural guess for a useful definition of temperature: Remember: the box is hotter when the slope is smaller. We use E int instead of q 1 to be more general.

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).