This document discusses temperature and thermal equilibrium. It begins by explaining that temperature represents the average kinetic energy of atoms in a substance. When two substances are in thermal contact, heat will flow between them until they reach the same temperature and thermal equilibrium. Several temperature scales are then described, including Celsius, Fahrenheit, and Kelvin. The document also covers thermal expansion, explaining that all materials expand slightly when heated through the transfer of kinetic energy between atoms. Examples of applications that rely on thermal expansion, such as bimetallic strips and railway tracks, are provided.

1. Lecture 10
Temperature. Thermometers.
Thermal expansion.

2. Matter is very complex
A simple cup of coffee contains ~ 1023 atoms
Numerically impossible to follow trajectory of each
atom, use Newton’s laws for force, acceleration etc..
Two options
1. Statistical: probability distribution of molecule velocities
2. Macroscopic: a few variables (temperature, pressure,
volume…) characterize the bulk properties of matter
These variables are called state variables.

3. Temperature and thermal equilbrium
Temperature ~ average kinetic energy of atoms
Two pieces of matter in thermal contact exchange
energy until their temperatures (T ) are the same
(0th law of thermodynamics)
Example:
Molecules in coffee transfer energy to the air molecules.
Eventually all coffee molecules have the same average
kinetic energy as the molecules of air in this room
Reaching equilibrium (same T ) requires transfer of energy
This energy in transit is called heat Q.

4. ACT: Temperature
When a series of blocks are connected thermally, heat starts to flow
between the blocks as shown. What does this tell us about relative
temperatures of the blocks?
1
Q
A. T1 = T2 = T3 = T4
B. T1 > T2 > T3 > T4
C. T1 < T2 < T3 < T4
2
Q’
3
Q’’
4
This is how the sequence is
established empirically.
Also: You just applied the 2nd
law of thermodynamics....

5. Temperature scales: Celsius, Fahrenheit
Celsius
Based on the boiling and freezing points of water.
0°C = freezing point of water
100°C = boiling point of water
Fahrenheit
Based on the boiling and freezing points of alcohol.
Connection to Celsius: 0°C = 32°F
100°C = 212°F
9
TF = TC + 32
5

6. Constant volume gas thermometers
For gases, p is proportional toT for constant volume (Charles’ law).
p
p
= constant
T
T
gas
(He)
Liquid
(Hg)

7. Temperature scales: Kelvin
p(T) for different gases:
p
Extrapolation of ALL
lines points to
T = −273.15°C
T (°C)
Kelvin scale
0 K = −273.15°C (lowest energy state, quantum motion only)
2nd fixed point: 273.16 K (=0.01°C) is the triple-point for H 2O (ice, water,
steam coexist)
TK =TC + 273.15
With this choice, 1°C = 1K (equal increments)

8. Thermal expansion of a bar
A bar of length L0 expands ΔL when temperature is increased by ΔT.
Experimentally,
∆L = α L0 ∆T
α = coefficient of linear expansion (depends on material)
Basis for many thermometers
e.g. liquid mercury
Critically important in many engineering projects (expansion
joints)

9. In-class example: Bimetallic strips
Which of the following bimetallic strips will bend the furthest to the right
when heated from room temperature to 100°C?
α (µJ/m K)
Invar
Al
Al
Invar
Brass
C
Al
Brass
D
Ag
Au
E
Al/Invar have the largest difference in α. Aluminum
expands more than invar, so A will bend to the right.
1.3
Brass
B
A
24
Invar
Al
Al
21
Au
14
Ag
19
DEMO:
Bimetallic
strips

10. A couple of applications
Bimetallic strips can be used as mechanical thermometers.
About invar: Fe-Ni alloy with very small α
Train tracks have expansion joints (gaps) to prevent buckling in
hot weather.
(origin of the “clickety-clac, clickety-clac”)
High speed trains cannot afford the vibrations produced by
these gaps. Alloys with small α to build tracks are a key
development.

11. Area Thermal Expansion
b
b +Δb
ΔT
a
a +Δa
Afinal = (a + ∆a )(b + ∆b )
= ab + a ∆b + b ∆a + very small terms
≈ ab + a (bα∆T ) + b (a α∆T )
= ab + 2ab α∆T
∆
A
∆A = 2α A0 ∆T

12. ACT: Washer
DEMO:
Balls and rings
A circular piece of metal with a round hole is
heated so that its temperature increases. Which
diagram best represents the final shape of the
initial shape
metal?
A. Both inner, outer radii larger
B. Inner radius smaller, outer
radius larger
C. Same size
All lengths increase: a 2-D object grows in length and breadth

13. Opening tight jar lids
αglass = 0.4-0.9 × 10-5 K−1
αbrass = 2.0 × 10-5 K−1
If you place the jar top under the hot water faucet,
the brass expands more than the glass.

14. Volume Thermal Expansion
c
a
c +Δc
ΔT
b
a +Δa
b +Δb
Vfinal = (a + ∆a )(b + ∆b )(c + ∆c )
= abc + ab ∆c + ac ∆b + bc ∆a + smaller terms
≈ abc + 3abc (α∆T )
≈ V0 + 3 0α∆T
V
∆V
∆V = βV0 ∆T
β = 3α
Coefficient of volume expansion

15. The special case of water
Most materials expand when temperature increases.
Water between 0 and 4°C is the exception.
V
Maximum
density
Ice is less dense than
cold (< 4°C ) water.
Ice floats
α has an important dependence on T
0
4°C
T
This prevents lakes from freezing from the bottom up,
which would kill all forms of life.

16. Microscopic Model of Solids
Consider atoms on a lattice, interacting
with each other as if connected by a spring
movie

17. Microscopic Model of Expansion
(attempt 1)
Model potential
energy of atoms
U =
U
1
k (x − l0 )2 + const
2
vibration at high T
vibration at low T
l0
spacing l0
separation x
Vibrating atoms spend equal time at x > l0 as at x < l0
→ average separation of atoms = l0 , at both low and high T
→ material same size at low and high T
→ This spring model of matter does not describe expansion

18. Microscopic Model of Expansion (2)
Change model to non-ideal spring
U
U ≠
1
k (x − l 0 )2
2
asymmetric
parabola
vibration at high T
vibration at low T
spacing lT
lhighT
llowT
x
At high T, the average separation of atoms increases → solid expands
(Now we need another parameter in model, asymmetry of parabola)

19. Why two models?
Is this more useful compared to empirical ∆L = λ L0 ∆T ?
The empirical formula can be used for simple cases.
Example: for designing structures to take into account expansion.
The microscopic model of expansion (non-ideal spring) can be used
in wider number of cases.
Example: thermal expansion while also under high-pressure in a
chemical engineering plant

20. Thermal stress
A rod of length L0 and cross-sectional area A fits perfectly between
two walls. We want its length to remain constant when we increase the
temperature.
Constant length = zero net strain
 ∆L 
 ∆L 
 ∆L 
=
+
=0

÷
÷
÷
 L0  all  L0  thermal  L0  applied stress
α∆T +
F
=0
YA
F
= −αY ∆T
A
Thermal stress (stress
walls need to provide to
keep length constant)

21. Application of thermal stress
For very tight fitting of pieces (like wheels):
Wheel is heated
and then axle
inserted.
Wheel cools down and
shrinks around axle,
very tight.