3.012 Fundamentals of Materials Science
Fall 2003
Lecture 22: 11.21.03 Connecting events at the atomic/molecular level to
macroscopic thermodynamic behavior
Today:
LAST TIME .............................................................................................................................................................................................. 2
TEMPERATURE AND THE OCCUPATION OF STATES .................................................................................................................................. 3
The Boltzmann factor and Partition function ..................................................................................................................................... 3
All thermodynamic quantities can be calculated from the partition function .................................................................................... 5
The partition function of molecules vs. multi-molecular systems....................................................................................................... 5
THE EINSTEIN SOLID .............................................................................................................................................................................. 7
The complete partition function for an Einstein solid ........................................................................................................................ 7
REFERENCES ........................................................................................................................................................................................... 9
Reading:
Dill and Bromberg Ch. 10, ‘The Boltzmann Distribution Law,’ pp. 171-191
Supplementary Reading:
-
Planning Notes:
HEAT CAPACITY REFLECTS THERMAL FLUCTUATIONS- DILL P. 228

distribution functions that describe the ensemble average under constraints of constant energy, constant
temperature
 a common laboratory situation: constant µ,V, T
 Description of a magnetic material using a two-state energy model
Curie’s law of paramagnetism using the Schottky energy model (Dill p. 186)
HOMEWORK PROBLEMS:
Lecture 22 – Connectiing molecular events tu
3.012 Fundamentals of Materials Science
Last time
Lecture 22 – Connectiing molecular events tu
Fall 2003
3.012 Fundamentals of Materials Science
Fall 2003
Temperature and the occupation of states

As we described last time, the possible energies accessible by atoms and molecules in a material are quantized
(we indexed them by the ‘quantum integer’ n), and thus the state of a material can be characterized by how the
energy levels are occupied in the system. What we want to do now is determine how the microstates of a
material are occupied when we consider the common experimental case of a system with fixed temperature
instead of fixed energy.
The Boltzmann factor and Partition function



Last time we discussed the constant energy ensemble (fixed E, V, and N) for a Debye (harmonic oscillator) solid.
Let’s now turn to the more interesting question of how to make calculations for systems where we fix T, V, and N.
The ensemble for fixed (T, V, and N) includes all possible microstates for the solid that have the same
temperature; it is called the canonical ensemble (‘canonical’ because it is used so often to model real systems).
With the temperature fixed, the internal energy and pressure may fluctuate in the real thermodynamic systemremember that the physical situation is that the system is surrounded by a large ‘heat bath’ which passes heat
back and forth with the system to maintain a constant temperature. Thus, microstates with different total internal
energies may be included in the allowed microstates.
We used the first postulate of statistical mechanics (equal a priori probabilities) to determine the state probabilities
pj in the microcanonical ensemble. For the canonical ensemble, we will determine what values the pj’s take in
order to satisfy the second law for a system at constant (T,V,N). Recall that for systems with constant (T,V,N),
the second law is satisfied when the Helmholtz free energy (F = U - TS) is a minimum.
o To determine the equilibrium probability pj for each individual state j, we simply calculate what values of pj
(for each possible state j) minimize the Helmholtz free energy subject to the constraint that the pj’s act like
a probability and sum to 1:
all
microstates
p
(Eqn 1)
j
1
j1

At equilibrium at constant temperature and volume, F is minimized in order to satisfy the second law. We want to
calculate the minimum in F with respect to pj for all possible states j:

(Eqn 2)

(Eqn 3)
(Eqn 4)
for all states j
To satisfy the constraint that the pj sum to 1, we use the method of Lagrange multipliers. This simply requires that
we minimize with the constraint included:


F
0
p j
W

 
F    pi 
p j 

i1
for all states j
We can fill in U and S with our formulas from last time:

W
U  E   pjE j
j1

Lecture 22 – Connectiing molecular events tu
3.012 Fundamentals of Materials Science
Fall 2003
W
S  k  p j ln p j
(Eqn 5)
j1
o
Which gives us:

(Eqn 6)

(Eqn 7)
W
W

 W
 pi E i  kT pi ln pi    pi  0
p j i1

i1
i1
for each state j
E j    kT1 ln p j  0
for each state j
pj  e

Ej
kT
e


kT
1
for each state j

o
We can determine the Lagrange multiplier a by referring back to our constraint equation:

all
microstates

(Eqn 8)
all
microstates

pj 
j1

e
(Eqn 9)
 1
k
kT
e
j1

e


 1
k
e

 1
k
all
microstates

e

Ej
kT
1
j1
1
all
microstates


e


Ej
Ej
1
Q
kT
j1
o

We define the summation in the denominator as the partition function Q. The importance of this sum
will soon be apparent (for now, at least it simplifies our notation!).
all
microstates
Q
(Eqn 10)
  E i 
e kT 
 

j1
o

Substituting (Eqn 9) into (Eqn 7), we arrive at the solution for the probability for the system to be in some
arbitrary state j:
pj 
(Eqn 11)
e

