Einstein and Debye Models of Solids
April 11, 2022
1
, Classical Heat Capacity of Solids and the Dulong-Petit Law
Given a crystalline solid at high temperatures (including room temperature), its [total internal]
energy U , according to classical statistical mechanics, is equal to
U = 3N kB T (1)
where N is the number of atoms, kB = 1.38 × 10−23 J/K is the Boltzmann constant, and T is the
absolute temperature of the system. The expression in equation (1) is obtained from the Equipartition
Theorem, which states that each degree of freedom a system possesses contributes a total energy of
1
2 N kB T to the overall system. Assuming that each atom in the solid is connected to its neighboring
atom with a spring, each atom in the solid has three vibrational degrees of freedom, with three degrees of
freedom contributing to the mean kinetic energy and the remaining three degrees of freedom contributing
to the mean potential energy. This leads to a total of six degrees of freedom, giving us equation (1).
From the energy, we can obtain the expression for the constant-volume heat capacity CV of the
solid by using the expression
dU
CV = (2)
dT
Thus, the heat capacity of the solid is then given by
CV = 3N kB (3)
Equation (3) is known as the Dulong-Petit Law. The classical model predicts that the Dulong-Petit
Law is accurate at all temperatures. However, at low temperatures, the heat capacity of solids based on
experimental data are given by
CV = αT 3 + γT (4)
where α and γ are constants depending on the solid; γ = 0 for insulators and γ 6= 0 for conductors.
This is the shortcoming of the classical model. It is the direct violation of the experimental results and
the Third Law of Thermodynamics. Thus, we need to make use of quantum mechanics to explain the
behavior of the heat capacity of the solids at all temperatures.
2
April 11, 2022
1
, Classical Heat Capacity of Solids and the Dulong-Petit Law
Given a crystalline solid at high temperatures (including room temperature), its [total internal]
energy U , according to classical statistical mechanics, is equal to
U = 3N kB T (1)
where N is the number of atoms, kB = 1.38 × 10−23 J/K is the Boltzmann constant, and T is the
absolute temperature of the system. The expression in equation (1) is obtained from the Equipartition
Theorem, which states that each degree of freedom a system possesses contributes a total energy of
1
2 N kB T to the overall system. Assuming that each atom in the solid is connected to its neighboring
atom with a spring, each atom in the solid has three vibrational degrees of freedom, with three degrees of
freedom contributing to the mean kinetic energy and the remaining three degrees of freedom contributing
to the mean potential energy. This leads to a total of six degrees of freedom, giving us equation (1).
From the energy, we can obtain the expression for the constant-volume heat capacity CV of the
solid by using the expression
dU
CV = (2)
dT
Thus, the heat capacity of the solid is then given by
CV = 3N kB (3)
Equation (3) is known as the Dulong-Petit Law. The classical model predicts that the Dulong-Petit
Law is accurate at all temperatures. However, at low temperatures, the heat capacity of solids based on
experimental data are given by
CV = αT 3 + γT (4)
where α and γ are constants depending on the solid; γ = 0 for insulators and γ 6= 0 for conductors.
This is the shortcoming of the classical model. It is the direct violation of the experimental results and
the Third Law of Thermodynamics. Thus, we need to make use of quantum mechanics to explain the
behavior of the heat capacity of the solids at all temperatures.
2
