Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann...

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Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N particles obeying Maxwell- Boltzmann statistics, the occupation number for the jth energy level is given by T j j Z NkT N ln

Transcript of Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann...

Page 1: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

Summary

• Boltzman statistics:

• Fermi-Dirac statistics:

• Bose-Einstein statistics:

• Maxwell-Boltzmann statistics:

• Problem 13-4: Show that for a system of N particles obeying Maxwell-Boltzmann statistics, the occupation number for the jth energy level is given by

Tjj

ZNkTN

ln

Page 2: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

13.7 The Connection between Classical and Statistical thermodynamics

U = Σ NJ EJ where EJ = EJ (x)

x is an extensive property such as volume (see eqn

12.21!)

Differentiate the above equation

Since

Page 3: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

define

For two states with the same X

For classical thermodynamics

YdXTdS

or

PdVTdSdU

Page 4: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

Since dQ = T·dS and dW = PdV = YdX

The first equation illustrates that the heat transfer is energy resulting in a net redistribution of particles among the available energy levels involving NO WORK!

Page 5: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

The equation could be interpreted as: An increase in the system’s internal energy could be brought by a decrease in volume with an associated increase in the EJ. the energy levels are shifted to higher values with no redistribution of the particles among the levels.

 

For an open system, the change in the internal energy is

where μ is the chemical potential defined on a per particle.

Page 6: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

The Helmholtz function (F = U –TS) can be written as

For MB statistics

)!ln()ln()ln( jN

jMB NgkWkS j

Page 7: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.
Page 8: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

Now the MB distribution can be rewritten as

(similar to those derived in FD

and BE statistics)

kTuE

kTE

j

j

j

j

ee

Z

N

g

N)(

1

)1ln(ln NZNkTTSUF

Page 9: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

13.8 Comparison of the three Distributions

a = 1 for FD statistics

a = -1 for BE statistics

a = 0 for MB statistics aeg

N

kT

uEj

j

j

1

Page 10: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

• BE curve: The distribution is undefined for x < 0. particles tend to condense in regions where Ej is small, that is, in the lower energy state.

• FD curve: At the lower levels with Ej – u negative the quantum states are nearly uniformly populated with one particle per state.

• MB curve: lies between BE and FD curves and is only valid for the dilute gas region: many states are unoccupied.

• Statistical equilibrium is a balance between the randomizing forces of thermal agitation, tending to produce a uniform population of the energy levels, and the tendency of mechanical systems to sink to the states of lowest energy.

Page 11: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

Alternative Statistical Models

• Microcanonical ensemble: treats a single material sample of volume V consisting of an assembly of N particles with fixed total energy U. The independent variables are V, N, and U.

• The canonical ensemble: considers a collection of Na identical assemblies, each of volume V. A single assembly is assumed to be in contact through a diathermal wall with a heat reservoir of the remaining Na -1 assemblies. The independent variables are V, N, and T, where T is the temperature of the reservoir.

• Grand canonical ensemble: consists of open assemblies that can exchange both energies and particles with a reservoir. This is the most general and most abstract model. The independent variables are V, T and u, where u is the chemical potential.

Page 12: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

Further comments on degeneracy

• A microstate is said to be non-degenerate if no two particles have the same energy.

• Alternative way of defining a non-degenerate system is to say that the number of quantum states with E <kT, >> N, where k is Boltzmann constant, T is the temperature and N is the number of particles.

• Equation 12.24 can be employed to calculate the number of states contained within the octant.

Page 13: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

• Example: Calculate the total number of accessible microstates in a system where 100 units of energy have been distributed among three distinguishable particles of zero spin.

• Solution:

Page 14: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

• 13-10 (a) using results from chapter 9, show that

(b) It follows from the statistical definition of the entropy that

Consider a system with a chemical potential u = - 0.3eV. By what factor is the number of possible microstates of the system increased when a single particle is added to it at room temperature?

(k = 8.617 x 10-5 eVK-1)

• Solution: (a) using equation 13.53 directly

VUN

STu

,

NkT

uW ln

Page 15: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

Chapter 14: The Classical Statistical

Treatment of an Ideal Gas

Page 16: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

14.1 Thermodynamic properties from the Partition Function

• All the thermodynamic properties can be expressed in terms of the logarithm of the partition function and its derivatives.

• Thus, one only needs to evaluate the partition function to obtain its thermodynamic properties!

Page 17: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

• From chapter 13, we know that for dilute gas system, M-B statistics is applicable, where

S= U/T + Nk (ln Z – ln N +1)

F= -NkT (ln Z – ln N +1)

μ= = – kT (lnZ – lnN)

• Now one can derive expressions for other thermodynamic properties based on the above relationship.

Page 18: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

1. Internal Energy:

We have …

kT

j

jj

n

jj

j

eZ

N

g

NandNU

1

j

kTjj

jjj

kTjj

egZ

Nge

Z

NU

j

kTj

j

egZ

Page 19: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

differentiating the above Z equation with respect to T, we have …

= (keeping V constant means ε(V) is constant)

Therefore,

or

))1

((2

jj

kTj

j

j

kTj

V kTeg

dT

kTd

egT

Z jj

j

kTjj

j

egkT

2

1

VT

ZkT

Z

NU

)( 2

VT

ZNkTU

ln2

Page 20: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

2. Gibbs Function since G = μ N

G = - NkT (ln Z – ln N)

3. Enthalpy G = H – TS → H = G + TS

H =-NkT(lnZ - ln N) + T(U/T + NklnZ – NklnN + Nk)

= -NkTlnZ + NkTlnN + U + NkTlnZ – NkTlnN + NkT

= U + NkT = NkT2 + NkT

= NkT (1 + T · )

Page 21: Summary Boltzman statistics: Fermi-Dirac statistics: Bose-Einstein statistics: Maxwell-Boltzmann statistics: Problem 13-4: Show that for a system of N.

4. Pressure

TV

FP

TV

NZNkTP

))1ln(ln(

TV

NZNkTP

)1ln(ln

TV

ZNkTP

00)(ln

TV

ZNkTP

)(ln