1. Examples: Reif Show that C P and C V may be related to each other through quantities that may be determined from the equation of state (i.e. by knowing V as a function of P, T, and N) (the result to this question is something you will need for question 2 of assignment number 8).

2. Examples: Reif

3. Lecture 31– EXAM II on Friday Exam will cover chapters 6 through 10 NOTE: we did do a few things outside of the text, particularly around chapter 10: Maxwell Relations Jacobians Exam will start at 13:20, and end at 14:20 (60 minutes) 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 (2 sides) of your own creation. I will provide mathematical formulas you may need (e.g. summation result from the zipper problem, Taylor Expansions etc., Stirling’s approximation, certain definite integrals). Extra Office Hours: Thursday 3:30 to 5:00 Friday: 11:00 to 11:45

4. Review EXAM II Chapter 6: Converting sums to integrals (Density of States) for massive and massless particles Photon and Phonon Gases Debye and Planck models Occupation number for bosons Specific heat associated with atomic vibrations (Debye model) (note: this ignores Zero-point motion) Debye model for solids

5. Review EXAM II Chapter 7: Helmholtz free energy The exchange of particles (Chemical Potential) Gas columns, adsorbed layers are examples, but there are many others.     when equilibrium is established btn. A and B

6. Review EXAM II Chapter 8: Quantum gases and corrections to ideal gas law from “statistical” correlations. The occupation number formulation of many body systems Bose-Einstein and Fermi-Dirac Statistics and their occupation numbers (For non-relativistic particles with finite mass; a generalization of what we saw in chpt. 4)

7. Review EXAM II Chapter 9: Degenerate Quantum gases. The occupation number formulation of many body systems. Applications of degenerate Fermi systems (metals, White Dwarves, Neutron Stars) Physical meaning of the Fermi Energy (temperature) and Bose Temperature Bose-Einstein Condensation Temperature dependence of the chemical potential Fermion ideal gas Boson ideal gas T < T B Only for T<<T F

8. Review EXAM II Chapter 10: Natural Variables and the “fundamental relation” Thermodynamic potentials and manipulations Legendre Transformations (see next slide) Jacobians and their manipulations (see slide after next). Maxwell Relations Keep in mind that such relations may be derived for systems where work involves something other than PdV, and they come from equating the second-order mixed partial derivatives of one of the four major thermodynamic functions (E, H, F, G) [I copied the above formulae from Wikipedia, where A is used for the Helmholtz free energy). See Week 10 notes!!

9. Key Definitions: E=E(S,V,N) Internal energy (fundamental relation) H=H(S, P, N) = E + PV (Enthalpy) F=F(T, V, N) = E - TS (Helmholtz Free Energy) G=G(T, P, N) = E + PV –TS (Gibbs Free Energy) For hydro-static systems (volume the only external parameter). dE = TdS – PdV +  dN dH = TdS +VdP +  dN dF = -SdT –PdV +  dN dG = -SdT + VdP +  dN

10. Examples:

12. What is the fundamental difference between the thermodynamics of a gas of photons (the Planck radiation law and associated physics) and the thermodynamics of latttice vibrations in a solid (Debye model)? A 50 mW Ar Ion laser ( =488nm, beam diameter 5 mm) is directed toward a black sphere of radius R=1.0 cm in outer space. What will be the equilibrium temperature of the sphere? Can you neglect the microwave background?