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MINDMAP 4 - Coggle Diagram

- - - - first law of thermodynamics
        
        At constant volume
        
        At Constant Pressure
        
        Relation between Cp and Cv
        
        volume thermal expansion coefficient
        
        volume compressibility
        
        The experimental
        
        measurements of specific heats for solids are normally done at
        constant(atm) pressure, Cp. However calculates Cv
        since constant internuclear distances are used.
        
        Thermal Energy
        
        Kinetic Energy + Potential Energy of distortion of interatomic bonds
        
        Temperature increases
        
        The mean atomic velocity increases
        
        Amplitude of atomic vibrations increases
        
        Thermal energy increases
  - - - At high temperature, Cv is independent with temperature
        
        At low temperature, in insulator, the heat capacity drops and approaches zero as T^3
        
        Otherwise in conductors, the heat capacity will be constant
        
        At room temperature, 3Nk = 3R per mole = 25 J mol−1 deg−1. (DULONG-PETIT LAW)
  - - - Total energy for N molecule
    - - t2 = ½ mv2 kBT
        
        the integrals in the number and denominator will cancel each other
        
        for each degree of freedom, each particle has an energy of kBT/2
        
        1 more item...
  - - - where the first term on the right-hand side is the kinetic energy, involving the momentum p and mass m, and the second term is the potential energy, involving the displacement u and the force constant α.
    - - For N harmonic oscillators in 3D, the average internal thermal energy is 3NkBT Cv = 3NkB but we know that the Dulong and Petit value for a mole of oscillators is just 3R.
        
        Summary: Cv = = 3NkB = 3R, Independent of temperature, and classical physics offers no possibility of a temperature dependence. This result agrees with the Dulong and Petit Law at only high temperature but it failed to explain the Cv at low temperature.
- - - - <nq> is the thermal equilibrium occupancy of phonons of wavevector q which is given by Bose-Einstein distribution.
  - - - he assumed Planck’s (1990) quantization rule for each oscillator (vibrating oscillators (atoms) in a solid have quantized energies)
    - - The derivation of energy we obtained...
        
        after differentiating and substitute with Einstein temperature...
    - - At high temperature...
      - At low temperature...
    - - z = the number of atom in a unit formula
        
        So, CV approaches 0 when T appreaches 0 is true
  - - - The Debye model successfully describes the temperature dependence of the heat capacity not only at high temperatures but also at low temperatures
        
        T3 Debye law is known for the contribution of the phonon to the heat capacity at low temperatures
    - - Since there is a cut-off wave vector qD=ωD/vs, all the modes are confined within a sphere with radius qD.
    - - the velocity of sound vs is taken as constant for each polarization type, as it would be for a classical elastic continuum
        
        Replacing D, thus thermal energy for each polarization type is given by
        
        1 more item...
    - - experimental results of specific heats of four substances if they are scaled to T/TD
        
        look very similar and are very close to the Debye theory.
    - - value of x is very small throughout the range of the integral. The classical value of Dulong-Petit obtained
    - - Only long wavelength acoustic modes are thermally excited
        
        The energy of those short wavelength modes are too high to be populated significantly at low temperatures
    - - The approximation of isotropic continuum in the Debye model is appropriate for the phonos in the acoustic branch and the single frequency in the Einstein model suitable for the phonons in the optical branch