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THERMAL PROPERTIES - Coggle Diagram
THERMAL PROPERTIES
From the first law of thermodynamics (conservation of energy),dU = dQ - dWenergy of the system as described by thermodynamics variables, 
Heat capacity (a reversible infinitesimal process)at constant volume, where theoretical values are calculated with the usage of internuclear distances.at constant pressure, where experimental measures are done for solids.
where α is the thermal expansion coefficient and ϗ is the volume compressibility of a solid.
in most solids,
Cp > Cvbut when T → 0,
Cp → Cvfor insulators,
Cv drops as T³ is reached.at room temperature,
Cv = 3NK / mole = 29.94 J/mole
Cv of free particles,:star: only have translational KE :star: total energy for N molecule,
From Boltzmann's distribution,
average thermal particle energy, with then for each degree of freedom, thus, Cv = (3/2)R = 12.6 J/mole
(half of Dulong Petit value)
Harmonic oscillator potential,:star: energy of oscillator, where the first term is for KE and the second term is for PE.:star: average thermal energy, thus, Cv = 3NK = 3R
(Dulong Petit Law) at high T:star: failure to explain Cv at low T
EINSTEIN'S MODEL (1907)
Assumptions ::star: atoms vibrate as harmonic oscillators
:star: uses Planck's quantization rule for each oscillation where 0 ≤ n ≤ ∞
may be wrong to assume,
:star: 3N independent oscillators at frequency w.
:star: the use of Planck's quantization rule
occupation of energy level n, average total energy of solid, from both equations,
average total energy of solid is
since with Einstein Temperature, this yields, which is used to predict the limiting behaviour of Cv
High T limit where, θ/T << 1,Cv ≈ 3RLow T limit where θ/T >> 1, Cv yields, which is considered as an exponential falloff since the decline is more rapid than T³
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DEBYE'S MODEL (1912)
Assumptions:
:star: atoms are considered as harmonic oscillators producing elastic waves with frequencies of 0 ≤ w ≤ w(max)
:star: solid is a continuous elastic medium
:star: 3N normal modes
Changes in Cv caused by assumptions::star: each modes contribute to frequency-dependent C
:star: C has to be integrated over all w
:star: energy of phonon, with angular frequency,
volume for each allowed wavevector, all modes are confined to the sphere with cutoff wavevector, thus, the number of modes will be, all these equations will then define the Debye Temperature,
For the derivation for specific heat, we usedensity of state, Dw, average phonon energy at w(n), with these equations, the thermal energy becomes, then assuming that speed of phonon is independent of polarization and where, the thermal energy yield,
thus,
at high T where T>>T(D);Cv = 3Rat low T where T<<T(D);the value of Cv is,
where number of atoms, z = 1 and R is the gas constant.:star: known as the Debye approximation
:star: this is a better representation for low energy levels vibrations
:star: only long wavelength will be thermally excited.