Question

Einstein's law and Debye's law of specific heat merge with Dulong-Petit's law at:

• A. High T
• B. Low T
• C. All T
• D. \(T \to 0\,\mathrm{K}\)

Hint:

At high temperature the quantum modes are fully excited and \(C_V \to 3R\), the classical Dulong-Petit value.

Answer 1

The Correct Option is A

Step 1: The Dulong-Petit law states that the molar specific heat of a solid approaches the constant value \(C_V = 3R\), independent of temperature.

Step 2: Both the Einstein and Debye models reduce to \(3R\) in the high-temperature limit. For the Einstein model, when \(kT \gg h\nu\) the factor \(\left(\dfrac{h\nu}{kT}\right)^2 \dfrac{e^{h\nu/kT}}{(e^{h\nu/kT}-1)^2} \to 1\), giving \(C_V \to 3R\).

Step 3: For the Debye model, when \(T \gg \theta_D\) the Debye integral gives \(C_V \to 3R\) as well.

Step 4: Therefore both laws merge with Dulong-Petit at high temperature.
\[\boxed{\text{High } T}\]