Lect08 - 213 Midte com up rm ing Monday November 8, 7 pm...

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Lecture 8, p 1 213 Midterm coming up… Monday November 8, 7 pm (conflict exam at 5:15 pm) Covers: Lectures 1-10 + superficial 11 HW 1-4 Discussion 1-4 Labs 1-2 Review Session Sun. November 7, 3-5 PM, 228 Nat. Hist. Bldg.
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Lecture 8, p 2 Lecture 8 The Second Law of Thermodynamics; Energy Exchange The second law of thermodynamics Statistics of energy exchange General definition of temperature Why heat flows from hot to cold Reading for this Lecture: Elements Ch 7 Reading for Lecture 10: Elements Ch 8
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Lecture 8, p 3 Counting microstates of combined systems Volume exchange between systems Definition of Entropy and its role in equilibrium Last Time
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Lecture 8, p 4 Quick Review: The Scope of Thermodynamics When an isolated system can explore some number, , of microstates the microstates are equally likely . The probability that you observe macrostate A is: P A = Ω A / Ω All , i.e. , the fraction of all the microstates that look like A. Entropy: The entropy of a system is ln( Ω ) , where counts all possible states. Therefore, P A is proportional to e σ A , since σ A = ln( Ω A ). For a big system, the entropy of the most likely macrostate, σ A , is not much less than the total entropy, σ All . Example: 10 6 spins: σ All = ln(2 10 6 ) = 10 6 ln(2) = 693147 σ 5 × 10 5 up = ln(10 6 ! / 5 × 10 5 ! 5 × 10 5 !) = 693140 Thermodynamics applies to systems that are big enough so that this is true. Question: What is the probability that in a system of 10 spins, exactly 5 × 10 5 will be pointing up?
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Lecture 8, p 5 0 d dt σ 0 d dt σ The Entropy of an isolated system can only increase (or remain constant) as a function of time. The Second Law of Thermodynamics This is a consequence of probability. Macroscopic systems have very sharply peaked probability distributions (much sharper than shown here). If x is initially far from its most likely position, x e , then it will evolve towards x e (where most of the microstates are). If x is initially near its most likely position, x e , then it will remain there. σ has its maximum value. All available microstates are equally likely, but in big systems the vast majority of them correspond to very similar macrostates ( e.g. , about 50% spin up). Macroscopic quantity, x (e.g., position of partition) x e Ω (or 29 σ 0 d dt σ =
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Lecture 8, p 6 Lessons from Volume Exchange You learned last lecture: When the system consists of independent parts: The number of states of the whole was the product of the number of states of the parts. We define entropy to be the ln (# microstates). For a big system in equilibrium we almost certainly see the macrostate that maximizes TOT (or σ TOT ). To determine equilibrium, maximize the total entropy
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Lect08 - 213 Midte com up rm ing Monday November 8, 7 pm...

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