1. KINETIC THEORY 13-1
Richard Feynman considers the concept “Matter is made up of atoms” to be very significant. This concept is based on
Atomic Hypothesis: All things are made of atoms – the little particles that move around in perpetual motion,
attracting each other when they are a little distance apart but repelling when close enough to each other.
Atomic theory is credited to J. Dalton who proposed it to explain the laws of definite and multiple proportions obeyed
by the elements when they combine chemically to form compounds.
I law: Any given compound has a fixed proportion by mass of its constituents.
II law: 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.
According to Dalton’s atomic Theory:
1. The smallest constituents of an element are atoms.
2. Atoms of one element are identical in mass and chemical properties.
3. Atoms of different elements differ from one another.
4. Atoms a re indivisible particles, which cannot be created or destroyed in a chemical reaction.
5. Atoms combine in the ratio of small whole numbers to form compounds.
6. The relative number and kinds of atoms are constant in a given compound.
But now a days we know, atoms are not indivisible as Dalton stated. They consist of Proton, neutrons and electrons. The
proton and neutrons are again made up of quarks.
About the gases, like Dalton, Gay Lussac also stated that ‘when gases combine chemically to yield another gas then
their volumes are in the ratio of small integers’.
Avogadro’s hypothesis states, “Equal volumes of all gases at equal temperature and pressure have same
number of molecules”.
Since the substances are often in the form of molecules, Dalton’s atomic theory can also be referred to as the
molecular theory of matter that Molecules constitute Matter. The size of atom is about 10 – 10
m.
In solids, which are tightly packed, atoms are spaced about 2 Ǻ apart.
In liquids, the interatomic separation is almost the same as that in solids i.e. 2 Ǻ but the atoms are not rigidly fixed
and hence they can move around. This enables a liquid to flow.
In gases, the interatomic separation is very large, so, molecules of gases can move almost freely with out colliding
to each other. The average distance over which a molecule can travel without colliding is called the Mean free
Path. If the gas is not enclosed then its molecule may disperse away.
Due the closeness of the atoms in solids and liquids, the interatomic forces play important role to decide their
properties. When atoms are far away then attract each other while they repel each other if they are very close to
each other.
The gas can not acquire a static behavior like solids and liquids. The gas is full of activity and achieves dynamic
equilibrium instead of static equilibrium. In dynamic equilibrium, molecules collide and change their speed during
the collision. Average properties of the gas remain constant.
KINETIC THEORY OF GASES
Any sample of a gas is made of molecules. A molecule is the smallest unit having all the chemical properties of
the gas. The kinetic theory of gases attempts to develop a model of the molecular behavior of an ideal gas.
2. KINETIC THEORY 13-2
The kinetic theory of a gas is based on the following assumptions: -
1. All gases are made of molecules moving randomly in all directions.
2. The size of a molecule is much smaller than the average separation between the molecules.
3. The molecules exert no force on each other or on the walls of the container except during collisions.
4. All collisions between two molecules or between a molecule and a wall are perfectly elastic.
5. The time spent during a collision is negligibly small.
6. The molecules obey Newton’s laws of motion.
When a gas is left for sufficient time, it comes to a steady state. The density and the distribution of molecules with
different velocities are independent of position, direction and time.
IDEAL GAS & IDEAL GAS EQUATION
In a gas, molecules are far apart from each other and their mutual interactions are negligible except when two
molecules collide.
Gases at low pressures and high temperatures satisfy a simple relation P V = K T, where P = pressure, V =
volume and T = absolute temperature of the given mass of the gas.
K = a constant for the given sample of the gas but varies with its volume and also found to be proportional to the
number of molecules (N) of the gas, so, K = N k,
k is same for all gases and known as Boltzmann constant and its value is 1.38 x 10 – 23
J/K
So, P V = N k T ⇒
NT
PV
= constant = k
If N = μ NA, where NA = number of molecules contained by 22.4 litres of any gas at STP (273 K, 1 atm)
μ is the number of moles of the gas.
So, P V = μ NA k T ⇒ P V = μ R T,
this relation is known as perfect gas equation.
Any gas satisfying this equation exactly at all
pressures and temperatures is called an ideal
gas.
No real gas is truly ideal. The following graphs
show the deviation of a real gas from ideal gas
behaviour. Clearly from the graphs, all the curves
approach the ideal gas behaviour for low pressu-
res and high temperatures because under these
conditions the molecules are far apart and molecular interactions are negligible.
PRESSURE OF AN IDEAL GAS
Consider an ideal gas enclosed in a cubical vessel of edge ‘a’.
