# The Molar Specific Heat of Metals

I. Introduction
Already in 1819 Pierre Louis Dulong and Alexis Thérèse Petit found that the specific molar heat for metals is nearly a constant. The value turned out to be approximately 25 J/K. This is recognized as 3R, where R is the gas constant. R equals Nk = 8.314 J/K. N is Avogadro’s constant and k is Boltzmann’s constant. This phenomenon is shown in fig. 1.

Figure 1: A number of metals plotted with their molecular masses along the x-axis and the molar specific heat on the y-axis.

II. Einstein Model
In 1907  Einstein proposed the model for a solid based on two assumptions:

• Each atom is an independent 3D quantum harmonic oscillator
• All atoms oscillate with the same frequency

This implies that in the solid there are 3N oscillators. In the Einstein model all oscillators vibrate with the same frequency. The heat capacity, symbol Cp, of a system is defined as the ratio of heat transferred to or from the system and the resulting change in temperature in the system.

If the temperature change is sufficiently small the heat capacity may be assumed to be constant. Einstein calculated the average energy Q of an oscillator as:

here Z is the canonical partition function for an harmonic oscillator

Where 𝜖=ℏ𝜔 is the energy of a vibrational quantum. Insertion of the result of equation (4) into equation (2) yields for Q

After the insertion of 𝛽 = 1 / kT and multiplying with a factor 3N, and realizing that R = Nk one obtains the final result for Cp

Upon substituting ℏ𝜔 with kTE, where TE is the Einstein temperature equation (5) can be simplified to

III. Debye Model
Although the trend indicated by the Einstein model was correct, especially the behaviour at low temperature required improvement. In 1913 Peter Debye, following a reasoning based on determination of available phase space, similar to Rayleigh and Jeans for thermal radiation, derived the density of states D(ω) in a metal as:

where V is the unit volume, ω is the angular frequency and vs is the velocity of sound in the material. The average energy Q is given by

where 𝑛(𝜔)ℏ𝜔 is the average energy of an oscillator. It can be shown that Q can be written as

where xD = TD/T, where TD is the Debye temperature, which is characteristic for each metal. TD is given as

Which leads to the following expression for Cp

In figure 2 a comparison is shown for the molar specific heat for the Einstein and the Debye model for a Debye and Einstein temperature TD and TE of 200 K

Figure 2: Molar specific heat for the Einstein and Debye models. To show the difference the characteristic Einstein and Debye temperature have been taken identical at 200K. Note the difference at low temperature.

IV. References
[1]. Petit, A.-T.; Dulong, P.-L. “Recherches sur quelques points importants de la Théorie de la Chaleur”. Annales de Chimie et de Physique 10: 395–413 (1819).
[2]. Einstein Albert, “Theorie der Strahlung und die Theorie der Spezifische Wärme”, Annalen der Physik. 4. 22: 180-190, 800 (1906).
[3]. Debye, Peter “Zur Theorie der spezifischen Wärme”. Annalen der Physik  39 (4): 789–839 (1912).

V. Author
N. van Veen