This theory was proposed by Albert Einstein in 1907 to describes the quantum nature of vibrational energy in a crystalline solid and its effect on heat capacity. It successfully explains the decrease in heat capacity at low temperatures, a limitation in the earlier classical Dulong–Petit law.
Assumptions and Model
- Each atom vibrates independently as a quantum harmonic oscillator.
- All atoms vibrate with the same frequency (\(\nu_E\)).
- Three vibrational degrees of freedom per atom.
- The energy levels are quantized, not continuous.
Mathematical Formulation
The heat capacity at constant volume (\(C_V\)) is given by:
\(C_V = 3Nk \left( \frac{T_E}{T} \right)^2 \frac{e^{T_E/T}}{(e^{T_E/T} - 1)^2}\)
Where \(N\) is the number of atoms, \(k\) is Boltzmann's constant, \(T_E\) is Einstein temperature, and \(T\) is absolute temperature.
Features & Limitations
- At high temperatures: \(C_V\) approaches \(3Nk\) (matches classical theory).
- At low temperatures: \(C_V\) approaches zero (agrees with experiment and third law of thermodynamics).
- Limitation: Assumes a single vibrational frequency for all atoms; the Debye model offers improved accuracy at low temperatures.