Examples: ReifShow that CP and CV 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)

Note, what Reif labels a, is what we have already introduced as bP.

Examples: Reif

Iis the work you calculate in c equal in magnitude to the change in the Internal energy of the film? If not why, why not?d) Calculate the change in the free energy. How is this related to the work done?

Mountain windsWhen wind blows over a range of mountains onto the plains below, if the flow is not too turbulent, you can consider the process to be one that occurs without heat exchange with the ground or the rest of the atmosphere. Assuming also that the process is quasistatic, determine the relevant partial derivative to describe this process and show that the air warms up as a result (you can treat the air as a diatomic ideal gas).This process plays an important role in the famous Santa Ana winds of California, the Chinook’s of western North America, the Zonda winds of Argentina and many others.

Let’s think about this homework problem for a minute or two:

Examples: Reif

Heat Capacity of diatomic gases

http://www.phys.unsw.edu.au/COURSES/FIRST_YEAR/pdf%20files/x.%20Equipartion.pdf

Note: the temperature scale is hypothetical

Internal degrees of Freedom(diatomic molecules)

Interaction between spin and rotation for homonuclear mol.

See the following applet to see the effect of nuclear statistics on the heat capacity of hydrogen:http://demonstrations.wolfram.com/LowTemperatureHeatCapacityOfHydrogenMolecules/

Low-Temperature CV of Metals(review of exam II)

From:T. W. Tsang et al. Phys. Rev. B31, 235 (1985)

Lecture XX– EXAM II on Monday• Exam will cover chapters 6 through 10.8 (NOTE earlier I had thought it

would include some of chpt. 11, but I decided to drop that question).• 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 3 questions some with multiple parts. • Total number of “parts will be 7 or so.• 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).

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• Black body radiation• Specific heat associated with atomic vibrations

(note: this ignores Zero-point motion)(also note connection to chpt. 8)

Debye model for solids

Pwr=AsBT4

(For T<<qD)

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.

m A =mB

when equilibrium is established btn. A and B

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)

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 << TB

Only for T<<TF

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!!

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 + mdNdH = TdS +VdP + mdNdF = -SdT –PdV + mdNdG = -SdT + VdP + mdN

Examples: • What is the fundamental difference between the

thermodynamics of a gas of photons (the Planck radiation law and associated physics) and the thermodynamics of lattice vibrations in a solid (Debye model)?

• A 50 mW Ar Ion laser (l=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? [sB=5.67x10-8 W/m2.K4]

Examples:

An important industrial process is one in which gas at high pressure is allowed to expand through a flow restriction (porous plug, or partially closed valve) to a lower pressure. This process, called the Joule-Thomson expansion, is used to liquify gases, takes place at constant Enthalpy. Derive an expression for the relevant thermodynamic derivative in terms of a heat capacity and quantities derivable from the equation of state.

Example 11.9 from Baierlein• A gas of the HBr is in thermal equilibrium. At what temperature will the

population of molecules with J=3 be equal to the population with J=2? (NOTE: HBr has Qr=12.2K. )