1

Prof Andy Buffler

Room 503 RW James

andy.buffler@uct.ac.za

PHY2009S 2010

Intermediate Physics

Thermodynamics and

Statistical models in physics

2

60 lectures: Monday to Friday, 5th period.

12 Tuesday afternoon sessions: Wednesdays, 14:00 to 17:00.

(laboratory practicals, theory tutorials and VPython tuts)

13 weekly problem sets

2 class tests

1 final examination (November)

Use the class tutor …

Also check the course website regularly for resources …

… click from www.phy.uct.ac.za

PHY2009S 2009

3

These lecture notes are based upon

Matter and Interactions

2nd Edition, Volume 1

Ruth Chabay and Bruce Sherwood

Chapters …

6. Energy in Macroscopic Systems

7. Energy Quantization

11. Entropy: Limits on the Possible

12. Gases and Engines

PHY2009S 2009

Intermediate Physics

Thermodynamics and

statistical models in physics

… and are derived in some places from slides produced by

David Aschman and Indresan Govender.

4

PHY2009S

Thermodynamics and statistical models in physics

Part 1

The

conservation

laws

5

One fundamental idea in physics: conservation

In a closed system,

energy, momentum, angular momentum, charge, …

are conserved [with a bit of fine print to consider].

What is physics?

… a small number of abstract ideas (principles and

theories) can be applied to a wide range of physical

applications, which allow description / explanation /

predication of natural phenomena.

6

Conservation of energy and charge in e+e- production

7

The universe in which we live: the physics view

The universe started with a big bang, 13.8 billion years ago ...

… beginning of “spacetime”

The universe consists of …

… electromagnetic radiation and matter …

… held together by forces (interactions).

8

9

10

“Classical” physics (pre 1900) …

… includes …

… Newtonian mechanics

… Maxwell’s electromagnetic theory

Classical physics based on two (philosophical) assumptions:

… nature is continuous

… space and time are independent of each other

These views were superseded after 1900 …

… by Planck, Schrodinger, Heisenberg, Bohr, Einstein, …

[Quantum theory … nature is “quantised”]

… by Einstein

[Special and general relativity

… the speed of light is constant for all observers]

11

The momentum principle

… is a fundamental principle in physics …

… states that the change in momentum of a system is equal

to the net force acting on the system times the duration of

the interaction…net tp F

wheref ip p p

andnet 1 2 3 ...F F F F

Principle of

conservation of momentumsystem surroundings 0p p

12

The energy

principlesystem surroundingsE W Q

The change in energy ( ) of a system is equal to the

work done on the system by the surroundings ( ) , and to

energy flow between the system and the surroundings due to a

difference in temperature ( Q ).

systemE

surrW

The momentum principle (Newton’s 2nd law) tells us how

systems evolve, given the forces involved.

Energy is a concept that allows us to say what is possible …

… what are the constraints on behaviour.

The evolution of a system can be related to the transfer of energy

… leads to the conservation of energy as a central principle.

What is energy?

13

There are basically two modes of energy we need consider:

• Energy of particles.

• Energy arising from the interaction between particles.

All others are semantics.

Energy of a particle

What is the energy of a particle?

… what is a particle ..?

A particle is any object that does not undergo any

internal changes during the process of interest.

What we regard as a particle depends on our model …

… e.g. is the Earth a particle?

14

Newtonian mechanics works well at low speeds, but fails

when applied to objects whose speed approaches c.

Albert Einstein: Special theory of relativity 1905

General theory of relativity 1917

The special theory is based on two postulates …

Special Relativity

The relativity postulate: The laws of physics are the same for all

observers in all inertial reference frames.

(Inertial reference frames move at constant velocity with respect

to each other.)

The speed of light postulate: The speed of light in free space

(vacuum) has the same value (c = 3 108 m s-1) in all directions

and in all reference frames.

