Question

Q1. What are the Dulong-Petit law and Einstein model for the heat capacity of solids?

Answer 1

Dulong–Petit law

It states that the gram-atomic heat capacity (specific heat times atomic weight) of an element is a constant; that is, it is the same for all solid elements, about six calories per gram atom.

The value of the constant may be found from the law of equipartition of energy.
Let R be the molar gas constant and T the thermodynamic temperature.
The molar heat capacity for 3 degrees of freedom is 3R or

In modern terms, the mass m divided by atomic weight M gives the number of moles N.

Therefore, using uppercase C for the total heat capacity, and lowercase c for the specific heat capacity c :

or

Therefore the heat capacity of most solid crystalline substances is 3R per mole of a substance.

Einstein Model

The modern theory of the heat capacity of solids states that it is due to lattice vibrations in the solid and was first derived in crude form from this assumption by Albert Einstein. The Einstein solid model, for the first time gave a reason why the Dulong–Petit law should be stated in terms of the classical heat capacities for gases.

The Einstein solid is a model of a solid based on two crude assumptions:

1. Each atom in the lattice is an independent 3D quantum harmonic oscillator with discrete energy levels

where

• n is a non-negative integer
• h is the Planck-constant
• f is the frequency.

At low temperatures the small average thermal energy is not enough to excite the oscillators, that is the main reason for the decrease of the molar heat.

1. All atoms oscillate with the same frequency which is the classical harmonic oscillator frequency

where

• k is the spring-constant
• m is the mass.

It means that e.g. the lead has much larger frequencies than diamond, which explains why quantum effect in case of the lead is smaller and become important only at lower temperatures, while they cannot be neglected at room temperatures in case of a diamond.