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Unformatted text preview: Lecture 213 Midterm coming up… 213 Midterm coming up… Monday November 8 @ 7 pm (conflict exam @ 5:15pm) Covers: Lectures 110 + ½ of 11 HW 14 Discussion 14 Labs 12 Review Session Sunday November 7, 35 PM, 228 NHB Lecture Lecture 10 The Boltzmann Distribution • Concept of a thermal reservoir • The Boltzmann distribution • Paramagnetic Spins – MRI • Elasticity of a Polymer • Supplement: Proof of equipartition, showing its limits Reading for this Lecture: Elements Ch 8 Reading for Lecture 11: Elements Ch 9 Lecture Some Questions We’d Like to Answer These questions involve the interaction between a small system (atom or molecule) and a much larger system (the environment). This is a basic problem in statistical mechanics. What is the range of kinetic energies of an O 2 molecule? Under what conditions does O 2 break up into two O atoms? What is the probability that a DNA molecule will unfold and replicate? What is the capacity of a myoglobin molecule to carry oxygen to the muscles? What is the vapor pressure of a solid or liquid? When do molecular vibrations become important? Vitruvian Man, 1490, by Leonardo da Vinci Lecture Averages from Probabilities IF: 1 You could list every quantum state of some small system. This is realistic for small objects ( e.g. , oscillators or atoms) 2 And you knew the properties of each state Energy, magnetic moment, optical density, etc . 3 And you knew the probability of each state (P 1 , P 2 , ...P n …) THEN: You could calculate the average energy, magnetic moment, optical density, etc. for each part . For example, <E> = P 1 E 1 + P 2 E 2 + ... P n E n … Now if you have a big system, made up of simple little parts, to get <E>, <m>, etc. for the big system, just add all the parts! We can figure out how things behave, starting from scratch. The key step is 3 : T he Boltzmann factor tells us the probabilities. Lecture Concept of a Thermal Reservoir We will be considering situations like this: The two systems can exchange energy, volume, particles, etc. If the large system is much larger than the small one, then its temperature will not be significantly affected by the interaction. We define a thermal reservoir to be a system that is large enough so that its T is constant. The reservoir doesn’t have to be very large, just a lot larger than the small system. A large system A small system Note: The systems do not have to be collections of oscillators (with equally spaced energy levels). We only assume that to simplify the math. Lecture Thermal Reservoir Let’s start with a reservoir that isn’t very large. That makes the problem easier to solve. Consider one oscillator in thermal contact with a system of three oscillators: Total energy: U = E n + U R Ω R = # reservoir microstates Question : What is the probability that the small system has energy E n ?...
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 Spring '08
 Staff
 Energy, Statistical Mechanics, Entropy, Boltzmann Distribution, minimum entropy state

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