3.012 Fundamentals of Materials Science
Fall 2003
Lecture 23: 11.26.03 Molecular degrees of freedom that make up the Entropy
Today:
LAST TIME .............................................................................................................................................................................................. 2
THE EINSTEIN SOLID, CONTINUED .......................................................................................................................................................... 3
Internal energy and heat capacity of the Einstein solid ..................................................................................................................... 3
DEGREES OF FREEDOM IN MOLECULAR MODELS ..................................................................................................................................... 5
Excitations in materials ..................................................................................................................................................................... 5
Complete molecular partition functions............................................................................................................................................. 5
Contribution of molecular degrees of freedom to the internal energy and entropy of materials ....................................................... 6
REFERENCES ........................................................................................................................................................................................... 7
Reading:
Supplementary Reading:
-
Planning Notes:


thermal excitations in solids
a statistical mechanical model to predict the heat capacity of solids
o lattice model of periodic solid (M. Cima notes)
o comparison of the experimental Cp of diamond with the Einstein model
structural excitations: rotations, vibrations, and normal modes
HOMEWORK PROBLEMS:
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The Einstein Solid, continued
Internal energy and heat capacity of the Einstein solid

Now that we have the partition function, it is straightforward to determine thermodynamic quantities for the
Einstein solid. First, let’s derive the internal energy:
 ln Q

(Eqn 1)
U  E 
(Eqn 2)

ln Q 

3Nh 3Nhe h  3
e h  1  3 Nh  3Nh
 Nh h
U  
 h

e 1 2 e 1 2
e h 1
 2
(Eqn 3)


INTERPRET THE BEHAVIOR
Using the internal energy, we can calculate the heat capacity of the Einstein solid:

dq  dU 
CV      
dT V dT V
(Eqn 4)

Since  = 1/kT:

(Eqn 5)

d
1
 2
dT
kT
Thus we can write the derivative dU/dT in the more convenient form:

(Eqn 6)
dU  dU  d 
1 dU 
CV          2  
dT V d V dT  kT d V
CV  3Nk 2 h 

h
 1 2
e kT
2

3Nk
h





2
kT 
 h 2
1
e kT 1


e h
2
(Eqn 7)

3Nh
 3N ln e h 1
2
e
h
The Einstein solid heat capacity is plotted below as calculated for Diamond, compared to the experimentally
measured heat capacity- and we see quite good agreement over a broad range of temperatures. In particular, at
high temperatures, we see the limiting behavior of the heat capacity is:

CV T    3Nk 2 h 
2
(Eqn 8)
1 h  ...  3Nk 2 h 2 1  3Nk  3R
 
2
2
1 h  ...1
h 
…correctly predicting the limiting value of Cv observed experimentally for many solids.
o

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PHYSICAL INTERPRETATION
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Degrees of freedom in molecular models
Excitations in materials

We modeled the atomic vibrations in a crystalline solid using 3 degrees of freedom- harmonic oscillations in X, Y,
and Z. We saw that a model using only these 3 degrees of freedom provided reasonable predictions for
the behavior of the heat capacity in X. Other materials may have other important degrees of freedom that we
should account for to obtain good statistical mechanics predictions of their behavior. The important molecular
degrees of freedom include:
1. Translation
2. Rotation
3. Vibration
4. Electron excitation
Translation

Molecules that have freedom to move within their confining volume (container) have translational degrees of
freedom. For example, the molecules of a gas can occupy positions throughout the volume in which they are
enclosed.
Rotation

What
Vibration

What
Electron excitation

What

EQUIPARTITION THEOREM? DILL P. 212
Complete molecular partition functions

A complex system may have all of these degrees of freedom. To make calculations for a given model, we need
to know how to put these degrees of freedom together in the partition function.
Independent degrees of freedom

A common approximation is to assume that each degree of freedom in the molecules of the system is
independent, with a unique amount of energy for each possible state of that degree of freedom (let’s use DOF as
an abbreviation for degree of freedom). Thus a molecule with both vibrational and electronic DOFs has states
characterized by one total energy containing independent contributions from the vibration and electronic
excitations:
elec
E total
 E vib
j
j  Ej
(Eqn 9)
o

Lecture 23
The subscript j refers to the single state that has the given characteristic vibrational and electronic energy.
Because we assume they are independent, the value of Ejvib does not depend on the value of Ejelec, and
vice versa. The partition function of this system with independent DOFs is:
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all
states
Qmolecule 
(Eqn 10)
e

kT
j

all
states
E total
j

e
j

E vib
j
kT
e

all
states
E elec
j
kT

e

all
E vib
states
j
kT
j
e

E elec
j
kT
 qvibqelec
j
Where the independent energies have been split off into partition functions for each DOF, qvib
and qelec:

all
states
qvib 
(Eqn 11)
e

E vib
j
kT
j
all
states
(Eqn 12)

o
e
E elec
j
kT
j
In general, a complete molecular partition function made up of independent degrees of freedom can be
written as the product of the individual DOF partition functions:

(Eqn 13)
qelec 

Qmolecule  qtransqvibqrotqelec
o
Contribution of molecular degrees of freedom to the internal energy and entropy of materials


Explain how these degrees of freedom show up in the energy and entropy
o More DOF, more states = more entropy
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References
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