# Lecture 33 - Temperature. Thermometers by pharmphresh26

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```									                                                                                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..
Lecture 33
Temperature Thermometers
Temperature. Thermometers.
Two options
Thermal expansion.
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.

Temperature and thermal equilbrium                                          ACT: Temperature

Temperature ~ average kinetic energy of atoms                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?
Two pieces of matter in thermal contact exchange
energy until their temperatures (T ) are the same                                1          2             3           4
(0th law of thermodynamics)                                                           Q           Q’            Q’’

Example:
Molecules in coffee transfer energy to                       A. T1 = T2 = T3 = T4                    This is how the sequence is
the air molecules.
established empirically.
Eventually all coffee molecules have the                     B. T1 > T2 > T3 > T4
same average kinetic energy as the
molecules of air in this room                                C. T1 < T2 < T3 < T4
Also: You just applied the 2nd
law of thermodynamics....
Reaching equilibrium (same T ) requires transfer of energy

This energy in transit is called heat Q.

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Temperature scales: Celsius, Fahrenheit                                                 Constant volume gas thermometers

Celsius                                                                              For gases, p is proportional toT for constant volume (Charles’ law).
Based on the boiling and freezing points of water.                                 p
0°C = freezing point of water                                                                              p
100°C = boiling point of water                                                                               = constant
T

T              gas
Fahrenheit                                                                                                               (He)
Based on the boiling and freezing points of alcohol.

Connection to Celsius: 0°C = 32°F                        9
TF = TC + 32
100°C = 212°F                     5
DEMO:
He and N2                                     Liquid
balloons                                     (Hg)

Temperature scales: Kelvin                                                           Thermal expansion of a bar

p(T ) for different gases:                                                       A bar of length L0 expands ΔL when temperature is increased by ΔT.
p                                               Experimentally,
Extrapolation of ALL
lines points to                                                                                      ΔL = α L0 ΔT
T = −273.15°C
273.15 C
α = coefficient of linear expansion (depends on material)
T (°C)

Kelvin scale                                                                         Basis for many thermometers
0 K = −273.15°C (lowest energy state, quantum motion only)                           e.g. liquid mercury
2nd fixed point: 273.16 K (=0.01°C) is the triple-point for H2O (ice, water,
Critically important in many engineering projects (expansion
steam coexist)
joints)
With this choice, 1°C = 1K (equal increments)           TK =TC + 273.15

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In-class example: Bimetallic strips                                                               A couple of applications

Which of the following bimetallic strips will bend the furthest to the right
when heated from room temperature to 100°C?
Bimetallic strips can be used as thermostat switches.

α (1/mK)
Al       Invar        Invar        Al                    Al            24

Invar          1.3    About invar: Fe-Ni alloy with very small α

A                         B                        Brass          21      Train tracks have expansion joints (gaps) to prevent buckling in
hot weather.
Au            14                     (origin of the “clickety-clac, clickety-clac”)
Al       Brass        Brass        Al            Au       Ag
Ag            19
High speed trains cannot afford the vibrations produced by
these gaps. Alloys with small α to build tracks are a key
C                         D                      E              DEMO:              development.
Bimetallic
Al/Invar have the largest difference in α. Aluminum                    strips
expands more than invar, so A will bend to the right.

DEMO:
Area Thermal Expansion                                                                 ACT: Washer                   Balls and rings

A circular piece of metal with a round hole is heated so that
b +Δb                  its temperature increases. Which diagram best represents
b                                                              the final shape of the metal?
ΔT
a                                                                    A. Both inner, outer radii larger                            initial shape
a +Δa
Afinal = (a + Δa )(b + Δb )                                                   B. Inner radius smaller, outer radius
larger
= ab + a Δb + b Δa + very small terms
≈ ab + a (bαΔT ) + b (a αΔT )
C. Same size
= ab + 2abαΔT
ΔA
ΔA = 2α A0 ΔT                         Think about how the
piece that was cut off
the center would grow.

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Opening tight jar lids                                                       Volume Thermal Expansion

c                                       c +Δc
αglass = 0.4-0.9 × 10-5 K−1                                                                               ΔT

αbrass = 2.0 × 10-5 K−1                                                                     b                                        b +Δb
a                                        a +Δa
Vfinal = (a + Δa )(b + Δb )(c + Δc )
= abc + ab Δc + ac Δb + bc Δa + smaller terms
If you place the jar top under the hot water faucet,                           ≈ abc + 3abc (αΔT )
the brass expands more than the glass.
≈ V0 + 3 0αΔT
V                                   ΔV = βV0 ΔT
ΔV                                    β ≈ 3α
Coefficient of volume expansion

The special case of water                                                                    Thermal stress

Most materials expand when temperature increases.                          A rod of length L0 and cross-sectional area A fits perfectly between
Water between 0 and 4°C is the exception.
°                                                     two walls. We want its length to remain constant when we increase the
temperature.
Constant length = zero net strain
V                                                     Ice is less dense than
Maximum                                             °
cold (< 4°C ) water.           ΔLall = ΔLthermal + ΔLapplied stress = 0
pp
density
Ice floats                           FL0
L0αΔT +         =0
YA
α has an important dependence on T
F
T                                  = −αY ΔT
0       °
4°C                                                                        A
Thermal stress (stress
This prevents lakes from freezing from the bottom up,                  walls need to provide to
which would kill all forms of life.                                    keep length constant)

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Application of thermal stress

For very tight fitting of pieces (like wheels):

Wheel is heated           Wheel cools down and
and then axle             shrinks around axle,
inserted.                 very tight.

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