Ej
kT
all
microstates

e


Ei
kT
e

Ej
kT
Q
i1
o
The quantity exp(-Ei/kT) is known as the Boltzmann factor: it is the ‘thermal weighting factor’ that
determines how many atoms access a given state of
i. The Boltzmann factor indicates that
states with high energies will not be accessible at low temperature, but may occur with a high frequency
 in the ensemble at high temperature:
Lecture 22 – Connectiing molecular events tu
3.012 Fundamentals of Materials Science
Fall 2003
(Dill)
All thermodynamic quantities can be calculated from the partition function


The Boltzmann factor and partition function are the two most important quantities for making statistical
mechanical calculations. If we have a model for a material for which we can calculate the partition function,
we know everything there is to know about the thermodynamics of that model of the material.
All thermodynamic quantities of interest can be derived using the partition function. Some examples are:
 ln Q
T
(Eqn 12)
U  kT2
(Eqn 13)
S  k ln Q  kT

(Eqn 14)


F 
P   
V T ,N
 ln Q
T
 ln Q 
 kT

 V T ,N
…hence the utility of knowing Q for a given system.
 of molecules vs. multi-molecular systems
The partition function

Typically, it is straightforward to develop models at the molecular level for allowed energies/states, and to even
write the partition function for individual molecules. But how do we handle the case when we have a mole of
atoms in a system and we want to determine Q? It is not possible to enumerate all the possible states by hand.
o One way to deal with this complexity is to assume the molecules of the solid behave independently. If the
each molecule of the N molecules in a solid is independent, the energy of one molecule is not influenced
by that of its neighbors, and the total energy is simply the sum of the energies of each individual
molecule: FIX THIS DILL. P. 182
N
E total, j   E m, j
(Eqn 15)
m1
o
If we then look at the total partition function for the multi-molecular system, we have:

N
all
states
Q
(Eqn 16)
e

kT
j1
Lecture 22 – Connectiing molecular events tu

all
states
E total, j

e
j1
 E m, j
 m1
kT
all
states


j1
  E1, j  E 2, j  E N , j 
kT

e kT ...e kT 
e



3.012 Fundamentals of Materials Science
Fall 2003
o
Q

Q  molecule
N
(Eqn 17)

N!
This equation gives us a simple route to making calculations for macroscopic systems from molecular leveldetailed models.

Lecture 22 – Connectiing molecular events tu
3.012 Fundamentals of Materials Science
Fall 2003
The Einstein Solid

Now that we have the formula for the probabilities in a system at constant temperature, we can start making some
powerful predictions using our Einstein solid model.
The complete partition function for an Einstein solid

Recall that in the Einstein solid, the atoms are assumed to vibrate in a harmonic potential. The energy of this
confined oscillation is quantized:
 1 
E n  n  hv
 2 
(Eqn 18)

We performed microcanonical calculations (fixed E,V,N) for a very imaginary 3-atom, 1D-oscillating solid. If we
take the more realistic case of allowing each atom to oscillate in X, Y, and Z space, we have 3 quantized
energies:

(Eqn 19)

1 
E n x  nx  hv

2 
(Eqn 20)

1 
E n y  ny  hv

2 


1 
E n z  nz  hv

2 
(Eqn 21)


The total energy in one microstate (characterized by one set of values n x, ny, nz) for one atom of the solid is:

E molecule(nx ,ny ,nz )  E nx  E ny  Enz

(Eqn 22)
It follows that the molecular partition function for one atom of the solid is:

(Eqn 23)
Qatom 



  e

E molecule(n x ,n y ,n z )
kT
n x  0n y  0n z  0





  e
n x  0n y  0n z  0
Since the virbration in each of the 3 directions is equivalent (i.e.

E nx
kT
e

E ny
kT
e

E nz
kT


e
nx  0

E nx
kT

e
ny  0

E ny
kT

e

E nz
kT
nz  0
E nx (nx 1)  E ny (ny 1)  E nz (nz 1)  E n1),
the three sums in Qatom are the same. We can therefore write:

   E n
Qatom   e kT

n  0
(Eqn 24)





3

We showed above that the partition function for a system of N non-interacting distinguishable atoms or molecules
is given by Q = (Qatom)N. Thus, for the partition function of the entire Einstein solid, we have:

Lecture 22 – Connectiing molecular events tu
3.012 Fundamentals of Materials Science
Q  Qatom 
N
(Eqn 25)

Fall 2003
   E n
  e kT

n  0




3N
with En = (n + 1/2)h, n = 0, 1, 2, …, 
The infinite sum looks messy, but we can simplify this partition function. To shorten the following notation, let’s
use  = 1/kT:


e
(Eqn 26)
 E n

n 0

e
 1 
 h n  
 2 
e

h
n 0
2

ehn  e
n 0

h
2

e   
 h
n
n 0


Using the approximation

(Eqn 27)


1
  x n (for x2 < 1), we can obtain:
1 x n 0
e

h
2

 e 
 h n
n 0
e

h
2
1
e h

1 e h e h 1
Finally, substituting into (Eqn 25), we arrive at the simplified total partition function for the Einstein solid:

(Eqn 28)
 h 3N  h 3N
 e 2 
 e 2kT 
Q   h   h


e 1





e kT 1


Lecture 22 – Connectiing molecular events tu
3.012 Fundamentals of Materials Science
References
Lecture 22 – Connectiing molecular events tu
Fall 2003