Take a corner of the vessel as the origin O and X-, Y-, Z- axes
along the edges. Let A1 and A2 be the parallel faces of the cube
perpendicular to X-axis. Consider a molecule moving with velo-
city v
with the components along X-, Y- and Z- axes to be vx, vy
and vz. When the molecule collides the face A1, the vx is reversed
3. KINETIC THEORY 13-3
while vy and vz remain unchanged as the collisions are perfectly elastic.
The change in momentum of the molecule = ∆p = (- m vx) – (m vx) = - 2 m vx.
Linear momentum is conserved, the change in momentum of the wall is ∆p’ = 2 m vx.
This molecule then travel to face A2 from A1 with constant velocity – vx along X-axis as we assume no collision
takes place during this motion between A1 and A2. The molecule travels a distance a from A1 to A2.
So, time taken from A1 to A2 by the molecule is t = a/vx.
The molecule collides the wall A1 two times successively after a time interval t ’ = 2 t = 2a/vx.
So, the force exerted by the molecules on the wall A1 per unit time = F’x = ∆p’/t ‘ =
a
m
vx
2
So, total force exerted on the wall A1 due to all the molecules will be F =
a
m
∑2
xv
All directions are equivalent, so, we have ∑2
xv = ∑2
yv = ∑ 2
zv =
3
1
∑ ++ )222( zvyvxv =
3
1
∑2v
So, F =
3
1
a
m
∑2v =
3
1
a
mN
N
v∑ 2
, where N = number of molecules in the gas sample
The pressure on the wall is p = F ÷ area of A1 = 2a
F
, so, p =
3
1
3a
mN
N
v∑ 2
=
3
1
3a
M
N
v∑ 2
=
3
1
ρ
2v , where M = total mass of the gas and ρ = density of the gas, 2v = mean square speed of the gas.
So, gas pressure p =
3
1
ρ 2v or p =
3
1
V
M
2v , where V = a 3
= volume of the gas
RMS SPEED (vrms)
The square root of mean square speed is called rms speed. vrms = 2v = ρ
p3
TRANSLATIONAL KINETIC ENERGY OF A GAS
The total translational kinetic energy of all the molecules of a gas is
K = ∑ 2
1
mv2
=
2
1
mN
N
v∑ 2
⇒ K =
2
1
M vrms
2
But p =
3
1
V
M
2v = p =
3
1
V
M
vrms
2 ⇒ p V =
3
1
M vrms
2
=
3
1
Nm vrms
2
=
3
2
×
2
1
M vrms
2
=
3
2
K ⇒ K =
2
3
p V
The average kinetic energy of a molecule is K/N =
2
1
N
M
vrms
2
⇒ K/N=
2
1
m vrms
2
KINETIC INTERPRETATION OF TEMPERATURE
p V =
3
1
Nm vrms
2
=
3
2
×
2
1
N m vrms
2
=
3
2
× average translational kinetic energy of the molecules =
3
2
K
But for an ideal gas, p V = μ RT = μ k NAT, where μ = number of moles in the gas, R = universal gas constant,
T = absolute temperature of the gas, k = Boltzmann constant, NA = Avogadro’s number
So, combining the above two equations, we have,
3
2
K = μ k NAT = k NT,
4. KINETIC THEORY 13-4
N = k NA = total number of moles in gas ⇒ K =
2
3
k NT
So, average kinetic energy of the gas molecules = K/N =
2
3
kT
So, the average kinetic energy of a molecule is proportional to the absolute temperature of the gas; and
independent of pressure, volume or nature of the ideal gas. K α T
This is fundamental result relating kinetic energy and temperature of the gas.
DEDUCTIONS OF GAS LAWS FROM KINETIC THEORY
Boyle’s Law
At a given temperature, the pressure of a given mass of a gas is inversely proportional to its volume.
p V =
3
1
Nm vrms
2
, and K α vrms
2
α T, so, vrms
2
is constant at a given temperature as well as m and N are also
constant for a given mass of the gas, so, p V = constant ⇒
V
p
1
α , which is Boyle’s law.
Charle’s Law
At a given pressure, the volume of a given mass of a gas is proportional to its absolute temperature.
p V =
3
1
Nm vrms
2
, and V α vrms
2
(if p = constant), but vrms
2
α T ⇒ V α T, which is Charle’s law.
Charle’s Law of pressure
At a given volume, the pressure of a given mass of a gas is proportional to its absolute temperature.
p V =
3
1
Nm vrms
2
, and p α vrms
2
(if V = constant), but vrms
2
α T ⇒ p α T, which is Charle’s law of pressure.
