Temperature, Thermal Expansion, and the Ideal Gas Law

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Temperature, Thermal Expansion, and the Ideal Gas
Law
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• Atomic Theory of Matter
• Temperature and Thermometers
• Thermal Equilibrium and the Zeroth Law of
  Thermodynamics
• Thermal Expansion
• Thermal Stress
• The Gas Laws and Absolute Temperature
• The Ideal Gas Law

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• Problem Solving with the Ideal Gas Law
• Ideal Gas Law in Terms of Molecules Avogadros
  Number
• Ideal Gas Temperature Scalea Standard

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Atomic Theory of Matter
Atomic and molecular masses are measured in
unified atomic mass units (u). This unit is
defined so that the carbon-12 atom has a mass of
exactly 12.0000 u. Expressed in kilograms 1 u
1.6605 x 10-27 kg.
Brownian motion is the jittery motion of tiny
flecks in water these are the result of
collisions with individual water molecules.
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Atomic Theory of Matter
On a microscopic scale, the arrangements of
molecules in solids (a), liquids (b), and gases
(c) are quite different.
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Atomic Theory of Matter
Distance between atoms. The density of copper is
8.9 x 103 kg/m3, and each copper atom has a mass
of 63 u. Estimate the average distance between
the centers of neighboring copper atoms.
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Temperature and Thermometers
Temperature is a measure of how hot or cold
something is. Most materials expand when heated.
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Temperature and Thermometers
Thermometers are instruments designed to measure
temperature. In order to do this, they take
advantage of some property of matter that changes
with temperature. Early thermometers
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Temperature and Thermometers
Common thermometers used today include the
liquid-in-glass type and the bimetallic strip.
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Temperature and Thermometers
Temperature is generally measured using either
the Fahrenheit or the Celsius scale. The freezing
point of water is 0C, or 32F the boiling point
of water is 100C, or 212F.
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Temperature and Thermometers
Taking your temperature. Normal body temperature
is 98.6F. What is this on the Celsius scale?
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Temperature and Thermometers
A constant-volume gas thermometer depends only on
the properties of an ideal gas, which do not
change over a wide variety of temperatures.
Therefore, it is used to calibrate thermometers
based on other materials.
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Thermal Equilibrium and the Zeroth Law of
Thermodynamics
Two objects placed in thermal contact will
eventually come to the same temperature. When
they do, we say they are in thermal
equilibrium. The zeroth law of thermodynamics
says that if two objects are each in equilibrium
with a third object, they are also in thermal
equilibrium with each other. The zeroth law is
the foundation of temperature measurement.
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Thermal Expansion
Linear expansion occurs when an object is heated.
Here, a is the coefficient of linear expansion.
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Thermal Expansion
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Thermal Expansion
Bridge expansion. The steel bed of a suspension
bridge is 200 m long at 20C. If the extremes of
temperature to which it might be exposed are
-30C to 40C, how much will it contract and
expand?
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Thermal Expansion
Do holes expand or contract? If you heat a thin,
circular ring in the oven, does the rings hole
get larger or smaller?
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Thermal Expansion
Ring on a rod. An iron ring is to fit snugly on a
cylindrical iron rod. At 20C, the diameter of
the rod is 6.445 cm and the inside diameter of
the ring is 6.420 cm. To slip over the rod, the
ring must be slightly larger than the rod
diameter by about 0.008 cm. To what temperature
must the ring be brought if its hole is to be
large enough so it will slip over the rod?
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Thermal Expansion
Opening a tight jar lid. When the lid of a glass
jar is tight, holding the lid under hot water for
a short time will often make it easier to open.
Why?
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Thermal Expansion
Volume expansion is similar, except that it is
relevant for liquids and gases as well as solids
Here, ß is the coefficient of volume
expansion. For uniform solids, ß 3a.
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Thermal Expansion
For uniform solids, ß 3a.
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Thermal Expansion
Gas tank in the Sun. The 70-liter (L) steel gas
tank of a car is filled to the top with gasoline
at 20C. The car sits in the Sun and the tank
reaches a temperature of 40C (104F). How much
gasoline do you expect to overflow from the tank?
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Thermal Expansion
Volume-temperature relation for water
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Thermal Expansion
Water behaves differently from most other
solidsits minimum volume occurs when its
temperature is 4C. As it cools further, it
expands, as anyone who leaves a bottle in the
freezer to cool and then forgets about it can
testify. Another example is antimony ? (Sb).
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Pressure
Pressure is defined as the force per unit area.
Pressure is a scalar the units of pressure in
the SI system are pascals 1 Pa 1 N/m2.
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Pressure

• The two feet of a 60-kg person cover an area of
  500 cm2.
• Determine the pressure exerted by the two feet
  on the ground.
• (b) If the person stands on one foot, what will
  the pressure be under that foot?

