M I N D M A P 4
THERMAL PROPERTIES I
THERMAL PROPERTIES II
HEAT CAPACITY
CLASSICAL RESULTS
DULONG & PETIT LAW
Cv OF FREE PARTICLES
HARMONIC OSCILLATOR POTENTIAL
the amount of heat absorbed by a system per unit change in temperature 
at constant volume, 
at constant pressure, 
Volume thermal expansion coefficient, 
Volume compressibility of a solid, 
most solids in general Cp > Cv
but when T -> 0, Cp -> Cv
Heat capacity at constant constant volume vs temperature for solid 
At room temperature, the value of the heat capacity is 3Nk = 3R per mole = 25 J/mol.deg
At lower temperatures the heat capacity drops markedly and approaches zero as T^3
from Boltzmann's distribution 
if N free particles in 3D, 
energy of the oscillator, 
average thermal energy, 
for N harmonic oscillator in 3D, 
This agrees with the Dulong-Petit law at high temperature, but fails at low temperature
EINSTEIN'S MODEL (1907)
DEBYE'S MODEL (1912)
assumed that the atoms are vibrating as harmonic oscillators, but instead of taking the classical expression for the energy of an oscillator
assumed that Planck' constant (1990) quantization rule for each oscillator in a solid have quantized energies
high T limit, 
low T limit, 
heat capacity equation, 
Einstein model is deficient at very low T
solid treated as continuous elastic medium
relations between the energy of phonon, E, the angular frequency, w and the wave vector, q 
Debye wavelength, 
Debye temperature, 
heat capacity, 
at high temperature limit, T >> TD 
at low temperature limit, T << TD 
describes the temperature dependence of the heat capacity not only at high temperatures but also at low temperatures
Debye model work better at low T than the Einstein model because Debye model gives a better representation for the very low energy vibrations