15

The energy of a single particle system

2

particleE mc where .2 2

1

1 v c

“Rest energy” of a particle at rest = .2mc

Therefore the kinetic energy of a particle: 2 2

2 1

K mc mc

mc

or2

particleE mc K

Einstein (1905):

As v approaches c,

K (and hence E) become very large

(and approach each other in magnitude). 2mc

K

particleE

16

The unit of energy is 1 N m = 1 joule.

1 joule = 1 J = 1 kg m2 s-2.

Another unit in wide use is the electron volt.

1 eV = 1.602 × 10−19 J.

This is used mainly when dealing with (sub-)atomic particles.

Units for energy

17

Kinetic energy

At low speeds, the kinetic energy becomes a useful concept.

For : v c2 32 2 2 2

2

particle 2 2 22 2

1 3 51 ...

2 8 161

mc v v vE mc

c c cv c

Taking the lowest order terms:2

2 2

particle 2 2

1

21

mcE mc mv

v c

Thus kinetic energy K at low speeds is 2

21

2 2

pK mv

m2mc

K

particleE

when v c2K mc

18

Now we can write:2 2 K mc mc E

total energyrest energy

kinetic energy

Therefore2

0 E mc E

mass is a form of energy.

A small mass can produce an enormous amount of energy

It can be shown that . 2 2 2 2 4E p c m c

For an object at rest, p = 0 , so E = E0 = mc2

… total energy = rest energy

For particles which have zero mass, e.g. photons, E = pc.

Kinetic energy …2

19

Potential energy

Isolated particles have only the energy .

If a particle interacts with others, there is an energy associated

with the interactions: this is called potential energy.

2

2 21

mcE

v c

Suppose a system of several (or many) particles interact with

one another and external forces act as well.

Then 1 2 3 internal external...E E E W W

We can write this as1 2 3 internal external...E E E W W

1 2 3 external...E E E U Wor

Hence we define the work done by the internal forces

to be the change in potential energy U.internalW

internalU W

20

Conservation of energy

We can define the energy of the system to be

system 1 2 3( ...)E E E E U

For example, for a 3 particle system:2 2 2

system 1 2 3 1 2 3 12 23 13( )E m c m c m c K K K U U U

and

system surroundingsE WSince

surroundings surroundingsE W

Write principle of

conservation of energysystem surroundings 0E E

Thus the energy of a system is changed by

transferring energy to or from the surroundings.

21

Energy diagrams

Energy versus separation

distance between a spacecraft

and a planet.

22

Energy in macroscopic systems

Macroscopic systems are composed of many interacting

particles.

The potential energy associated with stretching or

compressing interatomic bonds is similar to the potential

energy of macroscopic springs.

Thermal energy is the kinetic and potential energy of the

atoms and interatomic bonds within an object.

23

Mass-spring oscillator

Hooke’s Law:

Restoring force,

For horizontal forces on the mass:

Start with the

momentum principle:

2

2

skd xx

dt m

restore skF x

0x x xwhere

and ks is the “spring constant”

[N m-1]

or

x

x

x

unstretched

compessed

extended

externalF

externalFrestoreF

0x

x

x

restoreF

ks

m

net

d

dt

pF

xs

dpk x

dt( )x

s

d mvk x

dts

d dxm k x

dt dt

24

t

t

t

0( ) cos( )x t A t

0 0( ) sin( )v t A t

2

0 0( ) cos( )a t A t

A

0

0

0

0A

2

0A

Mass-spring oscillator …2

0sk

m

25

Mass-spring oscillator: an energy approach

Equilibrium

x

Suppose that the mass

has a speed v when it

has displacement x

v

m

m

Kinetic energy of mass =

Potential energy of spring =

212mv

' 212

0 0

' '

x x

s sFdx k x dx k x

There are no dissipative mechanisms in our model (no friction).

… the total energy of the mass-spring system is conserved.

