Lecture #2.4
1. Ideal gas law
As we already know, classical thermodynamics is a macroscopic science which
describes physical systems by means of macroscopic variables, such as pressure, volume,
density and temperature. These quantities are obtained by averaging over large numbers
of molecules from which the system consists. Introducing these variables, we never
discussed microscopic structure of substances and the relations between this structure and
aforementioned variables.
Today we are going to consider one particular example of a thermodynamic system,
which includes the so-called ideal gas. We shall consider this gas not just from the
standpoint of classical thermodynamics, but also using microscopic approach, known as
the kinetic theory of gases. We shall find interconnections between the two approaches. It
is obvious from the beginning that macroscopic pressure of gas exists due to collisions
between the gas molecules and the walls of the container. Temperature of gas is for sure
related to kinetic energy of its molecules.
When talking about molecules, it is efficient to introduce a special unit to measure
the amount of substance. Of course, we could do so in terms of numbers of molecules.
But the number of molecules filling the volume of several cubic meters is so huge that it
will be very inconvenient to operate with such large numbers. Instead, we shall use a
special unit, mole, to measure the amount of substance. Mole is also one of the
fundamental units of SI. It is defined as the number of atoms in a 12g sample of carbon12. This number of atoms can be measured experimentally and it is known as the
Avogadro's number which is
N A  6.02  1023 mol 1 .
(2.4.1)
So, the number of moles, n, contained in a sample of any substance is equal
n
N
,
NA
(2.4.2)
where N is the number of molecules in the sample. The amount of substance can be
calculated if one knows the mass of one mole for the substance (the molar mass M) or the
mass of one molecule m, so
n
M sam M sam
,

M
mN A
(2.4.3)
where M sam is mass of the sample.
Now we are ready to discuss the “ideal gas”. We can see even from its name, that
this substance does not actually exist in nature, it is an idealization. However, at certain
conditions any gas can be considered as almost ideal. This idealization assumes that there
are no any intermolecular forces acting between the molecules of this gas. In other words,
these molecules are neither attracted, not repelled by each other. The potential energy of
their interaction is equal to zero. The only type of energy which they can posses is the
kinetic energy of their motion, and it changes only by means of the rear collisions
between the molecules and the walls of the container. This idealization is almost valid for
any gas as long as it has low density. In such a case the average distance between the
molecules is so large, that we can ignore potential energy of their interaction compared to
their kinetic energy. It is experimental result for gases at low density, as well as
theoretical result for ideal gas, that
pV  nRT .
(2.4.4)
You will actually try to verify that statement in the lab tomorrow. This equation is known
as the ideal gas law. In that equation, p is the absolute pressure of gas, V is its volume, T
is temperature (in Kelvin scale), n is the amount of substance (in moles) and R is the
universal gas constant, which is
R  8.31 J mol  K .
(2.4.5)
This equation can also be represented in a different form as
pV  NkT ,
(2.4.6)
where
k
R
J
 138
.  1023
NA
K
(2.4.7)
is the Boltzmann constant.
Let us consider isothermal expansion of ideal gas. Gas expands from original
volume V1 to the final volume V2 and at the same time it is kept at constant temperature.
This can be arranged, if the expansion process is quite slow, so this gas has enough time
to reach thermal equilibrium with its environment all the time during this process.
Pressure of this ideal gas can then be found from equation 2.4.4 as
p
nRT
,
V
In the case of the constant temperature and the constant amount of substance, this
dependence will look like hyperbola in the (p, V) coordinate plane. This gives
p1V1  p2V2 ,
(2.4.8)
which is known as the Boyle’s Law. In the case if we have a constant volume-process,
which means V2  V1 , the ideal gas law gives
p
nRT
 const T ,
V
or
p1 p2
,

T1 T2
(2.4.9)
which is known as the Gay-Lussac law. In the case if we have constant pressure process
p1  p2  p then
V
nRT
 const T ,
p
or
V1 V2
 ,
T1 T2
(2.4.10)
known as the Charles’s law.
Example 2.4.1. A balloon contains 2.0 liters of nitrogen gas at a temperature of 77
K and a pressure of 101 kPa. If the temperature of the gas is allowed to increase to 23
deg C and the pressure remains constant, what volume will the gas occupy?
Let us consider this nitrogen as an ideal gas. In that case the ideal gas law gives
pV  nRT . One can rewrite this equation as
V nR
. The right hand side of the equation

T
p
stays constant for the system considered here, which means that the left hand side is also
conserved and
has V2  2.0 L
V1 V2
T
or V2  V1 2 . Taking into account that T2  23o C  296K , one

