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Statistical Mechanics statistical

distributions Maxwell-Boltzmann

Statistics molecular energies in an ideal

gas quantum statistics

Mar. 25 go to9.

Apr. 4 go to24.

Since Ive been unjustly picking on chemists

All science is either Physics or stamp

collecting.Ernest Rutherford, physicist and

1908 Nobel Prize winner in Chemistry.

Chapter 9 Statistical Mechanics

spinach

good stuff!

get it outta here

spinach

As a grade schooler, I went to a Catholic school.

They served lots of stewed spinach. Some of us

got sick just from the fumes wafting up from the

cafeteria 2 floors down.

The nuns made us clean our plate at lunch. It

had something to do with the starving children in

China.

They would inspect our trays as we passed through

the dump the trash line.

What to do on spinach day

Stuff it in your empty milk carton and hope the

nuns didnt inspect it

Sit next to the one kid in class who liked stewed

spinach, and see how much you could pass off to

him

The most valuable kid in school that day.

What does this have to do with statistical

mechanics

Physics faculty tend to think of thermodynamics

(including statistical mechanics) as the stewed

spinach of college physics courses.

The wacko faculty member who actually likes

teaching that stuff is a treasured friend

whenever it comes time to give out teaching

assignments.

Just thought you might want to know that before

we start this chapter on statistical mechanics

(thermodynamics).

Fair warnings and all that. There are more

cautionary tales, but not for now. I dont want

to scare you all off at once.

Before we get into chapter 9, lets think about

this course a bit.

We started with relativity. A logical starting

point, because relativity heralded the beginning

of modern physics.

Relativity forced us to start accepting

previously unthinkable ideas.

Relativity is more fun than most of the rest

of the material, so it wont drive prospective

students away from the class.

Relativity shows us that photons have momentuma

particle property, and gets us thinking about

particle properties of waves.

Waves having particle properties leads us (e.g.,

de Broglie) to ask if particles have wave

properties.

After chapter 3, we backtracked to try to explain

the properties of hydrogen, the simplest atom.

This is backtracking, because we had to move

backwards in the time sequence of discoveries,

from de Broglie back to Rutherford.

It doesnt break the logical chain because we

find we cant explain hydrogen without invoking

wave properties.

The puzzle of hydrogen forces us to completely

re-think the fundamental ideas of physics.

If physics cant explain hydrogenthe simplest

of all atomsit is in dire shape. Something

drastic must be done.

Something drastic quantum mechanics

Once quantum mechanics is discovered we rush off

to find applications and confirmations.

A logical place to start testing quantum

mechanics (Schrödingers equation) is to start

with simple model systems (particle in box) and

move up from there.

Once weve practiced with model systems, we go

full circle and apply quantum mechanics to the

system that started this trouble in the first

placehydrogen.

The next step up from hydrogen is atoms (chapter

7).

The next step up from atoms is a few atoms bonded

together (chapter 8).

Whats the next step up from a few atoms bonded

together

Good! Lots of atoms. Lets start with them

interacting but not bonded together in large

masses (chapter 9). Then well be able to tackle

lots of atoms bonded together (chapter 10).

Theres a logic to this, isnt there. No wonder

most modern physics books follow the same

sequence.

9.1 Statistical Distributions

Statistical mechanics deals with the behavior of

systems of a large number of particles.

We give up trying to keep track of individual

particles. If we cant solve Schrödingers

equation in closed form for helium (4 particles)

what hope do we have of solving it for the gas

molecules in this room (10really big number

particles).

Statistical mechanics handles many particles by

calculating the most probable behavior of the

system as a whole, rather than by being concerned

with the behavior of individual particles.

not again

yes, again

In statistical mechanics, we assume that the more

ways there are to arrange the particles to give a

particular distribution of energies, the more

probable is that distribution. (Seems

reasonable)

6 units of energy, 3 particles to give it to

more likely

(repeating) In statistical mechanics, we assume

that the more ways there are to arrange the

particles to give a particular distribution of

energies, the more probable is that distribution.

(Seems reasonable.)

We begin with an assumption that we believe

describes nature.

We see if the consequences of the assumption

correspond in any way with reality.

It is not bad to begin with an assumption, as

long as we realize what we have done, and discard

(or modify) the assumption when it fails to

describe things we measure and observe.

A brief note...

