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 Wk where k represents the kth

macrostate.

A true probability Pk can be calculated as

Pk = Wk/Ω with Ω =

n

kkW

1

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 2# students in Room 104case 113case 222case 331case 440case 5

  # 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 Wk ~ k

Wk 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 N1 students in a room can be calculated from

WN1 =

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 N1 is the number of students in room 1 and N2 is the number of students in room 2; Let Njk be the number of students in room j for the kth case

kk

kkjk

j W

WNN

The value of thermodynamic probability WN1 will become extremely large as the values of N and N1 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: N1, N2 …Nn

Note that

N1 + N2 + N3 + … + Nn-1 + Nn = N

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

W= x x ... x

=

!!!!

!

321 nNNNN

N

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 Nj is the number of particles on the energy level j with the energy Ej.

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.

NNn

jj

1

UENn

jjj

1

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

33

0

j

jN

EENj

jj 33

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, W1 = 3, W2 = 6, and W3 = 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

Stotal = SA + SB (S is extensive)Or…

f (Wtotal) = f (WA) + f (WB)

The configuration of the total system can be calculated as Wtotal = WA x WB

thus: f (WA x WB) = f (WA) + f (WB)

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.