phy1004 thermal physics

PHY1004W 

THERMAL PHYSICS 

Lecturer: Dr Indresan Govender 

room 511, RW James Bldg. 

650 5554 

indresan.govender@uct.ac.za 

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phy1004W 

Thermal Physics 

Thermal Physics: Textbooks 

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Young & Freedman: University Physics 

Halliday & Resnick: Fundamentals of Physics 

Chabay & Sherwood: Matter and Interactions 

Schroeder: Intro to Thermal Physics 

Guenault: Statistical Physics 

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Thermal Physics: Websites, notes, . . . 

Various . . . 

See UCT PHY1004W thermal physics website 

Google ! 

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Einstein and thermodynamics 

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“A theory is the more impressive the greater the 

simplicity of its premises, the more different kinds 

of things it relates, and the more extended its area 

of applicability. Therefore the deep impression that 

classical thermodynamics made upon me. It is the 

only physical theory of universal content which I am 

convinced will never be overthrown, within the 

framework of applicability of its basic concepts.” 

– A. Einstein 

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Thermodynamics 

Statistical Physics 

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Thermodynamics: 

Framework for relating macroscopic properties of a system to one 

another, egs. 

1. How does pressure of a gas depend on the temperature & volume 

of its container? 

2. How does a refrigerator work? What is its maximum efficiency? 

3. How much energy do we need to add to a kettle of water to 

change it to steam? 

Applications: thermal engines, egs. 

1. Internal combustion engine 

2. Steam engine 

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Statistical Mechanics: 

The goal of statistical mechanics is to begin with the microscopic 

laws of physics that govern the behaviour of the constituents of 

the system and deduce the properties of the system as a whole, 

egs. 

1. Why are the properties of water different from those of steam, 

even though water and steam consist of the same type of 

molecules? 

2. How are the molecules arranged in a liquid? 

Statistical mechanics is the bridge between the microscopic and 

macroscopic worlds. 

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IG-10 

Chabay & Sherwood: 

Matter and interactions 

Ch 11:Entropy 

Ch 12:Gases and Engines 

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CS Chapter 11 Entropy: sections 

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11.1 Statistical issues 

11.2 A statistical model of solids 

11.3 Thermal equilibrium of two blocks in contact 

11.4 The second law of thermodynamics 

11.5 What is temperature? 

11.6 Heat capacity of a solid 

11.7 The Boltzmann distribution 

11.8 The Boltzmann distribution in a gas 

11.9 Summary: Fundamental physical principles 

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Q11.1.a 


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You put an ice cube into a styrofoam 

cup containing hot coffee. You would 

probably be surprised if the ice cube got 

colder and the coffee got hotter. 

Would this be a violation of the energy 

principle? 

1) Yes 

2) No 

3) The energy principle does not 

apply in this situation 

∆E sys = W ext + Q 

Q: energy flowing from surroundings into the system 

W ext : work done by surroundings on our system 

But W ext = Q = 0 … system (coffee & ice cube) is isolated 

Therefore: ∆E sys = 0 implies ∆E coffee + ∆E ice = 0 

NB: no constraint on direction w.r.t. change in energy 

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Q11.1.b 


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Consider a ball bouncing on the floor. 

If we consider the closed system to be the ball and floor, what is the 

(fundamental) cause of the ball eventually coming to rest? 

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Q11.1.c 

Consider a ball bouncing on the floor. 

If we consider the closed system to be 

the ball and floor, which of the following 

principles are violated at a fundamental 

level? 

1) Momentum conservation 

2) Energy conservation 

4) None of the above 

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Can the ball increase in height with each bounce? 

Yes 

No 

Maybe 

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Bouncing Ball 

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•Initially (before 1 st bounce) energy associated with 

centre of mass of ball, i.e. one degree of freedom 

•After several bounces energy associated with 

individual molecules of floor and ball, i.e. many 

degrees of freedom (Ball and floor feel warm!) 

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Bouncing Ball 

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1. How do we quantify these “degrees of freedom”? 

2. What is the relationship between 

a) the “degrees of freedom” available to the system, 

b) the “degrees of freedom” taken up by the energy, 

c) the likelihood of the ball increasing in height with each 

bounce? 

3. How does the likelihood of the ball decreasing in 

height with each bounce compare with 2(c)? 

4. Are we learning yet another esoteric concept?%! 

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Concepts needed to begin 

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•energy (kinetic and potential) 

•Temperature 

•Work 

•heat ( = thermal transfer of energy) 

•microscopic vs macroscopic 

•Reversibility 

•Probability 

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Statistical issues 

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•Very notion of temperature: average kinetic energy of 

Molecules 

•Large numbers of molecules: N A = 6.023 × 10 23 mole -1 

•Thermal energy flow: hotter to colder (on average) 

•Reversibility 

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Reversible processes 

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Many processes in physics are reversible, eg 

atomic collisions, or frictionless 2-d elastic collisions 

• Cannot tell forward movie from backward 

• Collisions possible in both directions 

• Laws of physics (strong, electromagnetic, 

gravitational interactions) completely reversible 

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Observe: energy flows from hot to cold 

Spontaneous 

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Ever observe energy flow from cold to hot?? 