Beiser mentions W, which is the number of ways to

arrange particles to give a particular

distribution of energies.

The idea is to calculate W, and find when W is

maximized. That gives us the most probable state

of the system.

W doesnt appear anywhere else in this chapter.

In previous editions, it was calculated in an

appendix, where Beiser derived all the

distribution functions we will use.

So you dont need to worry about W.

http//www.rozies.com/Zzzz/920/alpha.html

Here, in words, is the equation we will be

working with in this chapter

( of particles in a state of energy E)

( of particles in a state of energy E) ( of

states having energy E) x

( of particles in a state of energy E) ( of

states having energy E) x (probability that a

particle occupies the state of energy E).

If we know the distribution function, the

(probability that a particle occupies a state of

energy E), we can make a number of useful

calculations.

Mathematically, the equation is written

It is common to use epsilon to represent energy

I will call it E when I say it.

In systems such as atoms, only discrete energy

levels are occupied, and the distribution g() of

energies is not continuous.

On the other hand, it may be that the

distribution of energies is continuous, or at

least can be approximated as being continuous.

In that case, we replace g(e) by g(e)de, the

number of states between e and ede.

We will find that there are several possible

distributions f(e) which depend on whether

particles are distinguishable, and what their

spins are.

Beiser mentions them (Maxwell-Boltzmann,

Bose-Einstein, Fermi-Dirac) in this section.

Lets wait and introduce them one at a time.

9.2 Maxwell-Boltzmann Statistics

We take another step back in time from quantum

mechanics (1930s) to statistical mechanics (late

1800s).

Classical particles which are identical but far

enough apart to be distinguishable obey

Maxwell-Boltzmann statistics.

classical slow, wave functions dont overlap

distinguishable you would know if two particles

changed places (you could put your finger on one

and follow it as it moves about)

Two particles can be considered distinguishable

if their separation is large compared to their de

Broglie wavelength.

Example ideal gas molecules.

The Maxwell-Boltzmann distribution function is

Ill explain the various symbols in a minute.

Boltzmann discovered statistical mechanics and

was a pioneer of quantum mechanics.

His work contained elements of relativity and

quantum mechanics, including discrete atomic

energy levels.

In his statistical interpretation of the second

law of thermodynamics he introduced the theory of

probability into a fundamental law of physics and

thus broke with the classical prejudice, that

fundamental laws have to be strictly

deterministic. (Flamm, 1997.)

With Boltzmanns pioneering work the

probabilistic interpretation of quantum mechanics

had already a precedent.

Boltzmann constantly battled for acceptance of

his work. He also struggled with depression and

poor health. He committed suicide in 1906. Most

of us believe thermodynamics was the cause. See

a biography here.

Paul Eherenfest, who wrote Boltzmanns eulogy,

carried on (among other things) the development

of statistical thermodynamics for nearly three

decades.

Ehrenfest was filled with self-doubt and deeply

troubled by the disagreements between his friends

(Bohr, Einstein, etc.) which arose during the

development of quantum mechanics.

Ehrenfest shot himself in 1933.

US physicist Percy Bridgmann (the man on the

right, winner of the 1946 Nobel Prize) took up

the banner of thermodynamics, and studied the

physics of matter under high pressure.

Bridgman committed suicide in 1961.

Theres no need for you to worry Ive never lost

a student as a result of chapter 9 yet

Back to physics

The facts above accurate but rather selectively

presented for dramatic effect.

Maxwell-Boltzmann distribution function

The number of particles having energy e at

temperature T is

A is like a normalization constant we integrate

n(e) over all energies to get N, the total number

of particles. A is fixed to give the right

answer for the number of particles. For some

calculations, we may not care exactly what the

value of A is.

e is the particle energy, k is Boltzmanns

constant (k 1.38x10-23 J/K), and T is the

temperature in Kelvin.

Often k is written kB. When k and T appear

together, you can be sure that k is Boltzmanns

constant.

We still need g(e), the number of states having

energy e. We will find that g(e) depends on the

problem under study.

Beiser justifies this distribution in Chapter 9,

and but doesnt derive it in the current text. I

wont go through all this justification. You can

read it for yourself.

Before we do an example monatomic hydrogen is

less stable than H2, so are you more likely to

find H2 or H in nature

H2, of course!

