CHAPTER THIRTEEN
KINETIC THEORY
13.1 INTRODUCTION
Boyle discovered the law named after him in 1661. Boyle,
Newton and several others tried to explain the behaviour of
gases by considering that gases are made up of tiny atomic
particles. The actual atomic theory got established more than
150 years later. Kinetic theory explains the behaviour of gases
based on the idea that the gas consists of rapidly moving
atoms or molecules. This is possible as the inter-atomic forces,
which are short range forces that are important for solids
and liquids, can be neglected for gases. The kinetic theory
was developed in the nineteenth century by Maxwell,
Boltzmann and others. It has been remarkably successful. It
gives a molecular interpretation of pressure and temperature
of a gas, and is consistent with gas laws and Avogadro’s
hypothesis. It correctly explains specific heat capacities of
many gases. It also relates measurable properties of gases
such as viscosity, conduction and diffusion with molecular
parameters, yielding estimates of molecular sizes and masses.
This chapter gives an introduction to kinetic theory.
13.2 MOLECULAR NATURE OF MATTER
Richard Feynman, one of the great physicists of 20th century
considers the discovery that “Matter is made up of atoms” to
be a very significant one. Humanity may suffer annihilation
(due to nuclear catastrophe) or extinction (due to
environmental disasters) if we do not act wisely. If that
happens, and all of scientific knowledge were to be destroyed
then Feynman would like the ‘Atomic Hypothesis’ to be
communicated to the next generation of creatures in the
universe. Atomic Hypothesis: All things are made of atoms -
little particles that move around in perpetual motion,
attracting each other when they are a little distance apart,
but repelling upon being squeezed into one another.
Speculation that matter may not be continuous, existed in
many places and cultures. Kanada in India and Democritus
13.1 Introduction
13.2 Molecular nature of matter
13.3 Behaviour of gases
13.4 Kinetic theory of an ideal gas
13.5 Law of equipartition of energy
13.6 Specific heat capacity
13.7 Mean free path
Summary
Points to ponder
Exercises
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324 PHYSICS
in Greece had suggested that matter may consist
of indivisible constituents. The scientific ‘Atomic
Theory’ is usually credited to John Dalton. He
proposed the atomic theory to explain the laws
of definite and multiple proportions obeyed by
elements when they combine into compounds.
The first law says that any given compound has,
a fixed proportion by mass of its constituents.
The second law says that when two elements
form more than one compound, for a fixed mass
of one element, the masses of the other elements
are in ratio of small integers.
To explain the laws Dalton suggested, about
200 years ago, that the smallest constituents
of an element are atoms. Atoms of one element
are identical but differ from those of other
elements. A small number of atoms of each
element combine to form a molecule of the
compound. Gay Lussac’s law, also given in early
19
th
century, states: When gases combine
chemically to yield another gas, their volumes
are in the ratios of small integers. Avogadro’s
law (or hypothesis) says: Equal volumes of all
gases at equal temperature and pressure have
the same number of molecules. Avogadro’s law,
when combined with Dalton’s theory explains
Gay Lussac’s law. Since the elements are often
in the form of molecules, Dalton’s atomic theory
can also be referred to as the molecular theory
of matter. The theory is now well accepted by
scientists. However even at the end of the
nineteenth century there were famous scientists
who did not believe in atomic theory !
From many observations, in recent times we
now know that molecules (made up of one or
more atoms) constitute matter. Electron
microscopes and scanning tunnelling
microscopes enable us to even see them. The
size of an atom is about an angstrom (10
-10
m).
In solids, which ar
e tightly packed, atoms are
spaced about a few angstroms (2 Å) apart. In
liquids the separation between atoms is also
about the same. In liquids the atoms are not
as rigidly fixed as in solids, and can move
around. This enables a liquid to flow. In gases
the interatomic distances are in tens of
angstroms. The average distance a molecule
can travel without colliding is called the mean
free path. The mean free path, in gases, is of
the order of thousands of angstroms. The atoms
are much freer in gases and can travel long
distances without colliding. If they are not
enclosed, gases disperse away. In solids and
liquids the closeness makes the interatomic force
important. The force has a long range attraction
and a short range repulsion. The atoms attract
when they are at a few angstroms but repel when
they come closer. The static appearance of a gas
Atomic Hypothesis in Ancient India and Greece
Though John Dalton is credited with the introduction of atomic viewpoint in modern science, scholars in
ancient India and Greece conjectured long before the existence of atoms and molecules. In the Vaiseshika
school of thought in India founded by Kanada (Sixth century B.C.) the atomic picture was developed in
considerable detail. Atoms were thought to be eternal, indivisible, infinitesimal and ultimate parts of matter.
