Thermal Properties

Thermal Properties I

Thermal Energy & Lattice Vibration

Thermal Energy?

  • fundamental to obtain an understanding of basic properties of solids
  • fundamental role in determining the Thermal Properties of a Solid

Heat Capacity

  • first law of thermodynamics
  • Increase energy, dU of a system equal to amount of heat absorbed by the system, dQ, minus amount work dine by the system

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Definition:

  • amount of heat absorbed by system per unit charge in temperature

Constant volume
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Constant pressure
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Classical Result

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  • room temperature, value of heat capacity is 3Nk=3R per mole = 25 J mol-1 deg-1
  • Dulong and Petit law
  • lower temperature, heat capacity drops markedly ad approach zero, T^3

Cv free particles

  • Translational kinetic energy
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  • Total energy N molecule
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Boltzmann distribution
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Harmonic oscillator potential

Energy oscillator
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Average thermal energy
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  • harmonic oscillators in three dimensions the average internal thermal energy
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  • independent temperature and classical physic
    result agress with Dulong-Petit law at high temperature

Thermal Properties II

Einstein Model

  • assumed atoms a vibrating as harmonic oscillators
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  • assumed Planck quantization rule for each oscillators
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Average total energy of solid
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Using Planck constant:
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The prediction:
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Debye Model

Limiting Behavior Cv(T)

High T limit
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Low T limit
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These prediction:

  • Cv: 3R for large T
  • Cv: 0 as T
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Developed more sophisticated treatment of atomic vibration solids:

  • atom consider as harmonic oscillators that produce elastic waves varying frequency
  • treat solid as continuous elastic medium
  • 3N normal modes oscillations

Changes of expression for Cv because each mode of oscillation contributes

Debye temperature:
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Heat Capacity:
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At high Temperature

  • x is very small throughout the range of the intergral

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  • The classical Dulong-Petit result

At low temperature

  • only long wavelength acoustic modes are thermally excited
  • the energy short wavelength modes are too high to be populated significant
  • approximately Xd=Td/T

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Why Debye model better at low temperature than Einstein model?

  • gives better representation for the very low energy vibrations
  • vibrations matter most

Limits of the Debye model:
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