Lecture 15-- CALM What would you most like me to discuss tomorrow in preparation for the upcoming exam? proton)? Partition Function/Canonical Ensemble (3) Efficiency/Engine Cycles (3) Examples (3) Taylor expansions, looking at limits (2) Adiabatic processes(1) Density of States (1) Multiplicity (1) Lots of other little things (but not typically requested by >1 person). External parameters Boltzmann factor Spin

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 = Nk B [ln(V/(N Th 3 )) + 5/2]

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. Taylor expansions, series summations, certain definite integrals, etc.) as well as the values of universal constants (such as h, and k B ). OFFICE HOURS: Tuesday 1:00 – 2:00 4: :00 Wednesday: ~10:00 – 11:30 Exam will start at 1:20, end at 2:20

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? What is the partition function in the other limit T<<  r Derive a closed-form expression for the canonical partition function for such a molecule in the limit where T>>  r (Hint: in this limit you can directly replace the sum by an integral over J). 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?

Examples Consider the thermodynamic cycle that appears as a rectangle on a P-V diagram. Call the lower and upper values of the Pressure P1 and P2 respectively, and the small and larger volumes V1 and V2 respectively. a)What is the work done in this cycle (in terms of these four parameters)? b)Draw the diagram, and indicate on it the sections where the engine absorbs heat, where it releases heat, and where no heat is exchanged (if any of the three conditions actually take place). c)Write the efficiency in terms of the four above parameters and the two specific heat capacities (c p and c v, both of which you may take to be constant).