Chapter 2

Statistical Thermodynamics

1- Introduction

- The object of statistical thermodynamics is to present a particletheory leading to an interpretation of the equilibrium propertiesof macroscopic systems.

- The foundation upon which the theory rests is quantum mechanics.

- A satisfactory theory can be developed using only the quantummechanics concepts of quantum states, and energy levels.

- A thermodynamic system is regarded as an assembly ofsubmicroscopic entities in an enormous number of every-changingquantum states. We use the term assembly or system to denote anumber N of identical entities, such as molecules, atoms, electrons,photons, oscillators, etc.

1- Introduction

- The macrostate of a system, or configuration, is specified by thenumber of particles in each of the energy levels of the system.

Nj is the number of particles that occupy the jth energy level.

If there are n energy levels, then

- A microstate is specified by the number of particles in eachquantum state. In general, there will be more than one quantumstate for each energy level, a situation called degeneracy.

n

jj NN

1

),...,,...,,(by defined is macrostateA 21 nj NNNN

1- Introduction

- In general, there are many different microstates correspondingto a given macrostate.

- The number of microstates leading to a given macrostate is calledthe thermodynamic probability. It is the number of ways in whicha given macrostate can be achieved.

- The thermodynamic probability is an “unnormalized” probability,an integer between one and infinity, rather than a number betweenzero and one.

- For a kth macrostate, the thermodynamic probability is taken tobe ωk.

1- Introduction

- A true probability pk could be obtained as

where Ω is the total number of microstates available to the system.

k

kp

k

k

2- Coin model example and the most probable

distribution - We assume that we have N=4 coins that we toss on the floor andthen examine to determine the number of heads N1 and the numberof tails N2 = N-N1.

- Each macrostate is defined by the number of heads and the numberof tails.

- A microstate is specified by the state, heads or tails, of each coin.

- We are interested in the number of microstates for each macrostate,(i.e., thermodynamic probability).

- The coin-tossing model assumes that the coins are distinguishable.

Macrostate Label

Macrostate Microstate k Pk

k N1 N2 Coin 1 Coin 2 Coin 3 Coin 4

1 4 0 H H H H 1 1/16

2 3 1 H H H T 4 4/16

H H T H

H T H H

T H H H

3 2 2 H H T T 6 6/16

H T H T

H T T H

T T H H

T H T H

T H H T

4 1 3 H T T T 4 4/16

T H T T

T T H T

T T T H

5 0 4 T T T T 1 1/16

2- Coin model example and the most probable distribution

The average occupation number is

where Njk is the occupation number for the kth macrostate.

k

kjkk

kjk

kk

kkjk

j pNNN

N

For our example of coin-tossing experiment, the average numberof heads is therefore

2)10()41()62()43()14(16

11 N

NNNN 4Then .2Similarly, 212

2- Coin model example and the most probable distribution

Suppose we want to perform the coin-tossing experiment with alarger number of coins. We assume that we have N distinguishablecoins.

Question: How many ways are there to select from the N candidatesN1 heads and N-N1 tails?

N1 k

0 1

1 4

2 6

3 4

4 10

1

2

3

4

5

6

7

0 1 2 3 4

N1

Om

ega

k

Figure. Thermodynamic probabilityversus the number of heads for acoin-tossing experiment with 4 coins.

2- Coin model example and the most probable distribution

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7 8

N1

Om

ega

k

The answer is given by the binomial coefficient

)1()!(!

!

111 NNN

N

N

N

Figure. ThermodynamicProbability versus thenumber of heads for acoin-tossing experimentwith 8 coins.

N1 k

0 1

1 8

2 28

3 56

4 70

5 56

6 28

7 8

8 1

Example for N = 8

The peak has becomeconsiderably sharper

2- Coin model example and the most probable distribution

What is the maximum value of the thermodynamic probability (max)for N=8 and for N=1000?

The peak occurs at N1=N/2. Thus, Equation (1) gives

???!500!500

!10001000For

70!4!4

!88For

max

max

N

N

For such large numbers we can use Stirling’s approximation:

nnnn )ln()!ln(

2- Coin model example and the most probable distribution

)!500ln(2)!1000ln()ln(!500!500

!1000maxmax

500)500ln(50021000)1000ln(1000)ln( max

693500

1000ln1000)ln( max

3006934343.0)ln()(log)(log max10max10 e

300max 10

For N = 1000 we find that max is an astronomically large number

2- Coin model example and the most probable distribution

k coins 1000 NFor 30010

1N

region randomTotally

region ordered region ordered

Figure. Thermodynamicprobability versus thenumber of heads for acoin-tossing experimentwith 1000 coins.

