Chapter 13
Temperature & the Ideal Gas
Chapter 13: Temperature & the Ideal Gas
•
•
•
•
•
•
•
Temperature
Temperature scales
Thermal expansion of solids and liquids
Molecular picture of a gas
Absolute temperature and the ideal gas law
Kinetic theory of the ideal gas
Collisions between gas molecules
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Temperature
Heat is the flow of energy due to a temperature
difference. Heat always flows from objects at high
temperature to objects at low temperature.
When two objects have the same temperature, they are
in thermal equilibrium.
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The Zeroth Law of Thermodynamics
If two objects are each in thermal equilibrium with
a third object, then the two objects are in thermal
equilibrium with each other.
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Thermodynamics
“Thermodynamics is a funny subject. The first time you go through it, you
don’t understand it at all. The second time you go through it, you think you
understand it, except for one or two small points. The third time you go
through it, you know you don’t understand it, but by that time you are so
used to it, it doesn’t bother you any more.”
- Arnold Sommerfeld
Thermodynamics is the scientific study of work,
heat, and the related properties of chemical and
mechanical systems.
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Temperature Measurement
Requirement:
A substance that exhibits a measureable change in
one of its physical properties as the temperature
changes.
Examples:
Volume change with temperature is a common
physical property used for the measurement of
temperature
The change of electrical resistance with temperature
is also used to measure temperature.
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Not a Good Temperature
Measurement Instrument
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Mercury Thermometer
A mercury-in-glass thermometer, invented by German
physicist Daniel Gabriel Fahrenheit.
Bulb – reservoir for the thermometric liquid
Stem – glass capillary tube through which the mercury or organic
liquid fluctuates with changes in temperature
Scale – the scale is graduated in degrees, fractions or multiples of
degrees
Contraction Chamber– an enlarged capillary bore which serves to
reduce a long length of capillary, or prevent contraction of
the entire liquid column into the bulb
Expansion Chamber– an enlargement of the capillary bore at the top
of the thermometer to prevent build up of excessive
pressures in gas-filled thermometers
http://www.omega.com/Temperature/pdf/INTRO_GLASS_THERM.pdf
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Temperature Calibration
Celsius and Fahrenheit are two
different temperature scales but
they are calibrated to the same
reference points.
The two scales have different sized
degree units and they label the
steam point and the ice point with
different temperature values.
The “Zero” values on both scales
were picked for convenience and
can cause problems.
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Temperature Measurement
Calibration
This process ensures the accuracy of the measurement.
It requires the availability of a repeatable set of states
of matter. The boiling point of water and the freezing
point of water are two examples.
Transferability
The calibration needs to be transferable to other
measuring instruments.
Standards
There needs to be agreed upon calibration procedures
to allow transferability.
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Temperature Scale
Anders Celsius devised the Celsius scale. This was
published in The origin of the Celsius temperature scale in
1742.
Celsius originally defined his scale "upside-down”
• boiling point of pure water at 0 °C and
• the freezing point at 100 °C.
One year later Frenchman Jean Pierre Cristin proposed to
invert the scale. He named it Centigrade.
In later years, probably when the lower reference
temperature was switch to the Triple Point, the name
Celsius was restored.
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Fahrenheit Temperature Scale
It was in 1724 that Gabriel Fahrenheit, an instrument maker of Däanzig and
Amsterdam, used mercury as the thermometric liquid.
Fahrenheit described how he calibrated the scale of his mercury thermometer:
"placing the thermometer in a mixture of sal ammoniac or sea salt, ice, and water a
point on the scale will be found which is denoted as zero. A second point is obtained
if the same mixture is used without salt. Denote this position as 30. A third point,
designated as 96, is obtained if the thermometer is placed in the mouth so as to
acquire the heat of a healthy man." (D. G. Fahrenheit,Phil. Trans. (London) 33, 78,
1724)
On this scale, Fahrenheit measured the boiling point of water to be 212. Later he
adjusted the freezing point of water to 32 so that the interval between the boiling and
freezing points of water could be represented by the more rational number 180.
Temperatures measured on this scale are designated as degrees Fahrenheit (° F).
http://www.cartage.org.lb/en/themes/sciences/Physics/Thermodynamics/AboutTemperature/Development/Development.htm
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Temperature Scales
Fahrenheit
scale
212 F
Celsius scale
Water boils*
Absolute or
Kelvin scale
373.15 K
Water freezes*
273.15 K
32 F
0 C
Absolute zero
0K
459.67 F
273.15C
100 C
(*) Values given at 1 atmosphere of pressure.
• In general, as the temperature of a substance is lowered the
atomic motion and vibrations slow down.
• Temperature can be related to molecular energy. Negative
temperature values imply erroneous negative energy.
• The Kelvin temperature scale avoids this problem.
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Comparison of Temperature Scales
The temperature scales are related by:
Fahrenheit/Celsius
TF  1.8 F/CTC  32F
Output
Absolute/Celsius
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Intput
T  TC  273.15
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Example (text problem 13.3):
(a) At what temperature (if any) does the numerical value of
Celsius degrees equal the numerical value of Fahrenheit
degrees?
TF  1.8TC  32  TC
TC  40 C
Require TF = TC
(b) At what temperature (if any) does the numerical value of degree
kelvins (T) equal the numerical value of Fahrenheit degrees?
TF  1.8TC  32
 1.8T  273  32
 1.8TF  273  32
Require T = TF
TF  574 F
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Comparison of Temperature Scales
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Thermal Expansion of Solids and
Liquids
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Thermal Expansion of Solids and
Liquids
Most objects expand in volume when their
temperature increases and decrease in
volume when their temperatures decrease.
Exception:
Water expands in volume when it freezes
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Why Ice Cubes Float
Most fluids contract as
they cool making the solid
phase more dense than the
liquid.
Water behaves this way
until it reaches 4 oC.
Below this temperature it
begins to expand yielding
a solid phase that is less
dense than the liquid.
Therefore ice cubes float
and ice forms on the top of
lakes rather than the
bottom.
The fish say thank you!
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Why Ice Cubes Float
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Why Ice Cubes Float
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Thermal Linear Expansion
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Thermal Linear Expansion
If ΔT is negative then the rod would contract
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Thermal Linear Expansion
An object’s length after its temperature has changed is
L  1  T L0
 is the coefficient of
linear expansion
where T = TT0 and L0 is the length of the object
at a temperature T0.
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Thermal Expansion in Multiple Rods
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Example (text problem 13.84):
An iron bridge girder (Y = 2.01011 N/m2) is constrained
between two rock faces whose spacing doesn’t change.
At 20.0 C the girder is relaxed. How large a stress
develops in the iron if the sun heats the girder to 40.0 C?
 = 12106 K1 (from Table 13.2)
Using Hooke’s Law:
F
L
Y
A
L
 Y T 



