Physics 151: Lecture 37
Today’s Agenda
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Topics
Temperature and Zeroth Law of Thermodynamics
Temperature scales
Thermal expansion
Ideal gas
Physics 151: Lecture 37, Pg 1
Temperature
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Temperature: measure of the motion of the individual atoms and
molecules in a gas, liquid, or solid.
related to average kinetic energy of constituents
High temperature: constituents are moving around energetically
In a gas at high temperature the individual gas molecules are
moving about independently at high speeds.
In a solid at high temperature the individual atoms of the solid
are vibrating energetically in place.
The converse is true for a "cold" object.
In a gas at low temperature the individual gas molecules are
moving about sluggishly.
There is an absolute zero temperature at which the motions of
atoms and molecules practically stop.
There is an absolute zero temperature at which the classical
motions of atoms and molecules practically stop
Animation
Physics 151: Lecture 37, Pg 2
Heat
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Solids, liquids or gases have internal energy
Kinetic energy from random motion of molecules

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translation, rotation, vibration
At equilibrium, it is related to temperature
Heat: transfer of energy from one object to another as a
result of their different temperatures
Thermal contact: energy can flow between objects
T2
T1
>
U1
U2
Physics 151: Lecture 37, Pg 3
Zeroth Law of Thermodynamics
T1

Thermal equilibrium:
when objects in
thermal contact cease
heat transfer
same temperature
=
T2
U2
U1
Animation
If objects A and B are separately in thermal equilibrium
with a third object C, then objects A and B are in
thermal equilibrium with each other.
C
A
B
Physics 151: Lecture 37, Pg 4
Thermometers
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A thermometer is a device that is used to measure the
temperature of a system
Thermometers are based on the principle that some physical
property of a system changes as the system’s temperature
changes
These properties include:
The volume of a liquid
The dimensions of a solid
The pressure of a gas at a constant volume
The volume of a gas at a constant pressure
The electric resistance of a conductor
The color of an object
A temperature scale can be established on the basis of any of
these physical properties
Physics 151: Lecture 37, Pg 5
Thermometer, Liquid in Glass
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A common type of
thermometer is a
liquid-in-glass
The material in the
capillary tube expands
as it is heated
The liquid is usually
mercury or alcohol
A thermometer can be
calibrated by placing it
in contact with some
natural systems that
remain at constant
temperature
Physics 151: Lecture 37, Pg 6
Thermometer, Celsius Scale
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A thermometer can be calibrated by placing it in contact
with some natural systems that remain at constant
temperature
Common systems involve water
A mixture of ice and water at atmospheric pressure
» Called the ice point of water
A mixture of water and steam in equilibrium
» Called the steam point of water
Celsius Scale :
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The ice point of water is defined to be 0o C
The steam point of water is defined to be 100o C
Physics 151: Lecture 37, Pg 7
Constant Volume Gas Thermometer
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The physical change exploited
is the variation of pressure of a
fixed volume gas as its
temperature changes

The volume of the gas is kept
constant by raising or lowering
the reservoir B to keep the
mercury level at A constant
Physics 151: Lecture 37, Pg 8
Absolute Zero
The thermometer readings are virtually independent of the gas used
If the lines for various gases are extended, the pressure is always
zero when the temperature is –273.15o C
This temperature is called
absolute zero
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Absolute zero is used as the basis of the absolute temperature
scale
The size of the degree on the absolute scale is the same as the
size of the degree on the Celsius scale
To convert: TC = T – 273.15
Physics 151: Lecture 37, Pg 9
Temperature scales
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Three main scales
Farenheit
Celcius
Kelvin
212
100
373.15
32
0
273.15
-459.67
-273.15
0
Water boils
Water freezes
Absolute Zero
Physics 151: Lecture 37, Pg 10
T (K)
Some interesting facts
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In 1724, Gabriel Fahrenheit made thermometers
using mercury. The zero point of his scale is
attained by mixing equal parts of water, ice, and
salt. A second point was obtained when pure
water froze (originally set at 30oF), and a third
(set at 96oF) “when placing the thermometer in
the mouth of a healthy man”.
