MSEG 803
Equilibria in Material Systems
10: Heat Capacity of Materials
Prof. Juejun (JJ) Hu
hujuejun@udel.edu
Heat capacity: origin

Molar heat capacity:
 u 
cV  
 ~ cP
 T V

Internal energy of solids:

Lattice vibration: collective
motion of interacting atoms
 Electron energy (metals)
 Other contributions: magnetic
polarization, electric
polarization, chemical/hydrogen
bonds, etc.
This mole has a large
molar heat capacity
Material
Molar heat capacity
cv (J/mol K)
cv/R
He
12.5
1.5
Ne
12.5
1.5
Ar
12.5
1.5
H2
20.2
2.43
O2
20.2
2.43
N2
19.9
2.39
H2S
26.7
3.22
CO2
28.5
3.43
H2O (100 °C)
28.0
3.37
Arsenic
24.6
2.96
Antimony
25.2
3.03
Diamond
6.1
0.74
Copper
24.5
2.95
Silver
24.9
3.00
Mercury
28.0
3.36
H2O
75.3
9.06
Gasoline
229
27.6
Degrees of
freedom
Type
Gases
Nonmetal
solids
Monatomic
gas
3 translational
Total 3
Diatomic gas
3 translational
2 rotational
Total 5
Triatomic gas
Depends on
molecular
geometry
Atomic solid
3 translational
3 vibrational
Total 6
Liquid
?
Metal
solids
Liquids
The values are quoted for 25 °C and 1 atm pressure for gases unless otherwise noted
Heat capacity of a harmonic oscillator
p 2 kq 2 m 2


q   2q2
 Energy: Er 
2m
2
2

Er  nr 


Classical
Quantum mechanical
Partition function:

1
1
Z

1  exp     
nr  0 1  exp    nr  
 ln Z


 Mean energy: E  

exp      1
 E 
exp    
2

k




 Heat capacity: CV  



2

T
exp      1

V
Heat capacity of a harmonic oscillator

High T limit
kT  
CV  k

Low T limit
kT  
CV ~
k  
2
exp    
0
Heat capacity of polyatomic gas

“Freeze-out” temperature of harmonic oscillators:

T f  0.1





k
When T < Tf, the DOF hardly contributes to Cv
Generally, Tf is defined as the temperature at which kT
is much smaller than the energy level separation
Translational degrees of freedom: energy level very
closely spaced (particles in a box) T f ~ 0 K
Rotational degrees of freedom: T f ~ 10 K  100 K
Bond stretching degrees of freedom: T f ~ 1000 K
At RT, only translational and rotational DOFs contributes
to Cv
Lattice vibration energy in solids
Construct generalized coordinates
The energy (Hamiltonian) is decomposed into a set of independent harmonic oscillators
Solve the partition function
The product of n harmonic oscillator partition functions, where n = 3N is the DOF
Calculate mean energy and heat capacity
High temperature and low temperature limits
Apply models of phonon density of states
Debye approximation
Lattice vibration energy in solids

Consider a solid consisting of N identical atoms
mqi 2
p2
 
 Kinetic energy: Ek  
2
N 2m
N i x, y ,z

Potential energy:
E p  E0, p
 E p
 
N,i  qi
 2 Ep

1
  qi +  
2 N,i N,j  qi q j

 2 Ep
1
 E0, p +  
2 N,i N,j  qi q j

  qi q j  ...


  qi q j  ...

3N
Define generalized coordinates: Qr   Br ,i  qi
3N
Etot

m
 Ek  E p    Qr 2  r 2Qr 2
2 r 1

i 1
Normal modes
Normal modes (lattice waves)
Lattice waves can be
decomposed to different normal
modes: Fourier analysis
Normal modes of lattice wave:
in analogy to “particle-in-a-box”
Energy associated with normal modes

3N harmonic oscillators: Etot

m 3N
   Qr 2  r 2Qr 2
2 r 1


1
 Energy of each mode:  r   nr    r
2


phonons
3N
1

   r    nr    r  E0   nr r
2
r 1
r 1 
r 1
3N
3N

Total energy: Etot

3N


Partition function: Z   exp      nr r 
nr  0
r 1





1
   exp     nr r    

r 1 nr  0
r 1 
1  exp    r  
3N

3N
Partition function and heat capacity

Define the phonon density of state    d : the number
of normal modes with frequency between  and  + d



1
ln Z   ln 
   0 ln 1  exp    r       d
r 1
1  exp    r  
3N

Mean energy:

 ln Z

E

    d
0

exp      1

Heat capacity:

 E 
exp    
2
CV  

k




    d 



2

0
exp      1
 T V
High temperature limit: the Dulong-Petit law


Heat capacity: CV  k  0

When
exp    
exp      1
         d 
2
2
  kT

CV  k      d  3Nk
0
Total number normal modes:



0
   d   3 N
Molar heat capacity: 3R (the Dulong-Petit law)
Debye approximation

Normal modes are treated as acoustic waves in
continuum mechanics
r  v  k

DOS of acoustic waves:
   d   3 
   D 
   d   0

v : sound wave velocity
V
2 2 v 3
  2 d
  D 
Debye frequency

D
0

N
   d  3N  D  v  6 2 
V 

1
3
k : wave vector
Debye heat capacity

Debye heat capacity
CV  k 
3V
 v 
 3Nk  f D  D T 
2
2
3

exp  x 
 D
0
exp  x   1
3
 Debye function: f D  y   3
y

y
0
2
 x 4 dx
exp  x 
exp  x   1
2
 x 4 dx

v 
N
Debye temperature: k D  D   D    6 2 
k 
V 
At high T   D , CV  3Nk

At low T   D , CV  T 3

1
3
Debye heat capacity
• Heat capacity
-- increases with temperature
-- for solids it reaches a limiting value of 3R (Dulong-Petit law)
-- at low temperature, it scales with T3
Cv = constant
R = gas constant 3R
= 8.31 J/mol-K
0
0
D
T (K)
Debye temperature
(usually less than RT)
Electron heat capacity
1
exp    Er      1

Fermi-Dirac distribution: nr 

Mean energy of electron gas:
Er
E   nr  Er  
r
r exp 
   Er      1
 2 

0
 E0 
Er
     d
exp    Er      1
2
3
 kT    0 
2
  E     E  dE
Factor 2: spin degeneracy
0 : Fermi surface at 0 K
E0 : electron gas energy at 0 K
 E 2 2 2
kT

k N
 Heat capacity: CV 
T
3
0

Only significant at very low temperature
3   2 kT 
cV  R  


2  3 0 
Other contributions
N 2H 1

 Magnetization in paramagnetic materials: M 
k
T
 S 
 S 
 S   M  
CH  T 
  T  
 
 
 

T

T

M

T

H
M 
T 
H 

 S 
N  2H 2
 H   M  
 T  
 
 
   CM 
2

T

T

T
kT






M
M
H 


Hydrogen bonds

Hydrogen-containing polar molecules like ethanol, ammonia,
and water have intermolecular hydrogen bonds when in their
liquid phase. These bonds provide another place where heat
may be stored as potential energy of vibration, even at
comparatively low temperatures
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