PHY300

1. Giambattista Physics
Chapter 13
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Chapter 13: Temperature and the Ideal Gas
13.1 Temperature and Thermal Equilibrium
13.2 Temperature Scales
13.3 Thermal Expansion of Solids and Liquids
13.4 Molecular Picture of a Gas
13.5 Absolute Temperature and the Ideal Gas Law
13.6 Kinetic Theory of the Ideal Gas
13.7 Temperature and Reaction Rates
13.8 Diffusion

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13.1 Temperature and Thermal Equilibrium
The definition of temperature is based on the concept of
thermal equilibrium.
Suppose two objects or systems are allowed to exchange energy.
The net flow of energy is always from the object at the higher
temperature to the object at the lower temperature. As energy
flows, the temperatures of the two objects approach one
another. When the temperatures are the same, there is no
longer any net flow of energy; the objects are now said to be in
thermal equilibrium.
Thus, temperature is a quantity that determines when objects
are in thermal equilibrium.

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Heat and Thermal Contact
The energy that flows between two objects or systems due to a
temperature difference between them is called heat .
If heat can flow between two objects or systems, the objects or
systems are said to be in thermal contact.
To measure the temperature of an object, we put a thermometer
into thermal contact with the object.
Temperature measurement relies on the zeroth law of
thermodynamics.

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Zeroth Law of Thermodynamics
If two objects are each in thermal equilibrium with a third
object, then the two are in thermal equilibrium with one
another.

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13.2 Temperature Scales
The SI unit of temperature is the
kelvin.
0 K represents absolute zero—there
are no temperatures below 0 K.

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Some Reference Temperatures 1

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Some Reference Temperatures 2

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Example 13.1
A friend suffering from the flu feels like she has a fever; her
body temperature is 38.6°C.
What is her temperature in (a) K and (b) °F?
Solution
(a)
(b)

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13.3 Thermal Expansion of Solids and Liquids
Linear Expansion
If the length of a wire, rod, or pipe is L0 at temperature T0, then
where ΔL = L − L0 and ΔT = T − T0. The length at temperature T is
The constant of proportionality α is called the coefficient of
linear expansion of the substance.

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Some Coefficients of Linear Expansion

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Thermal Expansion Continued
Over a limited temperature
range, the curve can be
approximated by a straight
line; the slope of the tangent
line is the coefficient  at the
temperature T0.

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Example 13.2
Two metal rods, one aluminum and one brass, are each clamped
at one end. At 0.0°C, the rods are each 50.0 cm long and are
separated by 0.024 cm at their unfastened ends.
At what temperature will the rods just come into contact?
(Assume that the base to which the rods are clamped undergoes
a negligibly small thermal expansion.)

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Example 13.2 Strategy

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Example 13.2 Solution

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Differential Expansion
When two strips made of different
metals are joined together and then
heated, one expands more than the
other (unless they have the same
coefficient of expansion).
This differential expansion can be
put to practical use: the joined strips
bend into a curve, allowing one strip
to expand more than the other.

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Area and Volume Expansion
Area Expansion
As you might suspect, each dimension of an object expands
when the object’s temperature increases.
For small temperature changes, the area of any flat surface of a
solid changes in proportion to the temperature change:
Volume Expansion
For solids,

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Expansion of Cavities
A hollow cavity in a solid expands exactly as if it were filled—the
interior of a steel gasoline container expands when its
temperature increases just as if it were a solid steel block.
The steel wall of the can does not expand inward to make the
cavity smaller.

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Example 13.3
A hollow copper cylinder is filled to the brim with water at
20.0°C.
If the water and the container are heated to a temperature of
91°C, what percentage of the water spills over the top of the
container?

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Example 13.3 Strategy

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Example 13.3 Solution
The percentage of water that spills is therefore 1.1 %.

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13.4 Molecular Picture of a Gas
The number of molecules per unit volume, N/V , is called the
number density to distinguish it from mass density.
In SI units, number density is the number of molecules per cubic
meter, m−3 (read “per cubic meter”).
If a gas has a total mass M, occupies a volume V, and each
molecule has a mass m, then the number of gas molecules is
and the average number density is:

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Moles and Avogadro’s Number
It is common to express the amount of a substance in units of
moles (abbreviated mol).
The mole is an SI base unit and is defined as follows: one mole of
anything contains the same number of units as there are atoms
in 12 grams (not kilograms) of carbon-12.
This number is called Avogadro’s number and has the value

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Molecular and Molar Mass
The mass of a molecule is often expressed in units other than
kg. The most common is the atomic mass unit (symbol u).
By definition, one atom of carbon-12 has a mass of 12 u
(exactly).
Using Avogadro’s number, the relationship between atomic
mass units and kilograms can be calculated
Instead of the mass of one molecule, tables commonly list the
molar mass—the mass of the substance per mole.

