MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Theoretical calculation of the heat capacity

Principle of equipartition of energy

Heat capacity of ideal and real gases

Heat capacity of solids: Dulong-Petit, Einstein, Debye models

Heat capacity of metals – electronic contribution

Reading: Chapter 6.2 of Gaskell

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Degrees of freedom and equipartition of energy

For each atom in a solid or gas phase, three coordinates have to bespecified to describe the atom’s position – a single atom has 3 degreesof freedom for its motion. A solid or a molecule composed of N atomshas 3N degrees of freedom.

We can also think about the number of degrees of freedom as thenumber of ways to absorb energy. The theorem of equipartition ofenergy (classical mechanics) states that in thermal equilibrium the sameaverage energy is associated with each independent degree of freedomand that the energy is ½ kBT. For the interacting atoms, e.g. liquid orsolid, for each atom we have ½ kBT for kinetic energy and ½ kBT forpotential energy - equality of kinetic and potential energy in harmonicapproximation is addressed by the virial theorem of classical mechanics.

Based on equipartition principle, we can calculate heat capacity of theideal gas of atoms - each atom has 3 degrees of freedom and internalenergy of 3/2kBT. The molar internal energy U=3/2NAkBT=3/2RT andthe molar heat capacity under conditions of constant volume iscv=[dU/dT]V=3/2R

In an ideal gas of molecules only internal vibrational degrees of freedomhave potential energy associated with them. For example, a diatomicmolecule has 3 translational + 2 rotational + 1 vibrational = 6 totaldegrees of freedom. Potential energy contributes ½ kBT only to theenergy of the vibrational degree of freedom, and Umolecule = 7/2kBT if alldegrees of freedom are “fully” excited.

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Temperature and velocities of atoms

At equilibrium velocity distribution is Maxwell-Boltzmann,

zyx

B

2z

2y

2x

23

B

dvdvdvT2k

vvvmexp

Tk 2π

mT,vdN

T/m3kv Bi

2

If system is not in equilibrium it is often difficult to separate differentcontributions to the kinetic energy and to define temperature.

Acoustic emissions in the fracture simulation in 2D model. Figure byB.L. Holian and R. Ravelo, Phys. Rev. B51, 11275 (1995). Atoms arecolored by velocities relative to the left-to-right local expansion velocity,

which causes the crack to advance from the bottom up.

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Heat capacity of molecules – straightforward application of equipartition principle does not work

Classical mechanics should be used with caution when dealing withphenomena that are inherently quantized.

For example, let’s try to use equipartition theory to calculate the heatcapacity of water vapor.

Motion Degrees of freedom U cv

Translational 3 3 ½ RT 1.5R

Rotational 3 3 ½ RT 1.5R

Vibrational 3 6 ½ RT 3R

Total cv = 6R

But experimental cv is much smaller.At T = 298 K H2O gas has cv = 3.038R.

What is the reason for the large discrepancy?

Rotation

VibrationTranslation

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Heat capacity of molecules – straightforward application of equipartition principle does not work (continued)

What is the reason for the large difference between the prediction of classical calculations, cv = 6R, and much smaller experimental cv = 3.038R at 25C?

(cm-1) Exp(-h/kBT)

3825 1.0 x 10-4

1654 1.9 x 10-2

3936 8.0 x 10-5

The table shows the vibrationalfrequencies of water along with thepopulation of the first excited state at600 K.

For the high frequency OH stretching motions, there should beessentially no molecules in the first vibrational state even at 600 K. Forthe lower frequency bending motion, there will be about 2% of themolecules excited.

Contributions to the heat capacity can be considered classically only ifEn ~ h << kBT. Energy levels with En kT contribute little, if at all, tothe heat capacity.

Only translational and rotational modes are excited, the contributionfrom vibrations is only 0.038R.

Rotation

VibrationTranslation

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 atroom temperature, cv 3R = 25 JK-1mol-1

This is consistent with equipartition theory: energy added to solids takesthe form of atomic vibrations and both kinetic and potential energy isassociated with the three degrees of freedom of each atom.

Tk2

3)t(K)t(P B

The molar internal energy is then U = 3NAkBT = 3RT and the molarconstant volume heat capacity is cv = [U/T]v =3R

Although cv for many elements at room T are indeed close to 3R, low-Tmeasurements found a strong temperature dependence of cv. Actually, cv

0 as T 0 K.