 2.0 1011 N/m 2 12 10 6 K 1 20 K 
 4.8 107 N/m 2
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How does the area of an object change when its
temperature changes?
The blue square has an area of L02.
L0
L0+ L
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With a temperature change T
each side of the square will have a
length change of L = TL0.
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The fractional change in area is:
new area  A  L0  TL0 L0  TL0 
 L  2TL   T L
2
0
2
0
2
2 2
0
 L20  2TL20
 A0 1  2T 
A
 2T
A0
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The fractional change in volume due to a temperature
change is:
V
  T
V0
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For solids  = 3
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Does the Hole Grow or Shrink With Higher Temperatures?
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The Hole Grows With Higher Temperatures
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The Hole Acts Like It Is Made of the Same Material
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Molecular Picture of a Gas
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Molecular Picture of a Gas
The number density of particles is N/V where N
is the total number of particles contained in a
volume V.
If a sample contains a single element, the
number of particles in the sample is N = M/m.
M is the total mass of the sample divided by
the mass per particle (m).
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Avogadro’s Number - NA
One mole of a substance contains the same number of
particles as there are atoms in 12 grams of 12C. The
number of atoms in 12 grams of 12C is Avogadro’s
number.
N A  6.022 1023 mol 1
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A carbon-12 atom by definition has a mass of exactly
12 atomic mass units (12 u).
 12 g  1 mole

12 u = 
23
 12 mole  6.022 10
 1 kg 
  1.66 1027 kg x 12

 1000 g 
This is the conversion factor between the atomic mass unit and kg
(1 u = 1.661027 kg). NA and the mole are defined so that a 1
gram sample of a substance with an atomic mass of 1 u contains
exactly NA particles.
The atomic mass MA (from the periodic table) is the gram weight
of a mole of that substance. “u” is the average nucleon mass.
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Example (text problem 13.37):
Air at room temperature and atmospheric pressure has a
mass density of 1.2 kg/m3. The average molecular mass
of air is 29.0 u.
How many air molecules are there in 1.0 cm3 of air?
total mass of air in 1.0 cm3
number of particles 
average mass per air molecule
The total mass of air in the given volume is:
3