Nevertheless, suppose we could make a cubic

meter of H atoms. How many atoms would we have

Example 9.1 A cubic meter of atomic H at 0 ºC and

atmospheric pressure contains about 2.7x1027 H

atoms. Find how many are in their first excited

state, n2.

Gas atoms at atmospheric pressure and temperature

behave like ideal gases. Furthermore, they are

far enough apart that Maxwell-Boltzmann

statistics can be used.

For the hydrogen atoms in the ground state,

For the hydrogen atoms in first excited state,

We can divide the equation for n(e2) by the

equation for n(e1) to get

Important this is temperature in K, not in C!

We know e1, e2, and T. We need to calculate the

g(e)s, which are the number of states of energy

e. We dont need to know A, because it divides

out.

We get g(e) for hydrogen by counting up the

number of allowed electron states corresponding

to each e.

Or we can simply recall that there are 2n2 states

corresponding to each n, so that g(e1)2(1)2 and

g(e2)2(2)2.

Plugging all of our numbers into the above

equation gives n(e2)/n(e1)1.3x10-188. In other

words, none of the atoms are in the n2 state.

Caution the solution to example 9.1 and

g(en)2(n)2 only works for energy levels in

atomic H and not for other assigned problems!

For example, to do it for H2 would require

knowledge of H2 molecular energy levels.

Skip example 9.2 I wont test you on it.

9.3 Molecular Energies in an Ideal Gas

The example in section 9.2 dealt with atomic

electronic energy levels in atomic hydrogen. In

this section, we apply Maxwell-Boltzmann

statistics to ideal gas molecules in general.

We use the Maxwell-Boltzmann distribution to

learn about the energies and speeds of molecules

in an ideal gas.

We already have f(). We assume a continuous

distribution of energies (why), so that

We need to calculate g(e), the number states

having an energy e in the range e to ede.

g(e) is called the density of states.

It turns out to be easier to find the number of

momentum states corresponding to a momentum p,

and transform back to energy states.

Why Every classical particle has a position and

momentum given by the usual equations of

classical mechanics.

Corresponding to every value of momentum is a

value of energy.

Momentum is a 3-dimensional vector quantity.

Every (px,py,pz) corresponds to some energy.

Think of as (px,py,pz) forming a 3D grid in

space. We count how many momentum states there

are in a region of space (the density of momentum

states) and then transform to the density of

energy states.

We need to find how many momentum states are in

this spherical shell.

The Maxwell-Boltzmann distribution is for

classical particles, so we write

The number of momentum states in a spherical

shell from p to pdp is proportional to 4pp2dp

(the volume of the shell).

Thus, we can write the number of states having

momentum between p and pdp as

where B is a proportionality constant, which we

will worry about later.

Because each p corresponds to a single e,

Now,

so that

and

The constant C contains B and all the other

proportionality constants lumped together.

If the derivation on the previous fours

went by rather fast and seems quite confusing

If the derivation on the previous fours

went by rather fast and seems quite confusing

dont worry, thats quite normal. Its only the

final result (which we havent got to yet) which

I want you to be able to use.

Here are a couple of links presenting the same

(or similar) derivation

hyperphysics

Britney Spears Guide to Semiconductor Physics

Density of States

To find the constant C, we evaluate

where N is the total number of particles in the

system.

Could you do the integral

Could you do the integral Could I do the

integral

Could you do the integral Could I do the

integral No, not any more.

Could you do the integral Could I do the

integral No, not any more. Could I look the

integral up in a table

Could you do the integral Could I do the

integral No, not any more. Could I look the

integral up in a table Absolutely!

The result is

so that

This is the number of molecules having energy

between e and ede in a sample containing N

molecules at temperature T.

Wikipedia says The Maxwell-Boltzmann

distribution is an important relationship that

finds many applications in physics and chemistry.

It forms the basis of the kinetic theory of

gases, which accurately explains many fundamental

gas properties, including pressure and diffusion.

The Maxwell-Boltzmann distribution also finds

important applications in electron transport and

other phenomena.

Webchem.net shows how the Maxwell-Boltzmann

distribution is important for its influence on

reaction rates and catalytic reactions.

Heres a plot of the distribution

Notice how no molecules have E0, few molecules

have high energy (a few kT or greater), and there

is no maximum of molecular energy.

k has units of energy/temperature so kT has

units of energy.

Heres how the distribution changes with

temperature (each vertical grid line corresponds

to 1 kT).

