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Lecture 15– EXAM I on Wed.

Lecture 15– EXAM I on Wed. Exam will cover chapters 1 through 5 NOTE: we did do a few things outside of the text: Binomial Distribution, Poisson Distr. (really 1/N 1/2 ) Thermometry Exam will have 4 questions some with multiple parts.

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Lecture 15– EXAM I on Wed.

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  1. Lecture 15– EXAM I on Wed. • Exam will cover chapters 1 through 5 • NOTE: we did do a few things outside of the text: • Binomial Distribution, Poisson Distr. (really 1/N1/2) • Thermometry • Exam will have 4 questions some with multiple parts. • Total number of “parts will be on the order of 8 or 9. • Most will be worth 10 points, a few will be worth 5. • You are allowed one formula sheet of your own creation. • I will provide mathematical formulas you may need (e.g. summation result from the zipper problem, Taylor Expansions etc., certain definite integrals).

  2. Lecture 15-- CALM • What would you most like me to discuss tomorrow in preparation for the upcoming exam? proton)? • Density of States (4) • Partition Function/Canonical Ensemble (3) • Examples (3) • Lots of other little things (but not typically requested by >1 person). • Averages: when to use what weights • Macro vs. micro states • What are the KEY concepts

  3. Lecture 15– Review • Chapter 1 • Describing thermodynamic systems • The definition of temperature (also thermometers in chpt. 4). • The Ideal Gas Law • Definition of Heat Capacity and Specific Heat * • The importance of imposed conditions (constant V, constant P, adiabatic etc.) • Adiabatic equation of State for an ideal gas: PVg=const. etc. • Internal Energy of a monatomic ideal gas: E=3/2NkBT • DISTINGUISH between ideal gas results and generalized results. • Chapter 2 • Micro-states and the second law. • Entropy, S=kBln(W) the tendency toward maximum entropy for isolated systems.* • Probability Distributions • (Poisson, Binomial), • computing weighted averages • 1/N1/2 distributions tend to get much sharper when averaged over many more instances or involving many more particles.* • For a reservoir: DS=Q/T* • For a finite system dS= dQ/T*

  4. Lecture 15-- Review • Chapter 3 • b= (dln(W)/dE)N,V = 1/kBT* • Efficiency of heat engines* • Reversibility (DSuniv=0) and the maximum efficiency of heat engines* • Chapter 4 • Sums over states can be recast as integrals over energy weighted by the Density of States • Types of thermometers and the ITS-90 • Chapter 5 : Systems at constant temperature* (i.e. everything here has a *). • Boltzmann Factor prob.~ exp(-E/kBT) • Canonical partition function: Z = Si exp(-Ei/kBT) • <E>=kBT2(dlnZ/dT)N,V • <P>= kBT2(dlnZ/dV)N,T • S= kBln(Z) + <E>/T • ZN=(Z1)N for distinguishable particles ZN=(Z1)N/N! for indistinguishable particles (these are semi-classical results which we will refine later). • For a monatomic ideal gas: • Z1=V/lTh3 where lTh=(h2/2pmkBT)1/2 • S = NkB [ln(V/(NlTh3)) + 5/2] • Extensive vs. Intensive quantities

  5. Examples • Simple model for rotations of diatomic molecule (say CO). A quantum rigid rotator has energy levels EJ = J(J+1)kBQr with degeneracy gJ=2J+1, where J is an integer. Take Qr=2.77K • At what temperature would the same number of such molecules be in each of the first two energy levels? • Derive a closed-form expression for the canonical partition function for such a molecule in the limit where T>>Qr • Suppose we have a solid whose heat capacity is given by the equation: • Cp= 3R (T/Q)3, where Q=300K and R=NAkB=8.314J/K. • How much energy is required to heat this solid from 10K to 300K? • What is the entropy change associated with that heat exchange?

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