Below is a step-by-step derivation of the classical specific heat formula for a solid, based on the equipartition theorem and the Dulong-Petit law, followed by behavior analysis.
Step 1: Model Assumptions
The classical model assumes a solid with \( N \) atoms, each acting as a 3D harmonic oscillator. Each atom has three degrees of freedom for kinetic energy (vibrational motion) and three for potential energy, giving \( 3N \) degrees of freedom. The equipartition theorem assigns \( \frac{1}{2} k_B T \) energy per degree of freedom.
Step 2: Total Internal Energy \( U \)
For \( 3N \) degrees of freedom (kinetic and potential):
The first term accounts for kinetic energy, and the second for potential energy.
Step 3: Specific Heat \( C_V \)
The specific heat at constant volume is:
Differentiate:
For one mole (\( N = N_A \), Avogadro’s number), where \( R = N_A k_B \):
This is the Dulong-Petit law, predicting a molar specific heat of \( 3R \approx 25 \, \text{J/(mol·K)} \).
Behavior Analysis
High Temperatures: The Dulong-Petit law holds at high temperatures, where quantum effects are negligible, and each degree of freedom is fully excited, yielding \( C_V = 3R \).
Low Temperatures: The classical model fails at low temperatures, where quantum effects reduce the specific heat. Models like Einstein or Debye account for the observed decrease (e.g., \( T^3 \) in Debye).
Explanation
The derivation uses the classical equipartition theorem to assign energy to each degree of freedom, leading to a constant specific heat. While accurate at high temperatures, it fails at low temperatures due to quantum effects, which are better described by the Einstein or Debye models.
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