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Molecular and Molar Mass Continued
The atomic mass unit is chosen so that the mass of a molecule in
“u” is numerically the same as the molar mass in g/mol.
For example, the molar mass of O2 is 32.0 g/mol and the mass of
one molecule is 32.0 u.

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Example 13.4
A helium balloon of volume 0.010 m3 contains 0.40 mol of He
gas.
(a) Find the number of atoms, the number density, and the
mass density.
(b) Estimate the average distance between He atoms.

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Example 13.4 Solution 1
(a)

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Example 13.4 Solution 2
(b)

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13.5 Absolute Temperature and
the Ideal Gas Law
Charles’s law:
V  T
(for constant P)
absolute zero

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Gay-Lussac’s Law
(for constant )
P T V

Both Charles’s law and Gay-Lussac’s law are valid only for a dilute
gas—a gas where the number density is low enough (and,
therefore, the average distance between gas molecules is large
enough) that interactions between the molecules are negligible
except when they collide.

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Other Dilute Gas Laws
Boyle’s law:
Avogadro’s law:

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Ideal Gas Law (Microscopic Form)
( number of molecules)
PV N T N
k
 
k = 1.38 × 10−23 J/K Boltzmann’s constant
Converting to macroscopic form:

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Ideal Gas Law (Macroscopic Form)
( number of moles)
PV nR n
T
 
Many problems deal with the changing pressure, volume, and
temperature in a gas with a constant number of molecules (and
a constant number of moles). In such problems, it is often easiest
to write the ideal gas law as follows:
1 1 2 2
1 2
PV PV
T T
nR 


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Example 13.5
Before starting out on a long drive, you check the air in your tires
to make sure they are properly inflated. The pressure gauge
reads 31.0 lb/in.2 (214 kPa), and the temperature is 15°C.
After a few hours of highway driving, you stop and check the
pressure again. Now the gauge reads 35.0 lb/in.2 (241 kPa).
What is the temperature of the air in the tires now?

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Example 13.5 Solution

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Example 13.6
A scuba diver needs air delivered at a pressure equal to the
pressure of the surrounding water—the pressure in the lungs
must match the water pressure on the diver’s body to prevent
the lungs from collapsing.
Since the pressure in the air tank is much higher, a regulator
delivers air to the diver at the appropriate pressure. The
compressed air in a diver’s tank lasts 80 min at the water’s
surface.
About how long does the same tank last at a depth of 30 m
under water? (Assume that the volume of air breathed per
minute does not change and ignore the small quantity of air left
in the tank when it is “empty.”)

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Example 13.6 Strategy
The compressed air in the tank is at a pressure much higher than
the pressure at which the diver breathes, whether at the surface
or at 30 m depth.
The constant quantity is N, the number of gas molecules in the
tank.
We also assume that the temperature of the gas remains the
same; it may change slightly, but much less than the pressure or
volume.

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Example 13.6 Solution
To match the pressure of the surrounding water, the pressure of
the compressed air is four times larger at a depth of 30 m; then
the volume of air is one fourth what it was at the surface. The
diver breathes the same volume per minute, so the tank will last
one fourth as long—20 min.

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Problem-Solving Tips for the Ideal Gas Law
• In most problems, some change occurs; decide which of the
four quantities (P, V, N or n, and T) remain constant during
the change.
• Use the microscopic form if the problem deals with the
number of molecules and the macroscopic form if the
problem deals with the number of moles.
• Use subscripts (i and f) to distinguish initial and final values.
• Work in terms of ratios so that constant factors cancel out.
• Write out the units when doing calculations.
• Remember that P stands for absolute pressure (not gauge
pressure) and T stands for absolute temperature (in kelvins,
not °C or °F).

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13.6 Kinetic Theory of the Ideal Gas
The ideal gas is a simplified model of a dilute gas in which we
think of the molecules as point-like particles that move
independently in free space with no interactions except for
elastic collisions.
This simplified model is a good approximation for many gases
under ordinary conditions.
Many properties of gases can be understood from this model;
the microscopic theory based on it is called the kinetic theory of
the ideal gas.