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Heat capacity of solids – Einstein model

The low-T behavior can be explained by quantum theory. The firstexplanation was proposed by Einstein in 1906. He considered a solid asan ensemble of independent quantum harmonic oscillators vibrating at afrequency . Quantum theory gives the energy of ith level of a harmonicquantum oscillator as

i = (i + ½) h where i = 0,1,2…, and h is Planck’s constant.

For a quantum harmonic oscillator the Einstein-Bose statistics must beapplied (rather than Maxwell-Boltzmann statistics and equipartition ofenergy for classical oscillators) and the statistical distribution of energyin the vibrational states gives average energy:

There are three degrees of freedom per oscillator, so the total internalenergy per mol is

1e

h)t(U Tkh B

1e

hN3U Tkh

A

B

2Tkh

Tkh

2

BBA

VV

1e

eTk

hkN3

T

Uc

B

B

The Einstein formula gives a temperature dependent cv that approaches3R as T , and approaching 0 as T 0.

Note: you do not need toremember all these scaryquantum mechanics equationsfor tests/exams but you doneed to understand the basicconcepts behind them.

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

The High Temperature Limit of the Einstein Specific Heat

Let’s show that Einstein’s formula approaches Dulong – Petit law athigh T. For high temperatures, a series expansion of the exponentialgives

The Einstein specific heat expression then becomes

2

B

B

2

BBA

2Tkh

Tkh

2

BBA

V

Tkh

Tkh

1Tk

hkN3

1e

eTk

hkN3

cB

B

Tk

h1e

B

Tkh B

R3kN3Tk

h1kN3 BA

BBA

In the Einstein treatment, the appropriate frequency in the expressionhad to be determined empirically by comparison with experiment foreach element. Although the general match with experiment wasreasonable, it was not exact. Einstein formula predicts faster decrease ofcv as compared with experimental data. Debye advanced the treatmentby treating the quantum oscillators as collective modes in the solid -phonons.

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Heat capacity of solids – Debye model

Debye assumed a continuum of frequencies with a distribution ofg() = a2, up to a maximum frequency, D, called the Debye frequency.

This leads to the followingexpression for the Debye specificheat capacity:

dx1e

exTkN9c

T/

02x

x43

DBAV

D

where x = h/kBT and D = hD/kB –Debye characteristic temperature3

D

BA4

V

T

5

kN12c

For low temperatures, Debye's model predicts

- good agreement with experimental results.

We can see that cv dependson T/D with D as thescaling factor for differentmaterials.

D ~ D ~ strength of theinteratomic interaction, ~1/(atomic weight).

For a harmonic oscillatormass reduced

constant force

2

1

T/D

Cv, J K-1 mol-1

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Heat capacity of metals – electronic contribution

cv = [U/T]v – therefore as soon as energy of electrons are changingwith T, they will make contribution to cv.

To contribute to bulk specific heat, the valence electrons would have toreceive energy from the thermal energy, ~kBT. But the Fermi energy ismuch greater than kBT and the overwhelming majority of the electronscannot receive such energy since there are no available energy levelswithin kBT of their energy.

The small fraction of electrons which are within kBT of the Fermi level(defined by Fermi-Dirac statistics) does make a small contribution to thespecific heat. This contribution is proportional to temperature, cv

el = Tand becomes significant at very low temperatures, when cv = T + AT3

(for metals only).

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Energy Band Structures

Semiconductors and Insulators

Metals

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Summary (1)

Make sure you understand language and concepts:

Degrees of freedom Equipartition of energy Heat capacity of ideal gas Heat capacity of solids: Dulong-Petit law Quantum mechanical corrections: Einstein and Debye models

Heat capacity of gas, solid or liquid tends to increase with temperature,due to the increasing number of excited degrees of freedom, requiringmore energy to cause the same temperature rise.

The theoretical approaches to heat capacities, discussed in this lecture,are based on rather rough approximations (anharmonicity is neglected,phonon spectrum is approximated by 2 in Debye model, etc.). Inpractice cp(T) in normally measured experimentally and the results aredescribed analytically, e.g. Cp = A + BT + CT-2 for a certain range oftemperatures.

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Summary (2)

Equipartition theory is only valid if all degrees of freedom are “fully”excited.

Low T, Cv → 0

High T, Cv → 3R

T

Cv,

JK

-1m

ol-1

Quantized energy levels

TΔE/kn

Be~P

ΔE << kBT - classical behavior

ΔE ≥ kBT - quantum behavior

24.9

Ene

rgy