 1 m 
1
.
2
kg
1
.
0
cm


6


m  V  

1
.
2

10
kg



3


 m  1  100 cm 
3
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Example continued:
total mass of air in 1.0 cm 3
number of particles 
average mass per air molecule
1.2 10 6 kg

29.0 u/particle  1.66 1027 kg/u


 2.5 1019 particles
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Absolute Temperature and the Ideal
Gas Law
Experiments done on dilute gases (a gas where
interactions between molecules can be ignored) show
that:
For constant pressure
V T
Charles’ Law
For constant volume
P T
Gay-Lussac’s Law
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For constant
temperature
1
P
V
Boyle’s Law
For constant pressure V  N
and temperature
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Avogadro’s Law
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The Ideal Gas Equation of State
Putting all of these statements together gives the ideal
gas law (microscopic form):
PV  NkT
k = 1.381023 J/K is
Boltzmann’s constant
The ideal gas law can also be written as (macroscopic
form):
PV  nRT
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R = NAk = 8.31 J/K/mole is
the universal gas constant and
n is the number of moles.
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Example (text problem 13.41):
A cylinder in a car engine takes Vi = 4.50102 m3 of air into the
chamber at 30 C and at atmospheric pressure. The piston then
compresses the air to one-ninth of the original volume and to 20.0
times the original pressure.
What is the new temperature of the air?
Here, Vf = Vi/9, Pf = 20.0Pi, and Ti = 30 C = 303 K.
PiVi  NkTi
Pf V f  NkTf
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The ideal gas law holds for
each set of parameters
(before compression and
after compression).
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Example continued:
Take the ratio:
Pf V f
PiVi
The final temperature is