2 21 12 2

constantsmv k x

sk : spring constant

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Mass-spring oscillator: an energy approach …2

2 21 12 2

constantsmv k xFor our mass-spring system:

2 21 12 2

0s

dmv k x

dt

0s

dv dxmv k x

dt dt

0s

dvmv k xv

dt

0s

dvm k x

dt2

2

skd xx

dt m

0( ) cos( )x t A tWhich has solution where 0

sk

m

27

For the mass-spring system: 0( ) cos( )x t A t

Potential energy U = 2 2 21 1

02 2cos ( )s sk x k A t

K = 2 2 2 2 21 1 1

0 0 0 02 2 2[ sin( )] sin ( )mv m A t mA t

Total energy = K + U

212 sk A 2( )E A

Mass-spring oscillator: an energy approach …3

2 2 2 2 21 10 0 02 2

cos ( ) sin ( )sk A t mA t

28

t

t

t

t

0

0

0

0

2 2102

p.e. cos ( )sk A t

2 2 210 02

k.e. sin ( )sk A t

212

total energy sk A

0cos( )x A t

Energy of the mass-spring simple harmonic oscillator

29

Young’s modulus

We take a wire of length L and stretch it by

applying an external force FT.

Say that the wire stretches by an amount ∆L.

Then the “strain” L

L

The tension force

per unit area is called the “stress” TF

A

Young’s modulus Ystress

strain

TF

A

L

L

L L

L

TF

30

Young’s modulus …2

A bar or wire can be regarded as a non-helical macroscopic spring.

Compare with for a spring

TF

AY

L

L

Write as T

YAF L

L

restore sF k x

Elastic oscillations in a wire

F

m

Here F is restoring force in wire

AYxma

L0

AY

mLand

31

Silicon atoms

32

Thermal energy in solid materials

Now considering the transfer of

energy to the atoms making up a

solid material. We model a solid as a

large number of tiny masses (the

atoms) connected to their neighbours

by springs (the inter-atomic bonds).

The “internal energy” of a solid can

be increased by increasing the kinetic

energy of the atoms or the potential

energy of the atom-spring system.

Since there are so many atoms we can only consider such

things in an averaged way.

But how do we measure this energy?

33

Temperature

Temperature and average energy of molecules are related.

(Joule 1842: “the mechanical equivalent of heat”).

A thermometer measures “temperature” T on some scale

which might be “degrees” or “joules”.

An ideal thermometer is the constant volume gas

thermometer, in which the gas pressure is measured as a

function of temperature.

By extrapolation, there is a temperature at which the

pressure becomes zero.

This is defined as the zero of the Kelvin scale.

The increments are the same as the Celsius scale.

0 K is also known as “absolute zero” in temperature.

0 K 273.15 C

34

Change of energy and temperature

How does change in temperature relate to change in

internal energy?

Joule found that 4.2 J was required to change the temperature

of 1 g of water by 1 K.

We define a (specific) heat capacity C by

thermalE mC T

Specific heat capacity of ethanol = 2.4 J g-1 K-1

Specific heat capacity of copper = 0.4 J g-1 K-1

or thermalEC

m Twith m in grams

35

Thermal energy transfer

If a hotter body is placed in contact with a colder body, then a

certain amount of energy passes from the hotter to the colder until

they are in thermal equilibrium, i.e. they are at the same

temperature.

This happens by work done at the microscopic level.

This thermal energy transfer is usually denoted by the symbol Q.

[It is sometimes (confusingly!) known as “heat”.]

High

temperatureLow

temperature

Thermal energy transfer Q

36

Thermal energy transfers

We normally separate the work done

in changing the energy of a system

into “macroscopic” work (W) and

“microscopic” work (Q).

This is an expression of the

“First law of thermodynamics”

… a version of conservation of energy.

systemE W Q

systemE

systemE

increases

decreases

Q > 0

Q < 0

37

A short history of the development of atomic theory

JJ Thomson (Nobel Prize in Physics, 1906)

“Plum pudding” model.

