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MSEG 803Equilibria in Material Systems
10: Heat Capacity of Materials
Prof. Juejun (JJ) Hu
Heat capacity: origin
Molar heat capacity:
Internal energy of solids: Lattice vibration: collective
motion of interacting atoms Electron energy (metals) Other contributions: magnetic
polarization, electric polarization, chemical/hydrogen bonds, etc.
~V PV
uc c
T
This mole has a large molar heat capacity
The values are quoted for 25 °C and 1 atm pressure for gases unless otherwise noted
MaterialMolar heat capacity
cv (J/mol K) cv/R Type Degrees of freedom
He 12.5 1.5Monatomic
gas3 translational
Total 3Ne 12.5 1.5
Ar 12.5 1.5
H2 20.2 2.43
Diatomic gas3 translational
2 rotationalTotal 5
O2 20.2 2.43
N2 19.9 2.39H2S 26.7 3.22
Triatomic gasDepends on molecular geometry
CO2 28.5 3.43
H2O (100 °C) 28.0 3.37Arsenic 24.6 2.96
Atomic solid3 translational 3 vibrational
Total 6
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
Liquid ?H2O 75.3 9.06
Gasoline 229 27.6
Gases
Non-metal solids
Metal solids
Liquids
Heat capacity of a harmonic oscillator
Energy:
Partition function:
Mean energy:
Heat capacity:
2 2
2 2 2
2 2 2r
p kq mE q q
m
0
1 1
1 exp 1 exprn r
Zn
r rE n
Classical
Quantum mechanical
ln
exp 1
ZE
22
exp
exp 1V
V
EC k
T
Heat capacity of a harmonic oscillator
High T limit
Low T limit
kT
VC k
kT
2
~ 0expV
kC
Heat capacity of polyatomic gas
“Freeze-out” temperature of harmonic oscillators:
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)
Rotational degrees of freedom: Bond stretching degrees of freedom: At RT, only translational and rotational DOFs contributes
to Cv
0.1fT k
~ 0fT K
~10 100fT K K~1000fT K
Lattice vibration energy in solids
Apply models of phonon density of states
Debye approximation
Calculate mean energy and heat capacity
High temperature and low temperature limits
Solve the partition functionThe product of n harmonic oscillator partition functions, where n = 3N is the DOF
Construct generalized coordinates
The energy (Hamiltonian) is decomposed into a set of independent harmonic oscillators
Lattice vibration energy in solids
Consider a solid consisting of N identical atoms
Kinetic energy:
Potential energy:
22
, ,2 2i
kN N i x y z
mqpE
m
2
0,
2
0,
1...
2
1...
2
p pp p i i j
N,i N,i N,ji i j
pp i j
N,i N,j i j
E EE E q + q q
q q q
EE + q q
q q
3
,1
N
r r i ii
Q B q
Define generalized coordinates:
3
2 2 2
12
N
tot k p r r rr
mE E E Q Q
Normal modes
Normal modes (lattice waves)
Normal modes of lattice wave:in analogy to “particle-in-a-box”
Lattice waves can be decomposed to different normal
modes: Fourier analysis
Energy associated with normal modes
3N harmonic oscillators:
Energy of each mode:
Total energy:
Partition function:
3
2 2 2
12
N
tot r r rr
mE Q Q
1
2r r rn
3 3 3
01 1 1
1
2
N N N
tot r r r r rr r r
E n E n
3
0 1
expr
N
r rn r
Z n
phonons
3 3
01 1
1exp
1 expr
N N
r rnr r r
n
Partition function and heat capacity
Define the phonon density of state : the number of normal modes with frequency between w and w + dw
Mean energy:
Heat capacity:
d
3
01
1ln ln ln 1 exp
1 exp
N
rr r
Z d
0
ln
exp 1
ZE d
2
20
exp
exp 1V
V
EC k d
T
High temperature limit: the Dulong-Petit law
Heat capacity:
When
Total number normal modes:
Molar heat capacity: 3R (the Dulong-Petit law)
2
20
exp
exp 1VC k d
kT
0
3VC k d Nk
03d N
Debye approximation
Normal modes are treated as acoustic waves in continuum mechanics
DOS of acoustic waves:
Debye frequency
r v k :v :ksound wave velocity wave vector
22 3
32
Vd d
v
D
0d D
1
32
03 6
D
D
Nd N v
V
Debye heat capacity
Debye heat capacity
Debye function:
Debye temperature:
At high ,
At low ,
43 22 0
exp3
2 exp 1
3
D
V
D D
xVC k x dx
v x
Nk f T
423 0
exp3
exp 1
y
D
xf y x dx
y x
D Dk
DT 3VC Nk
DT 3VC T
1
326D
v N
k V
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
R = gas constant 3R = 8.31 J/mol-K
Cv = constant
Debye temperature (usually less than RT)
T (K)QD0
0
Cv
Electron heat capacity
Fermi-Dirac distribution:
Mean energy of electron gas:
Heat capacity:
Only significant at very low temperature
1
exp 1r
r
nE
0
22
0 0
exp 1
exp 1
3
2
rr r
r r r
r
r
EE n E
E
Ed
E
E kT
E E dE
Factor 2: spin degeneracy
m0 : Fermi surface at 0 K
E0 : electron gas energy at 0 K
22
0
2
3V
E kTC k N
T
2
0
3
2 3V
kTc R
Other contributions
Magnetization in paramagnetic materials:
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
2 2
2
HH M T H
MM M H
S S S MC T T
T T M T
S H M N HT C
T T T kT
2 1N HM
k T
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ALL MSEG 803 PARTICIPANTS
RRESTRICTED
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