NkTf
NkTi

Tf
Ti
 Pf
T f  
 Pi
 V f 
 Ti
 Vi 
 Vi 
 20.0 Pi  9 

 
303 K   673 K
 Pi  Vi 


The final temperature is 673 K = 400 C.
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Constant Volume Gas Thermometer
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Constant Volume Gas Thermometer
negative-pressure
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Kinetic Theory of the Ideal Gas
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Kinetic Theory of the Ideal Gas
An ideal gas is a dilute gas where the particles act as
point particles with no interactions except for elastic
collisions.
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Gas particles have random motions. Each time a particle
collides with the walls of its container there is a force
exerted on the wall. The force per unit area on the wall is
equal to the pressure in the gas.
The pressure will depend on:
• The number of gas particles
• Frequency of collisions with the walls
• Amount of momentum transferred during each
collision
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The pressure in the gas is
2N
P
K tr
3V
Where <Ktr> is the root mean square (RMS) average of
the translational kinetic energy of the gas particles; it
depends on the temperature of the gas.
K tr
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3
 kT
2
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Root Mean Squared
The root mean squared (rms) method of averaging is
used when a variable will average to zero but its effect
will not average to zero.
Procedure
• Square it (make the negative values positive)
• Take the average (mean)
• Take the square root (undo the squaring opertion)
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Root Mean Squared Average
Sine Functions
1.5
1.0
0.5
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
-0.5
SIN(Theta)
SIN2(Theta)
RMS Value
-1.0
-1.5
Angle (Radians)
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The average kinetic energy also depends on the rms
speed of the gas
K tr
1
1 2
2
 m v  mvrms
2
2
where the rms speed is
3
1 2
K tr  kT  mvrms
2
2
3kT
vrms 
m
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The Maxwell-Boltzmann Distribution
The distribution of
speeds in a gas is
given by the MaxwellBoltzmann
Distribution.
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Example (text problem 13.60):
What is the temperature of an ideal gas whose molecules
have an average translational kinetic energy of
3.201020 J?
3
K tr  kT
2
2 K tr
T
 1550 K
3k
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Example (text problem 13.70):
What are the rms speeds of helium atoms, and nitrogen,
hydrogen, and oxygen molecules at 25 C?
vrms 
3kT
m
Element
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On the Kelvin scale T = 25 C = 298 K.
Mass (kg)
rms speed (m/s)
He
6.641027
1360
H2
3.321027
1930
N2
4.641026
515
O2
5.321026
482
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Molecular Velocity
How fast are the molecules moving ?
equation for rms velocity
k = Boltzmann's constant
T = temperature of the gas (K)
m = mass of the molecule
Not surprising:
The hotter it is, the faster they move.
The lighter they are, the faster they move.
At room temperature:
http://www.uccs.edu/~tchriste/courses/PHYS549/549lectures/gasses.html
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Most Probable Speed - Previous Slide Had Vrms
The speed of sound is about 340 m/s.
104 cm/s = 100 m/s = 223.7 mi/hr
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Mean Free Path
How far does a molecule travel before it collides with
another molecule ?
For air at room temperature, the mean free path can be
expressed as:
P = pressure in torr
Distance will be in cm.
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Summary
•
•
•
•
•
•
•
Temperature
Temperature scales
Thermal expansion of solids and liquids
Molecular picture of a gas
Absolute temperature and the ideal gas law
Kinetic theory of the ideal gas
Collisions between gas molecules
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Extra
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Centigrade to Celsius
In 1948 use of the Centigrade scale was dropped in favor of a
new scale using degrees Celsius (° C). The Celsius scale is
defined by the following two items:
• the triple point of water is defined to be 0.01 C
• a degree Celsius equals the same temperature change as
a degree on the ideal-gas scale.
On the Celsius scale the boiling point of water at standard
atmospheric pressure is 99.975 C in contrast to the 100 degrees
defined by the Centigrade scale.
http://www.cartage.org.lb/en/themes/sciences/Physics/Thermodynamics/AboutTemperature/Development/Development.htm
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Maximum Thermometer
A special kind of mercury thermometer, called a maximum
thermometer, works by having a constriction in the neck close
to the bulb. As the temperature rises the mercury is pushed up
through the constriction by the force of expansion. When the
temperature falls the column of mercury breaks at the
constriction and cannot return to the bulb thus remaining
stationary in the tube. The observer can then read the
maximum temperature over the set period of time. To reset
the thermometer it must be swung sharply. This is similar to
the design of a medical thermometer.
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Glass Thermometers: A Degree of History
Ever since the early 1600s, when Galileo invented the first crude instrument, called a thermoscope, to show
changes in temperature, the thermometer has evolved to become a crucial instrument for determining
temperature in a multitude of laboratory and industrial applications.
The earliest thermometers, considered a curiosity by many people, were used for medical and meteorological
purposes and shared basic features. All were glass tubes open at one end, partially filled with air, set in a basin
of water and graduated with primitive notches or dials.
It wasn't until 50 years later that the first closed-glass liquid thermometer was developed in Italy by Leopoldo,
Cardinal dei Medici, a co-founder of the Academia de Cimento. Known as Florentine thermometers, they were
graduated into 50, 100 or 300 degrees and roughly standardized by the heat of the sun and the cold of ice
water, and filled with distilled colored wine. Mercury and water thermometers were also tried by the
Florentines but their expansion was too small. This problem was later overcome by making thermometers with
finer bores, thereby increasing their sensitivity.
By the middle of the eighteenth century, mercury thermometers had superseded all others because of their
more uniform expansion. Today the glass thermometer is available in a variety of calibrations, immersion
styles, versions, lengths, divisions and backings and in °F and °C versions with columns filled with mercury or
safety liquid. Lodged as a staple in laboratories and industries worldwide, the glass thermometer is an
enduring symbol for the advancement
of knowledge.
Source: http://www.omega.com/Temperature/pdf/INTRO_GLASS_THERM.pdf
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Specific Heats in the Liquid and Solid States
Source: Elements of Chemistry, William Allen Miller
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Pressure Conversion Table
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Atomic Mass Unit & Nucleon Binding Energy
Mass of proton : 1.6726 x 10-27 kg
Mass of neutron: 1.6749 x 10-27 kg
Mass of electron: 0.00091x10-27 kg
Avg of proton & neutron mass: 1.6738 x 10-27 kg
1 u = 1.66x10-27 kg
Carbon Mass ( = 6p + 6n)
12u
Missing Mass
Missing Energy (Joules) (E = mc2)
Missing Energy (eV) (E = E(J)/1.602E-19)
Missing Energy (Mev) per nucleon
Missing Mass/u
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1.6726E-27
1.6749E-27
9.1000E-31
1.6738E-27
1.6600E-27
2.0085E-26
1.9920E-26
1.6500E-28
1.485E-11
92696629.21
9.27E+07
7.72E+06
9.94%
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