mseg 803 equilibria in material systems 10: heat capacity of materials prof. juejun (jj) hu...

MSEG 803Equilibria in Material Systems

10: Heat Capacity of Materials

Prof. Juejun (JJ) Hu

hujuejun@udel.edu

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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