Thermal Physics
This chapter deals with the concept of temperature. It covers temperature scales, the
ideal gas equation, and the kinetic theory of gases.
Temperature
Qualitatively, temperature is a measure of how ‘hot’ or ‘cold’ an object is. If two objects
are in contact, then energy (heat) will flow from the hot to the cold object. If they are in
thermal equilibrium with each other, they are at the same temperature and there will be
no heat flow.
The zeroth law of thermodynamics says that if two objects (A and B) are each in thermal
equilibrium with a third object (C), then the two objects are in thermal equilibrium with
each other. While this may seem a bit abstract, it essentially says that we can use a
thermometer (object C) to decide if two objects (A and B) are at the same temperature.
A thermometer is a device that has some property that changes with temperature so that,
when properly calibrated, it can be used to measure temperature. One such device could
be a mercury-in-glass thermometer. As the temperature of the mercury increases the
mercury expands and this expansion in the glass is a measure of the temperature.
Another more fundamental thermometer is the pressure of a constant volume of a dilute
gas. In the Celsius scale (also called Centigrade) we define the temperature of an icewater mixture as 0oC and the temperature of boiling water as 100oC. Knowing the
pressure of the constant volume gas thermometer at these two points would allow one to
measure any temperature by extrapolation.
Absolute zero temperature
If the ice point is defined at 0oC and boiling water as 100oC, then the pressure of the gas
thermometer would extrapolate to zero at -273.15oC. This is referred to as absolute zero.
In the Kelvin scale, absolute zero is defined as 0 K, and the difference between the ice
point and boiling water is assumed to be 100 K. This makes the ice point 273.15 K and
boiling water 373.15 K.
In the Fahrenheit scale, the ice point is 32oF and boiling water is 212oF.
Conversions:
TC  TK  273.15
TF  9 TC  32
5
TC  TK , TF  9 TC
5
Question: What temperature is the same in the Celsius and Fahrenheit scales?
Thermal Expansion
Most objects expand with increase in temperature. For small temperature changes, the
change in length is
L   L0 T
where  is the coefficient of linear expansion.
Area expansion is given by
A   A0 T
and volume expansion is given by
V   V0 T
It can be shown that  = 2 and  = 3.
Example:
A steel railroad track consists of 4-m sections. The track reaches a low temperature of
-40oC in the winter and a high of 100oC in the summer. What gap should be allowed
between the sections? For steel,  = 11 x 10-6/oC.
L  (11x10 6 / o C )( 4m)(140 C )  6.2 x10 3 m  6.2 mm
Expansion of water
The thermal expansion of water is different from most materials. As the temperature is
increased above 4oC, it expands. But between 0oC and 4oC, water contracts with
increasing temperature. Water is most dense at 4oC. This is the reason why the top of a
pond freezes before the bottom.
Ideal Gas Equation
Gases at low pressures are found to obey the ideal gas law:
PV  nRT
In the gas equation, the following are the symbols in SI units:
P = absolute pressure [Pa = N/m2]
V = volume [m3]
T = absolute temperature [K]
n = number of moles [unitless]
R = gas constant [8.31 J/mole·K]
If P is in atmospheres, V in liters, and T in K, then R = 0.0831 L·atm/mole·K
By absolute pressure, we mean pressure relative to a vacuum, where P = 0. For example,
sometimes gauge pressure is given, which means pressure above atmospheric pressure.
A tire gauge measures pressure above atmospheric pressure, not absolute pressure. Don’t
use gauge pressure in the gas equation. Likewise, don’t use Celsius or Fahrenheit, since
these are not absolute temperature scales. Convert Celsius or Fahrenheit to Kelvin.
Mole
A mole of particles (atoms or molecules) is essentially Avogadro’s number of particles,
where
NA = 6.02 x 1023 particles/mole
It is also the number of atoms in 12 g of C-12. All elements have an atomic mass
number. The atomic mass number is the mass in grams of one mole of the element.
n
mass
atomic mass
The mass of one atom of an element can be calculated from the atomic mass as
matom 
atomic mass
NA
Thus, the mass of a C-12 atom is
mC 12 
12 g / mole
6.02 x10
23
atoms / mole
 1.99 x10  23 g / atom
Example:
A 1 liter container is filled with helium at a (gauge) pressure of 2 atm and temperature of
20oC. How many moles helium are in the container?
P  Patm  Pguage  1 atm  2atm  3 atm  3x1.013x10 5 Pa  3.039 x10 5 Pa
TK  20  273  293K
V  (1 L)(10  3 m 3 / L)  10  3 m 3
PV (3.039 x10 5 Pa)(10  3 m 3 )
n

 0.125 mole
RT
(8.31 J / mole  K )( 293K )
The mass of the helium is
m  (0.125 mole)(4 g / mole)  0.5 g
Example:
The above container of helium is heated to 100oC while the volume of the container is
reduced to 0.5 liter. What is the resulting pressure? Since the number of moles doesn’t
change
P1V1 P2V2

T1
T2
P2 
P1V1T2 (3atm)(1L)(373K )

 7.64 atm
V2T2
(0.5 L)( 293K )
Molecular Interpretation of Temperature
In equilibrium, the molecules of a gas have a wide range of speeds. The average kinetic
energy of the molecules is given by
KEavg  1 mv 2  3 k BT
2
2
where kB = 1.38 x 10-23 J/K is the Boltzmann constant and T is the Kelvin temperature. kB
can also be written in terms of the gas constant as
kB 
R
NA
The quantity v 2 is the average of the square of the speed.
From the above equation for average kinetic energy we can get
vrms  v 2 
3k BT
m
Alternatively, using the relationship between kB and R and the relationship between the
mass of a molecule, m, and the molar mass, M, we can write
vrms 
3RT
M
The rms speed, vrms, is not the same as the average speed, although it may be similar.
Example:
What is the rms speed of nitrogen molecules (N2) in the air at 20oC?
molar mass
28 g
M


 4.65 x10  23 g  4.65 x10  26 kg
23
NA
N A 6.02 x10
T  273  20  293K
m
v rms 
3(1.38 x10  23 J / K )( 293K )
4.65 x10  26 kg
 511 m / s