1. Redo in class Cp and CV. (handout)
2. Probability Ps of being in energy eigenstate s.
3. Maximize F=-kBΣ Ps ln(Ps), with boundary conditions.
4. Lagrange multipliers. Class question, what are they? Example?
5. Canonical ensemble, Boltzmann factor.
6. Normalization: partition function.

• Diagnostic: State the second law of thermodynamics.
• Find maxima of x^4+y^4+z^4 on x^2+y^2=C.
• Minima of <ψ|H|ψ> with <ψ|ψ>=1.

Students did not know what to do with 0 ln(0). Needed explanation in terms of limits. There was the need to spend extra time on minimization with constraints. The formal procedure was explained using a 3d example, minimize f(x) with the extra condition g(x)=C. The target function was extended using a Lagrange multiplier, F(λ, x)= f(x) - λ ( g(x)-C). Minimizing over all λ and x now gives us the desired result. Class worked on the maxima of x^4+y^4+z^4 problem. The equations looked like 4x^3=λ2x, and evere group threw out the x=0 solution, which then only gives the minimum.

1. Maximize fairness with U. Canonical ensemble, Boltzmann factor.
2. Partition function is normalization.
3. Entropy is value of Fmax. Entropy analogue.
4. Consistent with thermodynamics.
   • Pn is Boltzmann factor.
   • Lambda1=kB beta.
   • U = - d/dbeta ln Z.
   • S = Fmax= kB beta U +kB ln Z.
   • U = -d/dbeta (S/kB – beta U) → 0=-d/dbeta(S/kB) + beta d/dbeta U.
   • d/dbeta U = 1/(kB beta) d/dbeta S.
   • Contstant V,N: En does not change. beta=f(T).
   • d/dT U = 1/(kB beta) d/dT S = T d/dT S.
5. What is the second law?
6. Equilibrium value. What if not in equilibrium?

• Diagnostic question: What is the meaning of the word “free” in the Helmholtz free energy?
• Three level system, -eps,0 (double degenerate), eps find Z, U, S.

Notation with nested sums is an issue. Σn e^epsn/Z with Z=Σj e^epsj, keep indices separate. The Boltzmann factor was not well known.

1. TD: Helmholtz free energy, F=U-TS. First law form. What does it mean?
2. Note: need three terms in first law, always add μdN.
3. SM: S = Fmax= kB beta U +kB ln Z hence beta F = - ln (Z).
4. New Maxwell relations.
5. First law forms in general, Laplace transformations.
6. Why different potentials? What is free?

• Diagnostic question: List all basic assumptions implied by an ideal gas.
• Meaning of enthalpy, H=U+pV. Solids! Cp. Throttling example.

Take more time for throttling work, needs better guidance. What is enthalpy? Measures amount of chemical work that can be done adiabatically at constant pressure, but also amount of heat needed at constant pressure.