THERMAL PROPERTIERS 1
Thermal Energy & Lattice Vibrations
The amplitude of this vibrational motion increases as the temperature increases
in a solid, the energy associated with these vibrations is called thermal energy
thermal energy plays a fundamental role in determining the Thermal Properties of a Solid
measurable property of a solid is it’s Specific Heat or Heat Capacity
Heat Capacity
The increase in energy, dU of a system= amount of heat absorbed by the system, dQ - amount of work done by the system, dW.
dU = dQ - dW
If the energy of the systems can be described by thermodynamic variables pressure, temperature and volume, then for an infinitesimal quasistatic reversible process
heat capacity of such a system is defined as the amount of heat absorbed by a system per unit change in temperature
classical results
The heat capacity at constant volume, CV as a function of temperature for solid
Cv of free particles
Only have translational kinetic energy
From Boltzmann distribution and calculate average thermal particle energy
This is just the classical result that each particle has an energy of kBT/2 for each degree of freedom
Harmonic Oscillator Potential
potential the atoms in a crystal will oscillate with amplitudes small compared to the internuclear distance.
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
Each of the first and second terms yields kBT/2 proves the contribution from each degree of freedom, momentum or position, to the average thermal energy is kBT/2
For Ν harmonic oscillators in three dimensions the average internal thermal energy is 3NkBT so CV = 3NkB, which for a mole of oscillators is just 3R, the Dulong and Petit value