On that scale, water boiled at 212.
Later, Fahrenheit moved the freesing point of
water to 32 (so that the scale had 180
increments).
In 1745, Carolus Linnaeus of Upsula, Sweden,
described a scale in which the freezing point of
water was zero, and the boiling point 100,
making it a centigrade (one hundred steps)
scale. Anders Celsius (1701-1744) used the
reverse scale in which 100 represented the
freezing point and zero the boiling point of water,
still, of course, with 100 degrees between the two
defining points.
108
Hydrogen bomb
107
Sun’s interior
106
Solar corona
105
104
103
100
Sun’s surface
Copper melts
Water freezes
Liquid nitrogen
10
1
Liquid hydrogen
Liquid helium
0.1
Lowest T
~ 10-9K
Physics 151: Lecture 37, Pg 11
Thermal expansion
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In most liquids or solids, when temperature rises
molecules have more kinetic energy
» they are moving faster, on the average
consequently, things tend to expand
amount of expansion depends on…
change in temperature T
L0
original length L
coefficient of thermal expansion
» L0 + L = L0 +  L0 T
» L =  L0 T (linear expansion)
V
» V =  L0 T (volume expansion)
L
V + V
Physics 151: Lecture 37, Pg 12
Thermal expansion
Physics 151: Lecture 37, Pg 13
Bimetallic strip
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A bimetallic strip is made of two
ribbons of dissimilar metals bonded
together.
Assume that the strip is originally
straight. As they are heated, the metal
with the greater average coefficient of
expansion (2) expands more than the
other, forcing the strip into an arc, with
the outer radius having a greater
circumference (as in the Fig). A
compact spiral bimetallic strip is used
in a home thermostat. Find the angle
(Q) through which the free end of the
strip turns when the temperature
changes by 1°C for such bimetalic strip
with the initial length L= 21.0 cm and
the separation of the centers of the
Q = (2 - 1 )L (dT/dr)
strips (r = r2 - r1 = 0.410 mm). The two
metals are bronze and invar.
Q = 1.06o
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QuickTime™ and a
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are needed to see this picture.
Physics 151: Lecture 37, Pg 14
Lecture 37, ACT 1
Thermal expansion
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As you heat a block of aluminum from 0 oC to 100 oC, its density
(a) increases
(b) decreases
• Solution
T=0C
(c) stays the same
T = 100 C
Here  is positive
Volume increases
Density decreases
M, V0
Answer: (b)
r0 = M / V0
M, V100
r100 = M / V100
< r0
Physics 151: Lecture 37, Pg 15
Lecture 37: ACT 2
Thermal expansion
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An aluminum plate has a circular hole cut in it. A copper ball (solid
sphere) has exactly the same diameter as the hole when both are
at room temperature, and hence can just barely be pushed
through it. If both the plate and the ball are now heated up to a
few hundred degrees Celsius, how will the ball and the hole fit ?
(a) ball won’t fit
(b) fits more easily
(c) same as before
Physics 151: Lecture 37, Pg 16
Lecture 37: ACT 2
Solution
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Both objects have positive thermal coefficients
all dimensions increase
plate and ball both get larger
After
Before
(b) fits more easily
Physics 151: Lecture 37, Pg 17
Special system: Water
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Most liquids increase in
volume with increasing T r(kg/m3)
1000.00
water is special
999.95
density increases from 999.90
0 to 4 oC !
999.85
999.80
ice is less dense than
999.75
liquid water at 4 oC:
999.70
hence it floats
999.65
water at the bottom of
999.60
a pond is the denser,
999.55
o
i.e. at 4 C
Density
0
2
4
6
8
10
T (oC)
Water has its maximum density at 4 degrees.
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Reason: chemical bonds of H20 (see your chemistry courses !)
Physics 151: Lecture 37, Pg 18
Lecture 37: ACT 3
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Not being a great athlete, and having lots of money to
spend, Gill Bates decides to keep the lake in his back yard
at the exact temperature which will maximize the buoyant
force on him when he swims. Which of the following would
be the best choice?
