Physics Chapter 4: Thermal Properties

Chapter 4: Various Thermal Properties

• Specific Heat: Specific heat (C) is the amount of heat required to raise the temperature of one mole of a substance by one degree at constant volume (Cv). Change in internal energy (DE) is equal to DQ (amount of heat) at constant volume.
• Experimental Observation: Dulong-Petit Law states a relationship between specific heat and the atomic weight of a solid; at high temperatures (C = 3R)

Models of Specific Heat

• Classical Model: Assumes that atoms in a solid vibrate harmonically around fixed positions when heated. The energy of one oscillator (in 1 dimension) is equal to kT. The energy of a solid consisting of one mole of N oscillators in 3 dimensions is E = 3NkT = 3RT. Specific heat, Cv, is constant (3R) and independent of temperature.

• Einstein Model: Treats atoms as independent oscillators. Explains specific heat behavior at high temperatures but fails at low. The average energy of an oscillator in 3 Dimensions is E = 3ħω/(e^(ħω/KT) - 1). Specific heat Cv depends on temperature.

• Debye Model: Considers interactions between oscillators, providing a more accurate description of specific heat behavior, especially at low temperatures. Expressed by the average energy of one oscillator E = 3ħω/(e^(ħω/KT) - 1) which ultimately relates to frequency.

High and Low Temperatures

• High Temperature: Experimental specific heat values match the classical model predictions. Cv = 3R
• Low Temperature: Specific heat approaches zero asymptotically for the Debye model, diverging from predictions given by classical models. Cv = 3R (T/ΘD)³

Chapter 5: Electron Theory

• Free Electrons: Electrons inside a metal sample move freely, except at the boundaries.
• Interatomic Interaction: The interaction between conduction electrons and ions or other electrons is weak due to the high speeds of electrons near ions.
• Collisions: The frequency of collisions between conduction electrons and ions is limited due to the short time electrons spend near ions and the resulting reduced likelihood of interactions
• Interaction: The interaction between conduction electrons themselves is weak, as described by the Pauli exclusion principle (parallel spin electrons cannot occupy the same quantum state).

Electrical Conductivity

• Current Density: The relationship between current density (J) and applied electric field (E) is the electrical conductivity
• Relaxation Time: The mean time between collisions of electrons
• Effective Mass: The electron's effective mass influences electrical conductivity.
• Temperature Dependence: Electrical conductivity depends on temperature and the concentration of electrons and effective mass,

Mathiessen's Rule

• Resistivity: The total resistivity of a material is the sum of the contributions from impurities and lattice vibrations (phonon scattering).
• Low Temperature: At low temperatures, the resistivity is dominated by impurities (non-linear)
• High Temperature: At high temperatures, resistivity is dominated by phonon scattering from lattice vibrations.