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Pressure
The pressure at a depth h below the surface of
the liquid is due to the weight of the liquid
above it. We can quickly calculate
This relation is valid for any liquid whose
density does not change with depth.
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The Gas Laws and Absolute Temperature
The relationship between the volume, pressure,
temperature, and mass of a gas is called an
equation of state. We will deal here with gases
that are not too dense.
Boyles law the volume of a given amount of gas
is inversely proportional to the pressure as long
as the temperature is constant.
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The Gas Laws and Absolute Temperature
The volume is linearly proportional to the
temperature, as long as the temperature is
somewhat above the condensation point and the
pressure is constant. Extrapolating, the volume
becomes zero at -273.15C this temperature is
called absolute zero.
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The Gas Laws and Absolute Temperature
The concept of absolute zero allows us to define
a third temperature scalethe absolute, or
Kelvin, scale. This scale starts with 0 K at
absolute zero, but otherwise is the same as the
Celsius scale. Therefore, the freezing point of
water is 273.15 K, and the boiling point is
373.15 K. Finally, when the volume is constant,
the pressure is directly proportional to the
temperature.
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The Gas Laws and Absolute Temperature
Why you should not throw a closed glass jar into
a campfire. What can happen if you did throw an
empty glass jar, with the lid on tight, into a
fire, and why?
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The Ideal Gas Law
We can combine the three relations just derived
into a single relation
What about the amount of gas present? If the
temperature and pressure are constant, the volume
is proportional to the amount of gas
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The Ideal Gas Law
A mole (mol) is defined as the number of grams of
a substance that is numerically equal to the
molecular mass of the substance 1 mol H2 has a
mass of 2 g. 1 mol Ne has a mass of 20 g. 1 mol
CO2 has a mass of 44 g. The number of moles in a
certain mass of material
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The Ideal Gas Law
We can now write the ideal gas law
where n is the number of moles and R is the
universal gas constant.
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Atmospheric Pressure
At sea level the atmospheric pressure is about
1.013 x 105 N/m2 this is called 1 atmosphere
(atm). Another unit of pressure is the bar 1 bar
1.00 x 105 N/m2. Standard atmospheric pressure
is just over 1 bar.
This pressure does not crush us, as our cells
maintain an internal pressure that balances it.
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Atmospheric Pressure
This is a mercury barometer, developed by
Torricelli to measure atmospheric pressure. The
height of the column of mercury is such that the
pressure in the tube at the surface level is 1
atm. Therefore, pressure is often quoted in
millimeters (or inches) of mercury.
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Problem Solving with the Ideal Gas Law
Standard temperature and pressure (STP) T 273
K (0C) P 1.00 atm 1.013 N/m2 101.3 kPa.
Volume of one mole at STP. Determine the volume
of 1.00 mol of any gas, assuming it behaves like
an ideal gas, at STP.
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Problem Solving with the Ideal Gas Law
Helium balloon. A helium party balloon, assumed
to be a perfect sphere, has a radius of 18.0 cm.
At room temperature (20C), its internal pressure
is 1.05 atm. Find the number of moles of helium
in the balloon and the mass of helium needed to
inflate the balloon to these values.
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Problem Solving with the Ideal Gas Law
Mass of air in a room. Estimate the mass of air
in a room whose dimensions are 5 m x 3 m x 2.5 m
high, at STP.
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Problem Solving with the Ideal Gas Law

• Volume of 1 mol of an ideal gas is 22.4 L
• If the amount of gas does not change

• Always measure T in kelvins
• P must be the absolute pressure

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Problem Solving with the Ideal Gas Law
Check tires cold. An automobile tire is filled to
a gauge pressure of 200 kPa at 10C. After a
drive of 100 km, the temperature within the tire
rises to 40C. What is the pressure within the
tire now?
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Ideal Gas Law in Terms of Molecules Avogadros
Number
Since the gas constant is universal, the number
of molecules in one mole is the same for all
gases. That number is called Avogadros number
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Ideal Gas Law in Terms of Molecules Avogadros
Number
Therefore we can write
or
where k is called Boltzmanns constant.
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Ideal Gas Law in Terms of Molecules Avogadros
Number
Hydrogen atom mass. Use Avogadros number to
determine the mass of a hydrogen atom.
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Ideal Gas Law in Terms of Molecules Avogadros
Number
Estimate how many molecules you breathe in with a
1.0-L breath of air.
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Ideal Gas Temperature Scalea Standard
This standard uses the constant-volume gas
thermometer and the ideal gas law. There are two
fixed points Absolute zerothe pressure is zero
here The triple point of water (where all three
phases coexist), defined to be 273.16 Kthe
pressure here is 4.58 torr.
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Ideal Gas Temperature Scalea Standard
Then the temperature is defined as
In order to determine temperature using a real
gas, the pressure must be as low as possible.
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Summary

• All matter is made of atoms.
• Atomic and molecular masses are measured in
  atomic mass units, u.
• Temperature is a measure of how hot or cold
  something is, and is measured by thermometers.
• There are three temperature scales in use
  Celsius, Fahrenheit, and Kelvin.
• When heated, a solid will get longer by a
  fraction given by the coefficient of linear
  expansion.