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Microscopic Basis of Pressure 1

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Microscopic Basis of Pressure 2

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Microscopic Basis of Pressure 3

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Temperature and Translational Kinetic Energy

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RMS Speed
The speed of a gas molecule that has the average kinetic energy
is called the rms (root mean square) speed.
The rms speed is not the same as the average speed. Instead, the
rms speed is the square root of the mean (average) of the speed
squared.

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Example 13.7
Find the average translational kinetic energy and the rms speed
of the O2 molecules in air at room temperature (20°C).
Strategy
The average translational kinetic energy depends only on
temperature. We must remember to use absolute temperature.
The rms speed is the speed of a molecule that has the average
kinetic energy.

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Example 13.7 Solution

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Maxwell-Boltzmann Distribution
Collisions keep the kinetic energy distributed among the gas
molecules in the most disordered way possible, which is the
Maxwell-Boltzmann distribution.
Example: The probability
distribution of kinetic
energies in O2 at two
temperatures. The area
under either curve for
any range of speeds is
proportional to the
number of molecules
whose speeds lie in that
range.

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Maxwell-Boltzmann Distribution Continued

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13.7 Temperature and Reaction Rates
Energy must be supplied to break a bond. Forming a bond
releases energy.
The minimum kinetic energy of the reactant molecules that
allows the reaction to proceed is called the
activation energy ( Ea ).
If a molecule of N2 collides with one of O2 , but their total kinetic
energy is less than the activation energy, then the two just
bounce off one another.

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Temperature and Reaction Rates Continued
When
then the only candidates for reaction are molecules far off in the
exponentially decaying, high-energy tail of the Maxwell-
Boltzmann distribution.
In this situation, a small increase in temperature can have a
dramatic effect on the reaction rate:

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Example 13.8
The activation energy for the reaction
2 2
N O N +O

is 4.0  10-19 J. By what percentage does the reaction rate
increase if the temperature is increased from 700.0 K to 707.0 K
(a 1% increase in absolute temperature)?

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Example 13.8 Strategy
We should first check that Ea >> (3/2)kT; otherwise,
does not apply.
Assuming that checks out, we can set up a ratio of the reaction
rates at the two temperatures.

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Example 13.8 Solution

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13.8 Diffusion
Mean Free Path
How far does a gas molecule move, on average, between
collisions?
The mean (average) length of the path traveled by a gas
molecule as a free particle (no interactions with other particles)
is called the mean free path (Λ, the Greek capital lambda).

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Random Walk
A gas molecule moves in
a straight line between
collisions.
The result is that a given molecule follows a random walk
trajectory.
After an elapsed time t, how far on average has a molecule
moved from its initial position? The answer to this question is
relevant when we consider diffusion.

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Random Walk Continued
The root mean squared displacement in one direction is

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Several Diffusion Constants

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Example 13.9
How long on average does it take an oxygen molecule in an
alveolus to diffuse into the blood?
Assume for simplicity that the diffusion constant for oxygen
passing through the two membranes (alveolus and capillary
walls) is the same: 1.8 × 10−11 m2/s.
The total thickness of the two membranes is 1.2 × 10−8 m.
Strategy
Take the x-direction to be through the membranes. Then we
want to know how much time elapses until xrms = 1.2 × 10−8 m.

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Example 13.9 Solution

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13.5 Absolute Temperature and
the Ideal Gas Law Appendix
Apparatus to verify Charles’s law. A beaker with water contains a
tube with some enclosed gas. Atop the gas in the tube is a fixed
quantity of mercury, and above that the tube is open to the
atmosphere. A graph of volume V of the enclosed gas versus
temperature T is a straight line with positive slope. If the same
graph is made for several different types of gases, the lines all
have different (positive) slopes but meet at the same x-intercept,
with is the limiting temperature at which the volume would be
extrapolated to equal zero. This intercept is 0 on the kelvin scale.

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Maxwell-Boltzmann Distribution Appendix
The graph looks roughly like a bell curve, except it starts at 0 for
v = 0. The relative number of molecules increases to a maximum
near the rms speed of the gas, and then decays exponentially to
zero as v increases. Curves are shown for oxygen gas at two
slightly different temperatures. The higher temperature curve is
shifted to the right (higher speeds) relative to the lower
temperature. The higher temperature distribution has a higher
rms speed and a larger area under the high-speed tail of the
curve.