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MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Heat capacity of solids – Dulong – Petit law
In 1819 Dulong and Petit found experimentally that for many solids at
room temperature,
c
v
≈
3R = 25 JK
1
mol
1
This is consistent with equipartition theory: energy added to solids takes
the form of atomic vibrations and both kinetic and potential energy is
associated with the three degrees of freedom of each atom.
T
k
2
3
)
t
(
K
)
t
(
P
B
=
=
The molar internal energy is then U = 3N
A
kT = 3RT and the molar
constant volume heat capacity is c
v
= [
∂
U/
∂
T]
v
=3R
Although c
v
for many elements at room T are indeed close to 3R, lowT
measurements found a strong temperature dependence of c
v
.
Actually,
c
v
→
0 as T
→
0 K.
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View Full DocumentMSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Heat capacity of solids – Einstein model
The lowT behavior can be explained by quantum theory. The first
explanation was proposed by Einstein in 1906.
He considered a solid as
an ensemble of independent quantum harmonic oscillators vibrating at a
frequency
υ
.
Quantum theory gives the energy of i
th
level of a harmonic
quantum oscillator as
ε
i
= (i + ½) h
υ
where i = 0,1,2…, and h is Planck’s constant.
For a quantum harmonic oscillator the EinsteinBose statistics must be
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 Spring '12
 Zhigilei

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