Poor agreement with experiment.

Energy quantizationM&I

Chapter 7

…but first …

38

The Rutherford model

Ernest Rutherford

(Nobel Prize in Chemistry, 1908)

39

Simulation of Rutherford scattering

off a gold nucleus

(a) Thomson’s atom

(b) Rutherford’s atom

Scattering of a 4He nucleus off …

The Rutherford model …2

40

Rutherford: “It was the most incredible event that has ever

happened to me in my life. It was almost as incredible as if you

fired a 15 inch shell at a piece of tissue paper and it came back

and hit you.”

Rutherford model (1911): The atom has a small hard central

core (nucleus) where all the positive charge is concentrated.

The negative charge inhabitants the nearly empty space around

the nucleus.

Thus most of the alpha particles migrate through the gold foil

with some or no (Coulomb) interaction, but some will

experience a “head-on” collision with a nucleus and return in a

backwards direction.

The Rutherford model …3

41

But there were still unanswered questions …

… why does the nucleus (all positive charge) not fly apart due

to Coulomb repulsion?

… why do the negative charges not radiate energy, spiral

inwards and collapse into the nucleus due to Coulomb

attraction?

… the model did also not explain existing experimental

observations.

The Rutherford model …4

42

The Bohr model

Niels Bohr (Noble Prize in Physics 1922) proposed a

revolutionary model which broke from classical theory.

For his model of the hydrogen atom, Bohr postulated that:

… the electron follows circular orbits around the positive

nucleus. (Centripetal force = Coulomb attraction)

… the electron moves in certain allowed orbits without radiating

energy. The electron is said to be in a “stationary state”.

… the electron “jumps” (undergoes a transition) between

stationary states by absorbing or emitting a quantum, hf, of energy.

Absorption: hf = Ef – Ei

Emission: hf = Ei – EfEi

Ef

Ef

Ei

43

2

13.6eV ; 1,2,3,...NE N

N

... which allowed the calculation of the frequencies of the photon

emitted when an electron undergoes a transition from an outer orbit to

an inner one ...

… which in turn explained the experimental observations perfectly.

For the hydrogen atom, Bohr was able to propose a formula

for the allowed energy levels for the electron:

Bohr’s model of the hydrogen atom was a great triumph.

He extended his model to other (ionized) single-electron atoms,

e.g. He ,Li ,Be

However ... the Bohr model was not satisfactory for other multi-

electron atoms and did not explain the quantum postulates.

The Bohr model …2

44

Electronic energy levels for a hydrogen atom

2

13.6eV ; 1,2,3,...NE N

N

M&I

7.1

45

Periodic Table of the Elements

46

X-rays : Roentgen (1896)

The electron: JJ Thomson (1895)

The neutron: Chadwick (1928)

Radioactivity: Pierre & Marie Curie (1897)

Other important experimental discoveries …

... Bohr’s model was to be superseded by

quantum mechanics …

47

Electromagnetic waves are propagating disturbances

involving time and space variations of coupled electric and

magnetic fields (which are perpendicular to each other).

The different types of electromagnetic radiation make up the

electromagnetic spectrum.

There is a range of and f …

… but c = f always, where

All electromagnetic waves can travel through “empty space”

… no medium is required.

8 13.0 10 msc

Electromagnetic radiationM&I

7.2

48

49

Electromagnetic radiation

Maxwell’s

equations:

0

E

d

dt

BE

0B

0 0

d

dt

EB J

It can be shown that in a vacuum:

2 2 2 2

0 0 0 02 2 2 2and

E E B B

x t x t

Compare this with the general wave equation:2 2

2 2 2

1f f

x v twhere v is the speed of the wave

and write 8 1

0 0

12.99792 10 msc

50

Blackbody radiation

Kirchoff showed that the most efficient radiator is also the

most efficient absorber. A perfect absorber absorbs all the

incident radiation and no light is reflected

called a blackbody.