T1
T1 T2
296 K
 7.7 L .
77 K
2. Kinetic Theory of Gases
So far we have interpreted ideal gas law from the macroscopic stand-point and
essentially took it as experimental result. Now we have to see what the microscopic
significance of this law is.
Our goal is to relate macroscopic characteristics of gas, such as its pressure or
temperature to microscopic ones, such as velocity of gas molecules and their energy. Let
us consider the gas in the amount of n moles which is placed in the cubic box of volume
V. This gas exerts pressure p on the walls of the box. The gas and the box are kept at
constant temperature T. We shall assume that this gas is an ideal gas. So, the typical gas

molecule of mass m moves with velocity v and then collides with the wall of the box.
We will treat this collision as absolutely elastic collision. So, the molecule will change
the direction of the component of its velocity perpendicular to the wall to the opposite
direction. Let us call this direction to be the x-direction. This axis will be directed
towards the wall, so the change of the molecule's momentum will be
px  mvx  mvx  2mvx
and the change of the wall's momentum, due to this collision, is 2mv x . The next collision
will occur after the time interval t for which this molecule travels to the opposite wall
reflects from it and comes back (we ignore collisions between molecules). This time is
t  2 L vx , where L is the distance between the walls. So, this molecule will deliver
momentum to the wall at average rate of
px 2mv x mv x2
.


t
2 L vx
L
According to the Newton's second law this rate equals to the average force acting on the
wall from this molecule. To find the total force acting on this wall we have to perform
summation over all the molecules, so finally we have pressure
N
F
F
p  x  2x 
A L
 mvxi2
L
i 1
L2

m N 2
 vxi ,
L3 i 1
where N  nN A is the number of molecules in the box. This pressure will be
p
m
nmN A 2
N (v x2 ) avg 
(v x ) avg ,
3
L
L3
where (v x2 ) avg is the average value for the square of the molecule’s velocity in the x
direction taken over all the molecules in the box. Since there are many molecules in the
box and they are moving in random directions, we can say that
v x2 avg
v 2 avg
the average

3
square of the velocity component in one direction is just 1/3 of the average square of the
molecule's speed. So the pressure will become
p
nM (v 2 ) avg
3V
.
One can introduce the root-mean-square speed, which is vrms  (v 2 ) avg , so
p
2
nMvrms
.
3V
Taking into account the ideal gas law we finally have the microscopic quantity vrms
related to macroscopic temperature as
vrms 
3RT
3kT

.
M
m
(2.4.11)
For most of the gases, this velocity at usual room temperature has a very high value of
several hundreds or even thousands of meters per second. Velocity of sound in gas cannot
be higher than velocity of its molecules. Experiment shows that it is in fact slightly
smaller than that.
Velocities of molecules are related to kinetic energy of translational motion of these
molecules, so that the average kinetic energy of one molecule will be
Kavg 
F mv I
GH 2 JK
2

avg
d i
m 2
v
2
avg

2
mvrms
m 3kT 3

 kT .
2
2 m
2
(2.4.12)
At given temperature T, all ideal gas molecules, have the same average translational
kinetic energy of 3kT 2 .
Of course, even though we have already calculated the root-mean square speed of
the molecules, it does not mean that all the molecules are moving with this speed. Some
of them are much faster, while others are much slower. The faster the molecule is above
the average, the fewer numbers of such fast molecules can exist. The same is true about
slow molecules. There are no molecules in gas which are not moving and only a few are
moving with sufficiently law speeds. In 1852 the Scottish physicist James Clerk Maxwell
derived the distribution of ideal gas molecules over their speeds. His result is known as
the Maxwell's speed distribution law, which is
F m IJ v e
.
(2.4.13)
Pbv g  4 G
H 2kT K
In this equation Pbv g is the probability density distribution function, the fraction of the
32
2  mv 2 2 kT
molecules having speeds in the given speed interval.
We have been talking about speed of molecules for so long, because this speed is
related to their kinetic energy. This we have already seen in the equation 2.4.12. In the
case of ideal gas, the kinetic energy is the only type of energy the gas molecules can
have. This means we can relate our results to the total internal energy of ideal gas. In the
case of monatomic gas, which consists of individual atoms rather than molecules, such as
He, Ne, Ar, the atoms can be considered as particle-like objects without size. So, they
cannot rotate and total internal energy of such gas is just the sum of translational kinetic
energies of its atoms. We have already proved that the average translational kinetic
energy of one molecule (atom in this case) is 3kT 2 . So, for the gas consisting of N
molecules, one has
E
3
3
3
NkT  nN A kT  nRT .
2
2
2
(2.4.11)
Thus, the internal energy of ideal gas is function of the gas’ temperature only. It does not
depend on any other variable.