Notice how the distribution for higher

temperature is skewed towards higher energies

(makes sense!) but all three curves have the same

total area (also makes sense).

Notice how the probability of a particle having

energy greater than 3kT (in this example)

increases as T increases.

If you arent interested enough in the derivation

of g(e) to visit Britney Spears Guide to

Semiconductor Physics Density of States

you might miss this graphic. (Can you tell

which one has the Ph.D.)

Continuing with the physics, the total energy of

the system is

Evaluation of the integral gives

This is the total energy for the N molecules, so

the average energy per molecule is

exactly the result you get from elementary

kinetic theory of gases.

Things to note about our ideal gas energy

The energy is independent of the molecular

mass.

Which gas molecules will move faster at a given

temperature lighter or heavier ones Why

at room temperature is about 40 meV, or

(1/25) eV. This is not a large amount of energy.

kT/2 of energy goes with each degree of

freedom.

Because e mv2/2, we can also calculate the

number of molecules having speeds between v and v

dv.

The result is

We (Beiser) call this n(v). The hyperphysics

web page calls it f(v).

Heres a plot (number having a given speed vs.

speed)

Looks like n( ) plotnothing at speed0, long

exponential tail.

The speed of a molecule having the average energy

comes from solving

for v. The result is

vrms is the speed of a molecule having the

average energy .

It is an rms speed because we took the square

root of the square of an average quantity.

The average speed can be calculated from

The result is

Comparing this with vrms, we find that

Because the velocity distribution curve is skewed

towards high energies, this result makes sense

(why).

You can also set dn(v) / dv 0 to find the most

probable speed. The result is

The subscript p means most probable.

Summarizing the different velocity results

Plot of velocity distribution again

This plot comes from the hyperphysics web site.

The Rs and Ms in the equations are a result of

a different scaling than we used. See here for

how it works (not testable material).

Example 9.4 Find the rms speed of oxygen

molecules at 0 ºC.

Would anybody (who hasnt done the calculation

yet) care to guess the rms speed before we do the

calculation

0 m/s

0 m/s 10 m/s

0 m/s 10 m/s 100 m/s

0 m/s 10 m/s 100 m/s 1,000 m/s

0 m/s 10 m/s 100 m/s 1,000 m/s 10,000 m/s

You need to know that an oxygen molecule is O2.

The atomic mass of O is 16 u (1 u 1 atomic mass

unit 1.66x10-27 kg).

Holy cow! You think youd feel all these

zillions of O2 molecules constantly crashing into

your skin at more than 1000 mph!

And why no sonic booms (Nothis is not a

question I expect you to answer.)

Click here and scroll down for a handy molecular

speed calculator.

Now, weve gone through a lot of math without

thinking much about the physics of our system of

gas molecules. We should step back and consider

what weve done.

A statistical approach has let us calculate the

properties of an ideal gas. Weve looked at a

few of these properties (energies and speeds).

Who cares about ideal gases Anybody interested

in the atmosphere, or things moving through it,

or machines moving gases around. Chemists.

Biologists. Engineers. Physicists. Etc.

This is modern physics, but not quantum

physics.

9.4 Quantum Statistics

Here we deal with ideal particles whose wave

functions overlap. We introduce quantum physics

because of this overlap.

Remember

The function f(e) for quantum statistics depends

on whether or not the particles obey the Pauli

exclusion principle.

The wierd thing about the half-integral spin

particles (also known as fermions) is that when

you rotate one of them by 360 degrees, its

wavefunction changes sign. For integral spin

particles (also known as bosons), the

wavefunction is unchanged. Phil Fraundorf of

UMSL, discussing why Balinese candle dancers have

understood quantum mechanics for centuries.

Links--- http//hyperphysics.phy-astr.gsu.edu/hbas

e/math/statcon.htmlc1 http//www.chem.uidaho.edu/

honors/boltz.html http//www.wikipedia.org/wiki/B

oltzmann_distribution http//www.webchem.net/notes

/how_far/kinetics/maxwell_boltzmann.htm http//www

.physics.nwu.edu/classes/2002Spring/Phyx103taylor/

mbdist.html http//mats.gmd.de/skaley/pwc/boltzma

nn/Boltzmann.html http//britneyspears.ac/physics/

dos/dos.htm

Cut and paste stuff l ö

h

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