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Summary

• The fractional change in volume of gases,
  liquids, and solids is given by the coefficient
  of volume expansion.
• Ideal gas law PV nRT.
• One mole of a substance is the number of grams
  equal to the atomic or molecular mass.
• Each mole contains Avogadros number of atoms or
  molecules.

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Kinetic Theory of Gases
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• The Ideal Gas Law and the Molecular
  Interpretation of Temperature
• Distribution of Molecular Speeds
• Real Gases and Changes of Phase
• Van der Waals Equation of State

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The Ideal Gas Law and the Molecular
Interpretation of Temperature

• Assumptions of kinetic theory
• large number of molecules, moving in random
  directions with a variety of speeds
• molecules are far apart, on average
• molecules obey laws of classical mechanics and
  interact only when colliding
• collisions are perfectly elastic

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The Ideal Gas Law and the Molecular
Interpretation of Temperature
The force exerted on the wall by the collision of
one molecule is
Then the force due to all molecules colliding
with that wall is
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The Ideal Gas Law and the Molecular
Interpretation of Temperature
The averages of the squares of the speeds in all
three directions are equal
So the pressure is
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The Ideal Gas Law and the Molecular
Interpretation of Temperature
Rewriting,
so
The average translational kinetic energy of the
molecules in an ideal gas is directly
proportional to the temperature of the gas.
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The Ideal Gas Law and the Molecular
Interpretation of Temperature
Molecular kinetic energy. What is the average
translational kinetic energy of molecules in an
ideal gas at 37C?
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The Ideal Gas Law and the Molecular
Interpretation of Temperature
We can now calculate the average speed of
molecules in a gas as a function of temperature
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The Ideal Gas Law and the Molecular
Interpretation of Temperature
Speeds of air molecules. What is the rms speed of
air molecules (O2 and N2) at room temperature
(20C)?
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The Ideal Gas Law and the Molecular
Interpretation of Temperature
Average speed and rms speed. Eight particles have
the following speeds, given in m/s 1.0, 6.0,
4.0, 2.0, 6.0, 3.0, 2.0, 5.0. Calculate (a) the
average speed and (b) the rms speed.
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Distribution of Molecular Speeds
The molecules in a gas will not all have the same
speed their distribution of speeds is called the
Maxwell distribution
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Distribution of Molecular Speeds
The Maxwell distribution depends only on the
absolute temperature. This figure shows
distributions for two different temperatures at
the higher temperature, the whole curve is
shifted to the right.
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Real Gases and Changes of Phase
The curves here represent the behavior of the gas
at different temperatures. The cooler it gets,
the further the gas is from ideal.
In curve D, the gas becomes liquid it begins
condensing at (b) and is entirely liquid at (a).
The point (c) is called the critical point.
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Real Gases and Changes of Phase
Below the critical temperature, the gas can
liquefy if the pressure is sufficient above it,
no amount of pressure will suffice.
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Real Gases and Changes of Phase
A PT diagram is called a phase diagram it shows
all three phases of matter. The solid-liquid
transition is melting or freezing the
liquid-vapor one is boiling or condensing and
the solid-vapor one is sublimation.
Phase diagram of water.
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Real Gases and Changes of Phase
The triple point is the only point where all
three phases can coexist in equilibrium.
Phase diagram of carbon dioxide.
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Van der Waals Equation of State
To get a more realistic model of a gas, we
include the finite size of the molecules and the
range of the intermolecular force beyond the size
of the molecule.
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Van der Waals Equation of State
We assume that some fraction b of the volume is
unavailable due to the finite size of the
molecules. We also expect that the pressure will
be reduced by a factor proportional to the square
of the density, due to interactions near the
walls. This gives the Van der Waals equation of
state the constants a and b are found
experimentally for each gas
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Van der Waals Equation of State
The PV diagram for a Van der Waals gas fits most
experimental data quite well.
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Summary

• The average kinetic energy of molecules in a gas
  is proportional to the temperature.
• Below the critical temperature, a gas can
  liquefy if the pressure is high enough.
• At the triple point, all three phases are in
  equilibrium.
• Evaporation occurs when the fastest moving
  molecules escape from the surface of a liquid.
• Saturated vapor pressure occurs when the two
  phases are in equilibrium.

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Summary

• Relative humidity is the ratio of the actual
  vapor pressure to the saturated vapor pressure.
• The Van der Waals equation of state takes into
  account the finite size of molecules.
• The mean free path is the average distance a
  molecule travels between collisions.
• Diffusion is the process whereby the
  concentration of a substance becomes uniform.