Not spontaneous 

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Irreversible processes 

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•No physics principle makes reverse 

process impossible 

• Chance of reverse process occurring 

may be VERY small 

• This process is called irreversible 

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Apparent Paradox in Thermodynamics 

•All fundamental processes exhibit time reversal invariance, 

i.e. Both forward and backward movie of fundamental 

interactions look possible (have seen them before!!!) 

•Macroscopic change has a preferred direction, egs. 

►Bouncing balls come to rest … & never reverse their motion 

►Energy moves from hot to cold … & never the reverse direction 

•If the sub-microscopic world exhibits time reversal invariance, 

surely the macroscopic world (which is made up of the submicroscopic) 

should also be time reversal invariant ! 

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•Better to think of macroscopic change statistically 

►Bouncing ball increasing in height with each bounce is 

NOT impossible, just wildly (ridiculously!!) improbable 

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Practical Example on Probability 

Consider the following collection of balls in a box: 

5000 red 

100 green 

3 blue 

4 yellow 

1. If you reached into the bag (without looking !!), 

which colour ball are you most likely to pick up? 

2. Can you estimate the likelihood of picking up a: 

a) Yellow ball? 

b) Green ball? 

c) Blue ball? 

d) Red ball? 

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Modelling strategy 

•Use a simple (we’ll use simplest) model of a solid to 

understand the distribution of energy between the atoms 

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•Determine probability of particular energy (speed) 

distributions 

•Use this probability to understand why certain 

distributions are VERY unlikely and others a sure bet 

•Test microscopic predictions with macroscopic 

measurements 

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Modelling strategy 

•Use a simple (we’ll use simplest) model of a solid to 

understand the distribution of energy between the atoms 

IG-10 

•Determine probability of particular energy (speed) 

distributions 

•Use this probability to understand why certain 

distributions are VERY unlikely and others a sure bet 

•Test microscopic predictions with macroscopic 

measurements 

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Model of solid: massive ball and springs 

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We will use model to ask detailed quantitative questions 

about energy distribution in solid 

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Simplified model: Einstein (1907) model of a solid 

Assumptions: 

•Each atom in the lattice is a 3D quantum mechanical oscillator (based on 

Planck’s quantisation assumption) 

•Each atom moves independently (connected to imaginary rigid walls; not 

other atoms) No mechanism for energy exchange between atoms 

•All atoms vibrate with the same frequency 

•Ignore collective motion of groups of atoms 

 

0 

 

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Atom as a 3-d oscillator 

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x, y and z motion independent, so 3-d oscillator is 

equivalent to three 1-d oscillators. 

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Energy of atom (as three 1-d oscillators) 

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Quantized energy levels of 1-d oscillator 

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n = 0, 1, 2, 3, etc 

Energy added to 1-d atomic oscillator only in multiples of: 

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Quantum Mechanical Oscillator 

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Three one dimensional Oscillators 

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Distributing energy amongst objects 

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Simple case: One atom equivalent to three oscillators. 

Imagine atom has 4 quanta of energy. 

Distribute 4 quanta over 3 oscillators. 

How much to each oscillator? 

How many ways to do it? 

Counting begins . . . 

(Finally, we shall derive a formula) 

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4 quanta over 3 osc: three ways 

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3 ways of distributing 4 quantum of vibrational energy 

amongst 3 one-dimensional oscillators 

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4 quanta over 3 osc: six more ways 

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6 more ways of distributing 4 quantum of vibrational 

energy amongst 3 one-dimensional oscillators 

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4 quanta over 3 osc: yet six more ways 

Still more ways of distributing 4 quantum of vibrational 

energy amongst 3 one-dimensional oscillators 

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Microstates and macrostates 

There are 15 microstates. 

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Each microstate is a way to distribute the energy. 

All microstates belong to the same macrostate. 

The macrostate has 4 quanta of energy. 

The 4-quanta macrostate can also be categorised into types: 

•(4,0,0) type 400 

•(2,1,1) type 211 

•(3,1,0) type 310 

•(2,2,0) type 220 

Which type is most likely to occur? 

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IG-10 

Q11.2.a 

Consider 4 quantized oscillators (a model for 1 and 1/3 

atoms!). They share 2 "quanta" of energy. List all the 

ways you can arrange these 2 quanta of energy among 

the 4 oscillators (such as "2000"); how many 

arrangements are there? 

1) 4 

2) 7 

3) 10 

4) 15 

5) 20 

2000 1100 1010 

0200 0110 1001 

0020 0011 0101 

0002 

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FUNDAMENTAL ASSUMPTION of 

STATISTICAL MECHANICS 

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An isolated system is equally likely, over time, to be 

found in any one of the fully-specified states 

accessible to it. 

NB: This assumption allows all possible 

configurations of the system (macrostate) if we 

observe it for long enough. 

