Day 13: Heat capacity of solids, general

  1. Definition of heat capacity, CV versus dU/dT and CP versus dH/dT
  2. How large are heat capacities, e.g. water.
  3. Ice to steam Q versus time, includes latent heat
Problems in class
  • Ingot in water example, calculate Cp metal
  • Give table of Al specific heats. What conclusions can you draw.
  • C for Al:
T (K) C (J/K/mole)
20 0.23
40 2.09
60 5.77
80 9.65
100 13.04
150 18.52
200 21.58
250 23.25
300 24.32
400 25.61

Gas constant R=8.31 J/K/mole


The idea of a calorie came out, and also that one needs to distinguish between Cp and Cv. At which temperature was a good question. One calorie is from 14.5 to 15.5 C (sometimes called a 15 degree calorie). The 4.2 J/cal was known. C for water per mole and per kg. Water has 18 grams per mole. Given the C versus T plot, students did guess the low and high temperature behavior (they read the notes, and knew about T3). They did not know how to show it, though. So we spent time discovering that we need to plot C versus 1/T to get the high T limit, and log C versus log T for the low T limit. Next, we discussed models we had seen. Ideal gas has right high T behavior with double the number of particles. Two level system has correct low T behavior, but not high T. Homework problem 4 looks best overall, a system of harmonic oscillators. So we assume that a solid is equivalent to a system of harmonic oscillators, with coupled modes like we did in ph435. We did not try to get an interpolating formula. I then mentioned that all solids show a similar behavior, with different constants, so the general model works for all.

Day 14: Einstein model

  1. Potential energy: independent atoms, not bad but not correct
  2. Partition function 3D, notes complicated for 1D
  3. F, S, U, CV
  4. Why do we have CV? Omega0 not changing!
  5. Einstein temperature.
  6. High temperature behavior OK, not low T.
  7. Physics: only one parameter!
  8. Melting criterium???
Problems in class
  • Calculate sum_n x^n
  • Calculate sum_nm x^n y^m
  • ½ M omega^2 ( 1/2 a)^2 = ½ k_B T

The derivation in the notes for the 1D case is not needed, since we can argue that we have Z=Z1^(3N). Z1 was done as a homework (apart from the zero point energy). It is useful to connect the model to new predictions, in this case the melting point.

Day 15: Debye model

  1. Ideal gas: stat mech plus atomic model
  2. Heat capacity solid: Einstein, stat mech plus atoms plus quantization
  3. Oscillations are coupled, normal modes. Omega = Vs times k
  4. Modes have to be quantized
  5. Number of modes at frequency omega D(omega) d omega
  6. Derive D(omega)
  7. Gives U(T), CV(T). Volume is now directly there.
  8. Debye temperature.
  9. Low T, T3 law indeed.
Problems in class
  • What are periodic boundary conditions?
  • Shell in omega space versus k space?
  • Why OmegaDebye? Keep 3N modes.
  • Since Einstein and Debye are so simlar, why bother? Because we are often at low T!

The idea of coupled modes was there, and also the idea of adding the contributions of different frequencies. Using the Einstein result was understood, but it needed some discussion how to add results. I am not sure how effective the derivation of a density of states was. Final results were OK.

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