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Statistical Physics 1

Topics

- Introduction
- The Boltzmann Distribution
- The Maxwell Distribution
- Summary

Introduction

- We believe we now have the basic laws that, in

principle, can be used to predict the detailed

behavior of an arbitrarily large assembly of

atoms and molecules - But even a tiny piece of matter consists of

millions of atoms - In practice, the complexity of the calculation is

far beyond the capability of any conceivable

computer and we need a different approach

Introduction

- Happily, for most applications we are not

interested in the precise behavior of each atom,

but only the collective behavior of the assembly,

which can be described with only a few variables,

such as temperature, pressure and volume - Statistical physics is the study of the

collective behavior of large assemblies of atoms

and molecules using probabilistic reasoning

The Boltzmann Distribution

The Austrian physicist Boltzmann asked the

following question in an assembly of atoms,

what is the probability that an atom has total

energy between E and EdE? His answer

where

Ludwig Boltzmann 1844 - 1906

The Boltzmann Distribution

is called the Boltzmann distribution, e-E/kT is

the Boltzmann factor and k 8.617 x 10-5

eV/K is the Boltzmann constant

The Boltzmann distribution applies to identical,

but distinguishable particles

Ludwig Boltzmann 1844 - 1906

The Boltzmann Distribution

The number of particles with energy E is given by

where g(E) is the statistical weight, i.e.,

the number of states with energy E. However, in

classical physics the energy is continuous so we

must replace g(E) by g(E)dE, which is the number

of states with energy between E and E dE. g(E)

is then referred to as the density of states.

The Boltzmann Distribution

Example 8-1 The Law of Atmospheres

Classically, the total energy of a gas

molecule of mass m, near the Earths surface, is

where z is the vertical distance above

the ground

z

Wanted the fraction of particles between z and

zdz

So we can write

The Boltzmann Distribution

Example 8-1 The Law of Atmospheres

A basic rule of probability is sum, or

integrate, over quantities whose values are

either unknown or not of interest. We are

interested only in z. After integrating the

Boltzmann distribution with respect to p we get

z

The Boltzmann Distribution

Example 8-1 The Law of Atmospheres

The fraction of molecules between z and z dz is

z

At T 300K, the ratio of the fraction at z

1000 m to that at z 0 m is just fB(1000) /

fB(0) 0.893

The Boltzmann Distribution

Example 8-2 H Atoms in First Excited State

At temperature T, the atoms of a gas will occupy

different energy levels. For hydrogen, the

energy difference E2 - E1 between the 1st excited

state and the ground state is 10.2 eV. What is

the ratio of the number of atoms in the 1st

excited state to the number in the ground state

at T 5800 K (the temperature of the Suns

surface)?

The Boltzmann Distribution

- Example 8-2
- Number of atoms in state E
- Ratio of number of atoms in E2 and E1

The Boltzmann Distribution

- Example 8-2
- Ratio of statistical weights. The degeneracy for

each orbital quantum number l is given by 2l1.

For the ground state of hydrogen, l 0, which

gives 1. For the 1st excited state l 0 and l

1, which gives 4. But for each of these states

the electron has 2 spin states. So we have g1 2

and g2 8. So

The Boltzmann Distribution

- Example 8-2
- 4. For T 5800 K (kT 0.5 and DE E2-E1

10.2 eV) we have

Even at the Suns surface there are relatively

few atoms in the 1st excited state. This

is because the energy gap DE gtgt kT

The Maxwell Distribution

An important application of the

Boltzmann distribution is the distribution of

molecular speeds v in a gas of N molecules

This distribution, in fact, was derived by James

Clerk Maxwell before Boltzmanns work. But it is

an important (and famous) special case.

The Maxwell Distribution

Most probable Average RMS

Different summaries of the molecular speed

computed from the Maxwell distribution

The Maxwell Distribution

Example The average speed of a nitrogen

molecule at T 300 K is given by

With k 1.39 x 10-23 J/K and m 4.68 x 10-26

kg one gets ltvgt 475 m/s 1700 km/h

Summary

- Statistical physics is the study of the

collective behavior of large assemblies of

particles - Ludwig Boltzmann derived the following energy

distribution for identical, but distinguishable,

particles - The Maxwell distribution of molecular speeds is a

famous application of Boltzmanns general formula