chapter 2 statistical thermodynamics

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

ii NN

1

),...,,...,,(by defined is macrostateA 21 ni 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 Nik is the occupation number for the kth macrostate.

k

kikk

kik

kk

kkik

i 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


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.


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


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(


)!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


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.


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:

iini

kni 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

ii NN

1

particles) ofon conservati(

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


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

iii 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


- 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


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


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

- In quantum theory, to each energy level there corresponds one ormore quantum states described by a wave function Ψ(r,t).

- When there are several quantum states that have the same energy,the states are said to be degenerate.

- The quantum states associated with the lowest energy level arecalled the ground states of the system; those that correspond tohigher energies are called excited states.

1 11 g

2 32 g

3 53 g

Figure. Energy levels and quantum states.


1 11 g

2 32 g

3 53 g

- For each energy level εi the number of quantum states is given bythe degeneracy gi (or the degeneracy gi is the number of quantumstates whose energy level is εi).

- The energy levels can be thought of as a set of shelves at differentheights, while the quantum states correspond to a set of boxes oneach shelf.


Example:

Consider a particle of mass m in aone-dimensional box withinfinitely high walls. (The particleis confined within the region0 ≤ x ≤ L.) Within the box the particleis free, subjected to no forces exceptthose associated with the walls of theContainer.

Time-independent Shrödinger equation:

0x Lx

V

x

V

m

dx

d

22

2 2

.point about the interval malinfinitesi

in the found be willparticle y that theprobabilit theis )(2

xdx

dxx


0 < x < L: V = 0

m

kkm

dx

d 2 where

2 222

2

General solution: )sin()( kxAx

)at continuity(0)sin()(

)0at continuity(0)0(

LxkLAL

x

,...3,2,10)sin( nnkLkLA

Using k = nπ/L, we find that the particle energies are quantized:

,...2,122

22

2222

nnmLm

kn


The integer n is the quantum number of the one-dimensional box.

- Stationary states for the particle in the box:

For each value of the quantum number n there is a specificwavefunction n(x) describing the state of the particle with energy εn.

,...2,1 and 0for sin)(

nLx

L

xnAxn

- For the case of a three-dimensional box with dimensions Lx, Ly, andLz , the energy becomes

Any particular quantum state is designated by three quantum numbers nx, ny, and nz.

2

2

2

2

2

222

2 z

z

y

y

x

x

L

n

L

n

L

n

m


If Lx = Ly = Lz = L, then 2222

22

2 zyx nnnmL

222222

22

where2 zyxiii nnnnn

mL

,...2,1

,...2,1

,...2,1

z

y

x

n

n

n

- ni is the total quantum number for states whose energy level is εi.

- εi depend only on the values of ni2 and not on the individual

values of the integers (nx,ny,nz).

- The volume V of a cubical box equals L3, so L2 = V2/3 and hence2

32

22

2 ii nmV

as V decreases, the value of εi increases.


Level Energy State (nx,ny,nz) ni2 gi

i = 1 Ground state (1,1,1) 3 1

i = 2 First excited state (1,1,2); (1,2,1); (2,1,1) 6 3

i = 3 Second excited state (1,2,2); (2,1,2); (2,2,1) 9 3

Table. First three states of a three-dimensional infinite potential well

6- Density of Quantum States

When the quantum numbers ni are large, the energy levels εi are veryclosely spaced and that the discrete spectrum may be treated as anenergy continuum.

Consequence:We can regard the n’s and the ε’s as forming a continuous functionrather than a discrete set of values.

We are interested in finding the density of states g(ε)

232

22

2 ii nmV

We have:

Dropping the subscripts in equation above, using , andsolving for n2, we obtain

2/h


22

322222 8

Rh

mVnnnn zyx

According to equation above, for a given value of ε, the values ofnx, ny, nz that satisfy this equation lie on the surface of a sphere ofradius R.

. to

range in the liesenergy whosestates quantum ofnumber )(

d

dg

xn

yn

0

d

g(ε)dε is the number of states whose quantumnumbers (nx,ny,nz) lie within theinfinitesimally thin shell of the octant of asphere with radius proportional to the squareroot of the energy.


d

d

dnndndg

)()()()(

Evidently,

n(ε) is the number of states contained within the octant of the sphereof radius R; that is,

2323

23 8

63

4

8

1)(

h

mVRn

dm

h

Vd

d

dndg 2123

3

24)()(


The last equation takes into account the translational motion only ofa particle of the system. But quantum particles may have spin as well.Thus we must multiply the above equation by a spin factor γs:

dmh

Vdg s

21233

24)(

where γs = 1 for spin zero bosons and γs = 2 for spin one-halffermions.

Bosons are particles with integer spin. Fermions are particles withhalf-integer spin.

For molecules, states associated with rotation and vibration mayexist in addition to translation and spin.

chapter 2 statistical thermodynamics

Documents

example of coin

number of heads n1

number of ways

enormous number

number betweenzero

alarger number of coins

given macrostate

therefore2 coin model