Avogadro’s Law
At the same temperature and pressure, equal volumes of all gases contain equal number of molecules.
Let m1, m2 = masses of one molecule of two gases, N1, N2 = number of molecules in the two gases respectively, p
= common pressure and V = common volume of two gases.
So, p V =
3
1
N1m 1v1
2
and p V =
3
1
N2m 2v2
2
⇒ N1m 1v1
2
= N2m 2v2
2
As temperature of the two gases are same, so, average kinetic energy of the gas molecules is the same for the
two gases, so,
2
1
m 1v1
2
=
2
1
m 2v2
2
⇒ N1 = N2, which is Avogadro’s law.
Graham’s Law of diffusion
When two gases at the same temperature and pressure are allowed to diffuse into each other, the rate of
diffusion of each gas is inversely proportional to the square root of the density of the gas.
Since, diffusion rate of gas is proportional the vrms of the gas. So, for two different gases, if r1 and r2 are the
diffusion rates then
2
1
2
1
v
v
r
r
= , but vrms = ρ
p3
, so, at given pressure of the gases,
1
2
2
1
ρ
ρ
v
v
= ,
⇒ =
2
1
r
r
1
2
ρ
ρ , which is Graham’s Law of diffusion.
Dalton’s Law of Partial Pressure
5. KINETIC THEORY 13-5
The pressure exerted by a mixture of several gases equals the sum of the pressures exerted by each gas
occupying the same volume as that of the mixture.
In kinetic theory, we assume that the pressure exerted by a gas on the wall of a container is due to the collisions
of the molecules with the walls. The total force on the wall is the sum of the forces exerted by the individual
molecules. Let N1 = molecules of gas 1, N2 = molecules of gas 2, … in the mixture.
Thus, the force on a wall of surface area A is F = force by N1 molecules of gas 1 + force by N2 molecules of
gas 2 + … = F1 + F2 + …
If the first gas alone is kept in the container, its N1 molecules will exert a force F1 on the wall. If the pressure in this
case is p1 then p1 = F1 / A, it is called the partial pressure of the gas.
This total pressure of the mixtures of the gas p = p1 + p2 + p3 + …, which is Dalton’s Law of Partial
pressure.
DEGREES OF FREEDOM & LAW OF EQUIPARTITION OF ENERGY
The kinetic energy of a single molecule of gas is given by ε =
2
3
m vrms
2
=
2
1
m vx
2
+
2
1
m vy
2
+
2
1
m
vz
2
Since, ε =
2
3
k T, so, for a gas in thermal equilibrium at temperature T,
kTxmv
2
12
2
1
= , kTymv
2
12
2
1
= , kTzmv
2
12
2
1
=
The motion of a body as a whole from one point to another is called translation. A single molecule free to move in
space needs 3 coordinates to specify its location. These three coordinates are known as degrees of freedom of
motion.
The number of degrees of freedom is 3 for motion of a single molecule in space, 2 for plane, 1 for a single line.
Thus, a molecule free to move in space has 3 translational degrees of freedom.
Each translational degree of freedom contributes a term that contains square of some variable of motion e.g. ½ m
vx
2
and similar terms in vy and vz. In thermal equilibrium, the average of each such term is ½ kT.
Molecules of monatomic gases like He have only translational degrees of freedom but a molecule of diatomic gas
like O2, N2 etc can also rotate about its centre of mass in addition to the translation. A diatomic molecule has two
interdependent axes of rotation both of which are normal to the line joining the two atoms such if X-axis is the line
joining the two atoms then it may rotate about the Y- and Z-axis respectively. Thus, the molecule has two
rotational degrees of freedom; each of which contributes a term to the total energy consisting of translational
energy. So, total energy of such a molecule is
ε =
2
1
m vx
2
+
2
1
m vy
2
+
2
1
m vz
2
+
2
1
I1 ω 2
1 +
2
1
I2 ω 2
2 , where symbols have their usual meanings.
In the above assumption, it is assumed that O2 molecule is a ‘rigid rotator’ an i.e. molecule does not vibrate. This
assumption is not found to be true for all diatomic molecules at moderate temperature like CO molecule. Its atoms
oscillate along the line joining them and contribute a vibration energy term to the total energy gibe by
ε vibration =
2
1
m
2
dt
dy
+
2
1
k y2
, where k is force constant of the oscillator and y is the vibration coordinate.
Clearly, unlike each vibration mode contributes two ‘squared terms’; kinetic and potential energies.
Thus, total energy of such a molecule is ε = ε t + ε r + ε v
6. KINETIC THEORY 13-6
According to classical statistical principle put forward by Maxwell, the total energy of a molecule is equally
distributed in all possible energy modes, with each mode having an average energy equal to ½ k T. This is
known as Law of Equipartition of Energy.