An ideal blackbody can be approximated

by a cavity with a very small opening.

A blackbody will emit radiation according

to the temperature of its walls.

51

The energy radiated by a

blackbody at each

wavelength (and therefore

the intensity) depends on

the temperature at which

the body is radiating.

The spectral distribution

for a blackbody:

Blackbody radiation …2

52

The ultraviolet catastrophe

In 1900 Rayleigh used the concept of classical physics to

derive a theoretical expression to describe the shape of the

blackbody spectral distribution (or “blackbody curve”).

This theory called the “Rayleigh-Jeans theory”, was a

complete failure as the fit of the theory to experiment was

very poor. The model assumed that atoms were oscillators that

emits all wavelengths of electromagnetic waves.

Rayleigh-Jeans Law: 4

2( , )

ckTI T

23 1: Boltzmann's constant 1.381 10 J Kk

53

Major disagreement between theory and experimental

data, especially at short wavelengths.

“ultraviolet catastrophe”

54

Planck’s theory (1900)

Max Planck found an empirical formula which agreed

with experiment at all wavelengths:2

5

2( , )

1hc kT

hcI T

e

h = Planck’s constant = 6.63 10-34 J s

In deriving this formula, Planck made two radical assumptions:

… the energy absorbed and emitted by the blackbody was

“quantised” or manifested in discrete amounts.

… atoms absorb or emit energy in discrete units (“quanta”)

of light energy (“photons”) by jumping from one quantum

state to the other.

55

The energy and frequency of the radiation are related by:

f iE E E nhf

Ef : final energy state of atom

Ei : initial energy state of atom

E : energy of absorbed or emitted photon in J (or eV)

n : quantum number

h : Planck’s constant in J s (or eV s)

f : frequency of atom’s oscillation = frequency of photon

hf : one “quantum” or packet of energy

3hf

2hf

hf

0

energyn = 3

n = 2

n = 0

n = 1

Planck’s theory …2

56

The photoelectric effect

Although Planck’s hypothesis worked, most physicists were

sure that a classical theory was still possible and that the

quantum hypothesis could be done away with. However, more

and more evidence of the quanta aspects of nature emerged.

In 1905 Einstein published a paper

showing that by treating the light in the

interior of a blackbody as a gas consisting

of particles of energy E =hf, one could

obtain Planck’s result. Furthermore, he

was able to explain a phenomenon known

as the photoelectric effect by using a

particle description of light.

57

Photoelectric effect:

… the ejection of electrons from

a metal by photons.

If photoelectrons reach

collector C, then a current

is measured by the

voltmeter, V.

Shine a beam of

monochromatic light

onto the metal plate.

For ultraviolet light, we

see that electrons

(called photoelectrons)

are emitted from E.

58

Einstein’s theory of the photoelectric effect

Einstein applied Planck’s quantum hypothesis. He assumed:

… that light propagates in discrete particle-like

quanta called photons.

… each photon has energy E = hf

… each photon gives all its energy to a single electron.

... in the photoelectric effect, an electron in the metal

completely absorbs a photon thus gaining energy hf.

Some of this energy goes into freeing the electron from the

rest of the metal while the remainder appears as the kinetic

energy of the electron .

maxhf K

maxK

Write:

59

Therefore the quantum hypothesis explained both the

blackbody spectrum and the photoelectric effect (and

a number of other phenomena) …

… formed the foundations for the development of

quantum mechanics …

At the same time, the classical understanding the

structure of the atom was also being challenged …

Read M&I 7.2 carefully …

“The quantum model of interaction of light and matter”

60

Quantum mechanics

Wave-particle duality

Consider light:

… diffraction and polarization, for example, imply that

light is a wave.

… photoelectric and Compton effects, for example, imply

that light is a stream of particles.

Louis de Broglie (1923) showed that the wave-

particle duality was not only associated with

light, but that particles had the same behaviour.