The most probable distribution is that of total randomness

(the most probable distribution isa macrostate for which we have amaximum number of microstates)

The “ordered regions” almost never occur; ω is extremely smallcompared with ωmax.

2- Coin model example and the most probable distribution

k

k maxFor N very large

Generalization of equation (1)

Question: How many ways can N distinguishable objects bearranged if they are divided into n groups with N1 objects in thefirst group, N2 in the second, etc?

Answer:

jjnj

knj N

N

NNNN

NNNNN

!

!

!!...!!...!!

!),...,...,,(

2121

3- System of distinguishable particles

The constituents of the system under study (a gas, liquid, or solid)are considered to be:- a fixed number N of distinguishable particles

- occupying a fixed volume V.

n

jj NN

1

particles) ofon conservati(

We seek the distribution (N1, N2,…, Nj,…, Nn) among energy levels(ε1, ε2,…, εj,…, εn) for an equilibrium state of the system.

3- System of distinguishable particles

We limit ourselves to isolated systems that do not exchange energyin any form with the surroundings. This implies that the internalenergy U is also fixed

energy) ofon conservati(1

n

jjj UN

Example: Consider three particles, labeled A, B, and C, distributedamong four energy levels, 0, ε, 2ε, 3ε, such that the total energy isU=3ε.a) Tabulate the 3 possible macrostates of the system.b) Calculate ωk (the number of microstates), and pk (True probability)for each of the 3 macrostates.c) What is the total number of available microstates, Ω, for the system

3- System of distinguishable particlesMacrostate

LabelMacrostate

SpecificationMicrostate

SpecificationThermod.

Prob.TrueProb.

k N0 N1 N2 N3 A B C ωk pk

1 2 0 0 1 003ε

03ε0

3ε00

3 0.3

2 1 1 1 0 00εε

2ε2ε

ε2ε02ε0ε

2εε

2ε0ε0

6 0.6

3 0 3 0 0 ε ε ε 1 0.1

10163

3- System of distinguishable particles

- The most “disordered” macrostate is the state of highestprobability.

- this state is sharply defined and is the observed equilibrium stateof the system (for the very large number of particles.)

4- Thermodynamic probability and Entropy

In classical thermodynamics: as a system proceeds toward a stateof equilibrium the entropy increases, and at equilibrium the entropyattains its maximum value.

In statistical thermodynamics (our statistical model): system tendsto change spontaneously from states with low thermodynamicprobability to states with high thermodynamic probability (largenumber of microstates).

It was Boltzmann who made the connection between the classicalconcept of entropy and the thermodynamic probability:

S and Ω are properties of the state of the system (state variables).

)( fS

4- Thermodynamic probability and Entropy

Consider two subsystems, A and B

)( AA fS

)( BB fS

The entropy is an extensive property, it is doubled when the mass ornumber of particles is doubled.Consequence: the combined entropy of the two subsystems issimply the sum of the entropies of each subsystem:

)3()()()( BAtotalBAtotal ffforSSS

Subsystem A Subsystem B

A

AS

B

BS

4- Thermodynamic probability and EntropyOne subsystem configuration can be combined with the other togive the configuration of the total system. That is,

Example of coin-tossing experiment: suppose that the twosubsystems each consist of two distinguishable coins.

)4(BAtotal

Macrostate Subsystem A Subsystem B

( N1 , N2 ) Coin 1 Coin 2 Coin 1 Coin 2 ωkA ωkB pkA pkB

( 2 , 0 ) H H H H 1 1 1/4 1/4

( 1 , 1 ) H T H T 2 2 2/4 2/4

T H T H

( 0 , 2 ) T T T T 1 1 1/4 1/4

1644 BAtotal

4- Thermodynamic probability and Entropy

Thus Equation (4) holds, and therefore

)5()()( BAtotal ff

Combining Equations (3) and (5), we obtain

)()()( BABA fff

The only function for which this statement is true is the logarithm.

Therefore )ln(kS

Where k is a constant with the units of entropy. It is, in fact,Boltzmann’s constant:

1231038.1 KJk

5- Quantum states and energy levels