Vol.:(0123456789)
International Journal of Thermophysics (2020) 41:84
https://doi.org/10.1007/s10765-020-02658-z
1 3
An Approximate Model for the Heat Capacity of Solids
M. Bouafa
1
· H. Sadat
2
Received: 30 January 2020 / Accepted: 25 March 2020
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
We present an explicit, simple and yet accurate expression of the heat capacity of
solids by approximating the Debye integral. This model can be used for tempera-
tures greater than one-tenth of Debye temperature. The results of this approximate
model are fnally compared with experimental data of Cu, MgO and ZnO and show
a very good agreement.
Keywords Debye function · Debye temperature · Heat capacity
1 Introduction
The heat capacity is a fundamental thermodynamic property related to internal
energy U, enthalpy H and entropy S of a system. It can be defned at constant pres-
sure: C
p
=
(
H
T
)
p
or at constant volume: C
v
=
(
U
T
)
v
. The relation between the two
defnitions is given as C
p
= C
v
+
2
VKT , where , V and K are the volume thermal
expansion coefcient, the molar volume and the isothermal bulk modulus. For most
solids, except metals at very low temperatures, the phonon contribution to the heat
capacity dominates. By assuming that the phonon density of states follows a quad-
ratic distribution up to a frequency
D
, at which it drops to zero, Debye developed a
model in which the molar specifc heat at constant volume can be expressed as fol-
lows [1]:
where N is the Avogadro number and k the Boltzmann constant. Debye tempera-
ture is defned as T
D
=
hc
2k
(
6
2
n
)
1
3
where h is the Planck constant, c the speed of
(1) C
M
v
= 9nNk
(
T
T
D
)
3
T
D
T
0
x
4
e
x
(e
x
- 1)
2
dx
* M. Bouafa
madiha.bouafa@univ-evry.fr
1
LMEE, Univ. Evry, Paris-Saclay, 91020 Evry, France
2
Université de Poitiers, 40 Avenue du Recteur Pineau, 86022 Poitiers, France