61

Particle with momentum p has associated wavelength :

h h hp mv

p mv

de Broglie wavelength

Experimental verification:

… by Davisson and Germer (1927)

… electrons scattered off a nickel crystal in the same

way as a diffraction grating maxima at m = d sin

… by G.P. Thomson (1927)

… demonstrated diffraction rings from electron

scattering in powered aluminium oxide.

62

Electron “double slit” experiment [ (a) to (c) ]

Light

63

The wave function

Maxwell considered electromagnetic waves to be

oscillating and fields …

… what about matter waves?

EB

Quantum theory (wave mechanics) reconciles the apparent

wave-particle duality by introducing the concept of the

wave function .

is the quantity whose variation with position

and time represents the wave aspect of a moving particle.

Given for a system, we can predict (by making appropriate

mathematical operations on ) what experiments on the

system should observe.

,tr

64

represents the probability of observing a

particle at position and time t

2 3, dt rrr

Copenhagen or “statistical” interpretation:

describes our knowledge of the particle.,tr

2 3, d 1t rrwhere

This shift away from a deterministic theory towards a

probabilistic theory was not liked by everyone.

Einstein: “God does not throw dice”

65

Correspondence principle:

… the equations of quantum physics must reduce to familiar

classical laws under conditions in which the classical laws are

known to agree with experiment.

Generally:Classical mechanics large bodies

Quantum mechanics atomic/molecularscale and smaller

(… quantum gravity …?)

It took more than 20 years to formulate a consistent quantum

theory. Many new fundamental concepts had to be developed first.

Today, 80 years later, virtually nothing in nature can be described

without quantum mechanics and relativity.

66

Heisenberg’s uncertainty principle

One of the important consequences which emerges from quantum

mechanics is the realization that nature imposes limitations on the

precision with which it is possible to make observations, even

using the most accurate instruments imaginable.

For example, if both the x-coordinate x, and the x-component of

the momentum px, of a particle are measured to the highest

precision theoretically possible, x and px respectively, then

from the Heisenberg Uncertainty Principle:

2

2

y

z

y p

z pand similarly:

2xx p

… if one quantity is precisely determined, then all knowledge

of the other is lost.

67

The Heisenberg Uncertainty Principle may also

be formulated in terms of the scalars energy and

time: 2E t

One interpretation is that energy conservation may

be violated by an amount E for a small period of

time t.

e.g. in “barrier tunneling”.

Heisenberg’s uncertainty principle …2

68

Quantized vibrational energy levelsM&I

7.4

Consider a model of solid

matter as a network of balls

and springs.

This “classical” model

explains many aspects of the

behaviour of solids … but

not all … for example the

behaviour of solids at low

temperatures and some

aspects of the interaction of

light with matter.

… therefore need a quantum treatment …

69

21total 2 sE K U k A

sk

s

The energy is “continuous.”

A classical harmonic oscillator

(mass-spring system) can vibrate

with any amplitude, and hence can

have any energy.

K+U of the system can have any

value in the coloured region.

70

Quantised energy levels of a 1D oscillator

0E

1 0 01E E

3 0 03E E

4 0 04E E

2 0 02E E

Quantum harmonic oscillator:

Energy can only be added in multiples of 0sk

m

…where34

346.67 10 Js1.05 10 Js

2 2

h

“ground state”

71

sk m

sk m

sk m

sk m

sk m

sk m

sk m

sk m

Quantum oscillator

with large ks

Quantum oscillator

with small ks

Effect of the “spring stiffness” ks

Note that the “ground state” E0 of the oscillator is not zero …

… from Heisenberg … usually 10 2 sE k m

72

Energy levels for the inter-atomic potential energy

Nearly uniform

energy spacing

… which describes the interaction of two neighbouring atoms

73

The energy bands of a

diatomic molecule

Electronic

levels

Rotational

levels

Vibrational

levels

vibE

rotE

elecE

M&I

7.5

74