It was argued that if matter could be subdivided without an end, there would be no difference between a
mustard seed and the Meru mountain. The four kinds of atoms (ParamanuSanskrit word for the
smallest particle) postulated were Bhoomi (Earth), Ap (water), Tejas (fire) and Vayu (air) that have characteristic
mass and other attributes, were propounded. Akasa (space) was thought to have no atomic structure and
was continuous and inert. Atoms combine to form different molecules (e.g. two atoms combine to form a
diatomic molecule dvyanuka, three atoms form a tryanuka or a triatomic molecule), their properties depending
upon the nature and ratio of the constituent atoms. The size of the atoms was also estimated, by conjecture
or by methods that are not known to us. The estimates vary. In Lalitavistara, a famous biography of the
Buddha written mainly in the second century B.C., the estimate is close to the modern estimate of atomic
size, of the order of 10
–10
m.
In ancient Greece, Democritus (Fourth century B.C.) is best known for his atomic hypothesis. The
word ‘atom’ means ‘indivisible’ in Greek. According to him, atoms differ from each other physically, in
shape, size and other properties and this resulted in the different properties of the substances formed
by their combination. The atoms of water were smooth and round and unable to ‘hook’ on to each
other, which is why liquid /water flows easily. The atoms of earth were rough and jagged, so they held
together to form hard substances. The atoms of fire were thorny which is why it caused painful burns.
These fascinating ideas, despite their ingenuity, could not evolve much further, perhaps because they
were intuitive conjectures and speculations not tested and modified by quantitative experiments - the
hallmark of modern science.
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KINETIC THEORY 325
is misleading. The gas is full of activity and the
equilibrium is a dynamic one. In dynamic
equilibrium, molecules collide and change their
speeds during the collision. Only the average
properties are constant.
Atomic theory is not the end of our quest, but
the beginning. We now know that atoms are not
indivisible or elementary. They consist of a
nucleus and electrons. The nucleus itself is made
up of protons and neutrons. The protons and
neutrons are again made up of quarks. Even
quarks may not be the end of the story. There
may be string like elementary entities. Nature
always has surprises for us, but the search for
truth is often enjoyable and the discoveries
beautiful. In this chapter, we shall limit ourselves
to understanding the behaviour of gases (and a
little bit of solids), as a collection of moving
molecules in incessant motion.
13.3 BEHAVIOUR OF GASES
Properties of gases are easier to understand than
those of solids and liquids. This is mainly
because in a gas, molecules are far from each
other and their mutual interactions are
negligible except when two molecules collide.
Gases at low pressures and high temperatures
much above that at which they liquefy (or
solidify) approximately satisfy a simple relation
between their pressure, temperature and volume
given by (see Chapter 11)
PV = KT (13.1)
for a given sample of the gas. Here T is the
temperature in kelvin or (absolute) scale. K is a
constant for the given sample but varies with
the volume of the gas. If we now bring in the
idea of atoms or molecules, then K is proportional
to the number of molecules, (say) N in the
sample. We can write K = N k . Observation tells
us that this k is same for all gases. It is called
Boltzmann constant and is denoted by k
B
.
As
1 1 2 2
1 1 2 2
P V P V
N T N T
=
= constant = k
B
(13.2)
if P, V and T are same, then N is also same for
all gases. This is Avogadro’s hypothesis, that the
number of molecules per unit volume is
the same for all gases at a fixed temperature and
pressure. The number in 22.4 litres of any gas
is 6.02 × 10
23
. This is known as Avogadro
number and is denoted by N
A
. The mass of 22.4
litres of any gas is equal to its molecular weight
in grams at S.T.P (standard temperature 273 K
and pressure 1 atm). This amount of substance
is called a mole (see Chapter 2 for a more precise
numbers in equal volumes of gas at a fixed
temperature and pressure from chemical
reactions. Kinetic theory justifies this hypothesis.
The perfect gas equation can be written as
PV =
µ
RT (13.3)
where
µ
is the number of moles and R = N
A
k
B
is a universal constant. The temperature T is
absolute temperature. Choosing kelvin scale for
John Dalton (1766 1844)
He was an English chemist. When different types of atoms combine,
they obey certain simple laws. Dalton’s atomic theory explains these
laws in a simple way. He also gave a theory of colour
blindness.
He made a brilliant guess that equal volumes of gases
have equal number of molecules at the same
temperature and pressure. This helped in
understanding the combination of different gases in
a very simple way. It is now called Avogadro’s hypothesis (or law). He also
suggested that the smallest constituent of gases like hydrogen, oxygen and
nitrogen are not atoms but diatomic molecules.
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