SPECIFIC HEAT CAPACITY OF GASES
Monatomic Gases:
Degrees of freedom = 3 (translational only) Thus, average energy of a molecule at temperature T is 3/2 k T
So, total energy of one mole of the gas U =
2
3
kT × NA =
2
3
RT
The molar specific heat at constant volume, CV =
dT
dU
=
2
3
R
For ideal gas CP – CV = R, thus, the molar specific heat at constant pressure, CP =
2
5
R
The ratio of specific heats γ =
VC
PC
=
3
5
Diatomic Gases:
1. For a rigid rotator
Degrees of freedom = 5 (3 translational and 2 rotational)
Total energy of one mole of the gas U =
2
5
kT × NA =
2
5
RT
Thus, CV =
2
5
R, CP =
2
7
R and the ratio of specific heats γ =
5
7
2. If molecule vibrate the axis then
Degrees of freedom = 7 (3 translational and 2 rotational, 2 vibration mode)
Total energy of one mole of the gas U =
2
7
kT × NA =
2
7
RT
Thus, CV =
2
7
R, CP =
2
9
R and the ratio of specific heats γ =
7
9
Polyatomic Gases:
Degrees of freedom = 3 translational, 3 rotational and f vibrational.
So, total energy of one mole of the gas U = (
2
3
kT +
2
3
kT + f k T) NA
The molar specific heat at constant volume, CV = (3 + f)R and CP = (4 + f)R
The ratio of specific heats γ =
VC
PC
= f
f
+
+
3
4
SPECIFIC HEAT CAPACITY OF SOLIDS
7. KINETIC THEORY 13-7
Let NA = number of atoms in one mole solid each vibrating about its mean position. An oscillation in one
dimension has an average energy of
2
1
m
2
dt
dy
+
2
1
k y2
= 2 x ½ k T = k T.
In 3 dimensions, the average energy = 3 k T, Thus, total energy of one mole of solid = U = 3 R T, At constant
pressure, ∆Q = ∆U + P ∆V, but for a solid, ∆V = negligible, so, ∆Q = ∆U. Hence, C =
dT
dQ
=
dT
dU
= 3 R
SPECIFIC HEAT CAPACITY OF WATER
Let’s treat the water as a solid. Each atom of a solid as average energy = 3 k T. A water molecule has 3 atoms.
So, a water molecule has total energy = 9 k T
Hence, specific heat capacity of water C = 9 R. This theoretical value is in close agreement of observed value of
specific heart capacity of water.
MEAN FREE PATH
The average distance between two successive collisions moved by a molecule of gas is called mean free path.
Molecules in a gas have large speeds almost of the order of speed of sound even then a gas leaking from a
container takes a considerable time to diffuse in the room. This is because the molecules in a gas have a finite
size, so, they are bound to undergo collisions. As a result, they cannot move straight and they get deflected from
their paths frequently.
Assume a molecule of the gas spherical in shape of diameter
d moving with average speed v. It will suffer collision any other
molecule that comes within a distance d between the centers of
two spheres. A molecule sweeps, a volume π d2
v ∆t,
within a time interval ∆t. If n = number of molecules per unit
volume of the gas then this molecule will suffer n π d2
v ∆t collisions in time interval ∆t.
So, on average, the number of collisions per unit time = n π d2
v
Thus, the time interval between two successive collisions = τ = 1 / n π d2
v
Thus, mean free path i.e. the distance traveled by the molecule between two successive collisions = v τ =1/ n π
d2
But in this derivation, it is assumed that during the motion of a molecule, the molecules are at rest. But in actual all
molecules are in motion equally. So, more exact treatment gives, mean free path = 1/ 2 n π d2
MAXWELL DISTRIBUTION FUNCTION
In a given mass of gas, the velocities of all molecules are not
the same, even when bulk parameters like pressure, volume
and temperature are fixed. Collisions change the direction and
the speed of molecules. However in a state of equilibrium, the
distribution of speeds is constant or fixed.
The molecular speed distribution gives the number of molecules
between the speeds v and v + dv.
d N(v) = 4 p N a 3 2bve− v2
dv = nv dv. This is called Maxwell distribution.
d
d
8. KINETIC THEORY 13-8
The plot of nv against v is shown in the figure. The fraction of the molecule with speeds v and v + dv is equal to the
area of the strip shown. The average of any quantity like v2
is defined by the integral <v2
> = (1/N) ∫v2
dN(v) =
m
kT3
which agrees with the result derived from more elementary considerations.
