# Statistical Thermodynamics

Mechanics

Oct 27, 2013 (4 years and 8 months ago)

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Statistical
Thermodynamics

Chapter 12

12.1 Introduction

The object: to present a particle theory which
can interpret the equilibrium thermal properties
of macroscopic systems.

Quantum mechanical concepts of quantum
states, energy levels and intermolecular forces
are useful here.

The basic postulate of statistical
thermodynamics is that all possible microstates
of an isolated assembly are equally probable.

A few new concepts

1)
Assembly
: denote a number N of
identical entities
,
such as molecules, atoms, electrons.

2)
Macrostate
: is specified by the number of particles in
each of the energy levels of the system.

3)
Microstate
: is specified by the number of particles in
each energy state.

4)
Degeneracy
: an energy level contains more than one
energy state.

5)
Thermodynamic probability
: the number of microstates
leading to a given macrostate. It is donated by W
k

where k represents the k
th

macrostate.

A true probability P
k

can be calculated as

P
k

= W
k
/
Ω

with
Ω

=

12.2 Coin
-
tossing Experiment

Example: assuming that there are four students to
be assigned into two classrooms, how many
possibilities to split them?

Using A1, A2, A3 and A4 to represent the identity
of these four students, there will be many
different combinations for each case (i.e.
macrostate).

# students in Room 1

Total

Case 1

A1, A2, A3 and A4

1

Case 2

A1, A2 and A3

4

A1, A2 and A4

A2, A3, A4

A1, A3 and A4

Case 3

A1 + A2; A1 + A3

6

A1 + A4; A2 + A3

A2 + A4; A3 + A4

Case 4

A1

4

A2

A3

A4

Case 5

0

1

In this example, the macrostate corresponds
to the case (i.e. the # students can be
found in each room), whereas the number
of possible arrangements for each case is
viewed as the corresponding
thermodynamic probability.

More specifically,

for the macrostate 2 (i.e. case 2), the

thermodynamic probability is 4.

The thermodynamics probability for

macrostate 3 equals 6.

Making a plot with W
k

~ k

W
k

maximum occurs at Case 3.

If the total number of students increased, the
peak in the above figure will become very
sharp. The center of the peak will remain
at where N is the total number of
students.

The number of combinations for having N
1

students in a room can be calculated from

W
N1

=

The average occupation
numbers

In the above case, it means the average
number of students in room 1 or room 2.

Let
j

= 1 or 2 , where
N
1

is the number of
students in room 1 and
N
2

is the number of
students in room 2; Let
N
jk

be the number of
students in room
j
for the
k
th case

The value of thermodynamic probability
W
N1

will become
extremely large as the values of N and N
1

are increased.

To facilitate the calculation
, Stirling’s approximation
becomes useful:

ln
(n!) = n
ln
(n)

n

When n is more than 50, the error in using Striling’s
approximation becomes very small.

Extend the above example into the situation where there
are multiple rooms (n rooms)

Now, the macrostate will be defined by the #
of students in each room, say: N
1
, N
2

…N
n

Note that

N
1

+ N
2

+ N
3

+ … + N
n
-
1

+ N
n

= N

The number of microstates for the above
macrostate can be calculated from

W= x x ... x

=

12.3 Assembly of distinguishable
particles

An isolated system consists of N
distinguishable particles.

The macrostate of the system is defined by (N,
V, U).

Particles interact sufficiently, despite very
weakly, so that the system is in thermal
equilibrium.

Two restrictive conditions apply here

(conservation of particles)

(conservation of energy)

where
N
j

is the number of particles on the energy
level
j

with the energy
E
j
.

Example:
Three distinguishable particles labeled A, B,
and C, are distributed among four energy levels, 0, E,
2E, and 3E. The total energy is 3E. Calculate the
possible microstates and macrostates.

Solution: The number of particles and their total
energy must satisfy

(
here the index
j

starts from 0)

# particles on
Level 0

# Particles on
Level 1 E

# particles on
Level 2E

# particles on
Level 3E

Case 1

2

0

0

1

Case 2

1

1

1

0

Case 3

0

3

0

0

So far, there are only THREE macrostates
satisfying the conditions provided.

Configurations for case 1

Thermodynamic probability for case 1 is 3

Level 0

Level 1E

Level 2E

Level 3E

A, B

C

A, C

B

B, C

A

Configurations for case 2

Configuration for case 3

Therefore, W
1

= 3, W
2

= 6, and W
3

= 1

.

Level 0

Level 1E

Level 2E

Level 3E

A

B

C

A

C

B

B

A

C

B

C

A

C

A

B

C

B

A

Level 0

Level

1E

Level 2E

Level 3E

A, B and C

The most “disordered” macrostate is the state with the
highest probability.

The macrostate with the highest thermodynamic
probability will be the observed equilibrium state of
the system.

The statistical model suggests that systems tend to
change spontaneously from states with low
thermodynamic probability to states with high
thermodynamic probability.

The second law of thermodynamics is a consequence
of the theory of probability: the world changes the
way it does because it seeks a state of probability.

12.4 Thermodynamic Probability
and Entropy

Boltzman made the connection between the classical
concept of entropy and the thermodynamic probability

S = f (w)

f (w) is a single
-
valued, monotonically increasing function
(because S increases monotonically)

For a system which consists of two subsystems A and B

S
total

= S
A

+ S
B

(S is extensive)

Or…

f (W
total
) = f (W
A
) + f (W
B
)

The configuration of the total system can be
calculated as W
total

= W
A

x W
B

thus:

f (W
A

x W
B
) = f (W
A
) + f (W
B
)

The only function for which the above
relationship is true Is the logarithm.
Therefore:

S = k

lnW

where k is the Boltzman constant with the
units of entropy.