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. 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).

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

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: PV  =const. etc. Internal Energy of a monatomic ideal gas: E=3/2Nk B T DISTINGUISH between ideal gas results and generalized results. Chapter 2 Micro-states and the second law. Entropy, S=k B ln(  ) the tendency toward maximum entropy for isolated systems.* Probability Distributions (Poisson, Binomial), computing weighted averages 1/N 1/2 distributions tend to get much sharper when averaged over many more instances or involving many more particles.* For a reservoir:  S=Q/T* For a finite system dS= dQ/T*

Lecture 15-- Review Chapter 3  = (dln(  )/dE) N,V = 1/k B T* Efficiency of heat engines* Reversibility (  S univ =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/k B T) Canonical partition function: Z =  i exp(-E i /k B T) =k B T 2 (dlnZ/dT) N,V = k B T 2 (dlnZ/dV) N,T S= k B ln(Z) + /T Z N =(Z 1 ) N for distinguishable particles Z N =(Z 1 ) N /N! for indistinguishable particles (these are semi-classical results which we will refine later). For a monatomic ideal gas: Z 1 =V/ Th 3 where Th =(h 2 /2  mk B T) 1/2 S = NkB [ln(V/(N Th 3 )) + 5/2] Extensive vs. Intensive quantities

Examples Simple model for rotations of diatomic molecule (say CO). A quantum rigid rotator has energy levels E J = J(J+1)k B  r with degeneracy g J =2J+1, where J is an integer. Take  r =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>>  r Suppose we have a solid whose heat capacity is given by the equation: C p = 3R (T/  ) 3, where  =300K and R=N A k B =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?