However, you might have to wait a very long time to 

observe certain configurations 

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Two blocks in thermal contact 

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Take two simple blocks. Take two atoms! (one from 

each block) 

Count ways to distribute 4 quanta of energy between 

the two atoms (six oscillators) 

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All 4 quanta to one atom, none to the other 

30 microstates. 

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3 to one atom, and 1 to the other 

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2 to one, and 2 to the other 

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Table of all 15 + 30 + 36 + 30 + 15 = 126 ways 

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Histogram of number of microstates 

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One system many times, or many systems once? 

Make many observations of our system. 

In 29% of these we have a 2|2 split. 

OR 

Make many copies of our system (macrostate). 

Observe each one. 

In 29% of these we have a 2|2 split. 

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IG-10 

Q11.2.a 

Two objects share a total energy E = E 1 +E 2 . There 

are 10 ways to arrange an amount of energy E 1 in 

the first object and 15 ways to arrange an amount 

of energy E 2 in the second object. How many 

different ways are there to arrange the total energy 

E = E 1 +E 2 so that there is E 1 in the first object and 

E 2 in the other? 

Ω(E) = Ω(E 1 ) Ω(E 2 ) 

= (10)(15) 

= 150 

1) 10 

2) 15 

3) 25 

4) 150 

5) 1x10 15 

Using Greek symbol Ω “Omega” to represent # of ways 

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IG-09 

A system of 300 oscillators contains 100 quanta of energy. What is the 

physical meaning of this model? 

1) one atom oscillating in 300 dimensions 

2) 300 atoms, each in the 100th energy level 

3) 300 atoms with 100 joules of energy distributed among them 

4) 100 atoms with 300 joules of energy distributed among them 

5) 100 atoms with 100* *sqrt(k s /m a ) joules of energy among them 

6) something else 

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IG-09 

A formula for the number of microstates 

We shall derive a formula 

for the number of microstates with q quanta 

divided amongst N oscillators 

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Sequences of removing five numbered balls 

from a bag 

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How many sequences? 

5 × 4 × 3 × 2 × 1 = 5! = 120 

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Permutations of three objects 

We list permutations of 3 objects: 

1 2 3 

1 3 2 

2 1 3 

2 3 1 

3 1 2 

3 2 1 

The number of permutations is 3 × 2 × 1 = 6 = 3! 

For n objects, there are n! permutations. Note 0! = 1 

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Making fewer distinctions 

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Of the five balls, imagine three are red, and two green. 

Say we do not care about the ball number, just the colour, 

eg RRRGG or RGRGR. 

There are 5! = 120 numbered sequences. 

There are 3! = 6 permutations of red balls, and 

there are 2! = 2 permutations of green balls. 

Thus the number of different colour sequences is: 

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IG-09 

Colour sequences: three red, two green 

RR GG R 

RR GR G 

RR RG G 

RG RR G 

RG RG R 

RG GR R 

GR RR G 

GR RG R 

GR GR R 

GG RR R 

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Arranging quanta amongst oscillators 

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6 objects: 4 quanta and 3 − 1 = 2 bars. 

Number of arrangements = 6!/(4! 2!) = 720/(24×2) = 15 

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IG-09 

A formula for the number of microstates 

number of microstates with q quanta divided 

amongst N oscillators 

gets very big very fast! 

For q = 100 and N = 300, Ω = . . . = 1.7 × 10 96 

Probability that all 100 quanta are on oscillator 

number 3 is 

P(0, 0, 100, 0, . . . , 0) = 1/ Ω ≈ 0.6 × 10 −96 

Not impossible, but VERY unlikely!! 

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How do we use formula? 

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Program syntax 

Program 

1 combin(a, b) Vpython 

2 combin(a, b) Excel 

3 exp(gammaln(a+1) + gammaln(b+1) + 

gammaln(a-b+1)) 

Matlab/Octave 

a = q + N - 1 

b = q 

a-b = N - 1 

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Stirling’s Theorem 

One form of Stirling’s theorem is that for large n 

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If n is very large we can simplify this by taking logs 

Noticing that is much smaller than other terms for large n, 

we can simply further as 

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IG-09 

A useful formulae for # of ways per macrostate type 

Given an isolated system of N oscillators and q quanta. 

Suppose the distribution of the quanta is such that n 1 , 

n 2 , n 3 , …, n m oscillators contain quanta q 1 , q 2 , …, q m 

respectively. This distribution is a type of macrostate. 

The total number of ways of constructing this 

macrostate type is: 

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Q11.2.d 


Which microstate is most probable? 

1) A 

2) B 

3) C 

4) D 

5) They’re equally probable 

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NB: over time, all microstates are equally probable! 

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Q11.2.d 


Which type of macrostate (A, B, C, D) 

is most probable? 

1) A 

2) B 

3) C 

4) D 

5) They’re equally probable 

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NB: over time, all microstates are equally probable! 

The most probable type of macrostate is the one with the most # of ways to 

exist (A, B, C & D are microstates from particular types of macrostates of 

which the B-type macrostate is the most probable) 

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phy1004 thermal physics

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