Derivation of Dulong-Petit’s Law

Derivation of Dulong-Petit’s Law.

The correct answer and explanation is :

Derivation of Dulong-Petit’s Law

Dulong-Petit’s Law Statement:
The molar heat capacity at constant volume ($C_V$) of a solid element is approximately 3R, where R is the universal gas constant:

$$
C_V \approx 3R \approx 24.9 \, \text{J/mol·K}
$$

Derivation (using Classical Physics):

1. Assumption (Classical Model):
   Atoms in a solid vibrate about fixed positions like harmonic oscillators.
2. Each atom has 3 degrees of freedom (x, y, z directions).
   In each direction, the vibration contributes:

• Kinetic energy: $\frac{1}{2}kT$
• Potential energy: $\frac{1}{2}kT$
  ⇒ Total energy per direction: $kT$

1. Total energy per atom:
   $3 \times kT = 3kT$
2. Total internal energy per mole of atoms:
   Multiply by Avogadro’s number $N_A$: $$
   U = 3kT \cdot N_A = 3RT
   $$
3. Molar heat capacity at constant volume: $$
   C_V = \left(\frac{dU}{dT}\right)_V = \frac{d}{dT}(3RT) = 3R
   $$

Explanation (300 Words):

Dulong-Petit’s Law is a classical result in thermodynamics and solid-state physics, stating that the molar heat capacity at constant volume for most solid elements is approximately 3R (about 25 J/mol·K). This law was derived under the assumption that atoms in a solid behave like harmonic oscillators vibrating in three dimensions.

In classical physics, each degree of freedom in a system contributes $\frac{1}{2}kT$ of kinetic energy, where $k$ is the Boltzmann constant and $T$ is the temperature in Kelvin. For harmonic oscillators (like atoms in a crystal), there is an equal contribution of potential energy, also $\frac{1}{2}kT$, per degree of freedom. Since each atom in a solid vibrates in three spatial directions (x, y, and z), the total energy per atom becomes $3kT$.

To find the molar energy, we multiply this energy per atom by Avogadro’s number $N_A$, giving $U = 3kT \cdot N_A = 3RT$. The molar heat capacity at constant volume $C_V$ is defined as the rate of change of internal energy with temperature, so:

$$
C_V = \frac{dU}{dT} = \frac{d}{dT}(3RT) = 3R
$$

This law holds well at high temperatures but fails at low temperatures, where quantum effects become significant, as shown by the more accurate Einstein and Debye models. Nevertheless, Dulong-Petit’s Law was crucial in the 19th century for estimating atomic masses and understanding thermal behavior in solids.