(a) 0 oC
(b) 4 oC (c) 32 oC (d) 100 oC
(e) 212 oC
Physics 151: Lecture 37, Pg 19
Lecture 37: ACT 3
SOLUTION
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The buoyancy force is
FB = rlVg
1000.00
999.95
since his volume and
g are constant, only r
is changing
999.90
999.85
999.80
Density
999.75
999.70
999.65
999.60
999.55
0
2
4
6
8
10
Water has its maximum density at 4 degrees.
(a) 0 oC
Physics 151: Lecture 37, Pg 20
Lecture 37: Problem 1
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A liquid with a coefficient of volume
expansion  just fills a spherical shell
of volume Vi at a temperature of Ti (see
Fig. to the right). The shell is made of a
material that has an average coefficient
of linear expansion . The liquid is free
to expand into an open capillary of
area A projecting from the top of the
sphere.
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When the temperature increases by T
how high does the liquid rises in the h =(V /A) ( - 3) T
i
capillary ( h ) ?
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For a typical system, such as a mercury thermometer, why
is it a good approximation to neglect the expansion of the
shell ?
(for glass) << (for mercury), so ( -3) within 5%.
Physics 151: Lecture 37, Pg 21
Ideal gas
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Consider a gas in a container of volume V, at pressure P,
and at temperature T
Equation of state
Links these quantities
Generally very complicated: but not for ideal gas
m=mass
M=mass of one mole
n = m/M : number of moles
One mole contains NA=6.022 X 1023 particles : Avogadro’s number
•
Equation of state for an ideal gas
PV = nRT
R is called the universal gas constant
In SI units, R =8.315 J / mol·K
Physics 151: Lecture 37, Pg 22
Boltzmann’s constant
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In terms of the total number of particles N
Animation (p,T)
PV = nRT = (N/NA ) RT
Animation (p,V)
PV = N kB T
Animation (p,N)
In SI units, with R =8.315 J / mol·K, we get
kB = R/NA = 1.38 X 10-23 J/K
kB is called the Boltzmann’s constant
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P, V, and T are the thermodynamics variables
Physics 151: Lecture 37, Pg 23
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Lecture 37: Problem 2
A vertical cylinder of crosssectional area A=0.001 m2 is
fitted with a tight-fitting,
frictionless piston of mass m =
20.0 kg (see the Fig. To the
right)
If n = 0.200 moles of an ideal
gas are in the cylinder at a
temperature of T = 350 K,
what is the height h at which
the piston is in equilibrium
under its own weight ?
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h = n R T/(mg + PA)
h = 1.94 m
Physics 151: Lecture 37, Pg 24
Lecture 37: Problem 3
To
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The mass of a hot-air balloon and its
cargo (not including the air inside) is
200 kg. The air outside is at 10.0°C
and 101 kPa. The volume of the
balloon is 400 m3. To what
temperature must the air in the
balloon be heated before the balloon
will lift off ?
(Air density at 10.0°C is 1.25 kg/m3.)
B = rTo V g
V, T
rTVg
m
mg
T = 472 K !
Physics 151: Lecture 37, Pg 25
Lecture 37: Problem 2b
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A cylinder is closed by a piston
connected to a spring of
constant 2.0 x 103 N/m (as on
the Fig.). With the spring
relaxed, the cylinder is filled with
5.00 L of gas at a pressure of
1.00 atm and a temperature of
20.0°C.
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poVo / To = pV / T
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If the piston has a crosssectional area of 0.010 m2 and a
negligible mass, how high will it
rise when the temperature is
raised to 250°C ?
p=po +kh / A
V=Vo +A h
h = 0.169 m
Physics 151: Lecture 37, Pg 26
Recap of today’s lecture
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Chap. 19: temperature
Temperature and Zeroth Law of Thermodynamics
Temperature scales
Thermal expansion
Ideal gas
Physics 151: Lecture 37, Pg 27