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MODULE - 3

Kinetic Theory of Gases

Thermal Physics

10
Notes

KINETIC THEORY OF GASES

As you have studied in the previous lessons, at standard temperature and pressure,
matter exists in three states – solid, liquid and gas
...
At room temperature,
these atoms/molecules have finite thermal energy
...
This state of matter is said to be the gaseous
state
...

Under different conditions of temperature, pressure and volume, gases exhibit
different properties
...
In this lesson you will learn the kinetic
theory of gases which is based on certain simplifying assumptions
...
Why the gases have two types of heat capacities and
concept of thermal expansion will also be explained in this lesson
...


10
...
How does this change
in temperature affect our day to day activities? How do things change their
properties with change in temperature? Is there any difference between
temperature and heat? All such questions will be discussed in the following
sections
...
In Physics, however, these two terms have very different meaning
...

10
...
1 Heat Capacity and Specific Heat
When heat is supplied to a solid (or liquid), its temperature increases
...
This simply implies
that the rise in the temperature of a solid, when a certain amount of heat is
supplied to it, depends upon the nature of the material of the solid
...
The specific heat of the material of a solid (or a liquid) may be
defined as the amount of heat required to raise the temperature of its unit mass
through 1°C or 1K
...
1
...
The specific heat
of a material and other physical quanties related to this heat transfer are
measured with the help of a device called calorimeter and the process of the
measurement is called calcorimetry
...
1
...
Then,
the heat will be transferred from the higher to the lower temperature and the
substances will acquire the same temperature θ
...

⇒

m1C1 (θ1 − θ) = m2C2 (θ − θ2 )

This is the principle of calorimetry
...
Also, by knowing θ1, θ2 and θ the specific
heat capacity of a substance can be determined if the specific heat capacity of
the other substance is known
...
1
...
This
is called thermal expansion
...

In linear expansion, the change in length is directly proportional to the original
length and change in temperature
...
It is given by
α=

Δl
l0 Δθ

If, Δθ = 1°C and l0 = 1m
Then α = Δl
PHYSICS

279

MODULE - 3
Thermal Physics

Kinetic Theory of Gases

Thus, α is defined as the change in length of unit length of the substance whose
temperature is increased by 1°C
...

In cubical expansion, the change in volume is directly proportional to the change
in temperature and original volume:
ΔV ∝ V0 Δθ
ΔV = γV0 Δθ

or

where γ in the temperature coefficient of cubical expansion
...

Relation between α, β and γ
Let there be a cube of side l whose temperature is increased by 1°C
...
10
...
We therefore have
β = 2α

ΔV l 3 (1 + α)3 − l 3
=
V
l3

similarly,

γ=

or,

γ = l 3 + α3 + 3α 2 + 3α − l 3

As α is very small, the term α2 and α3 may be neglected
...


10
...
5 Anomalous expansion in water and its effect
Generally, the volume of a liquid increases with increase in temperature
...
However the
volume of water does not increase with temperature between 0 to 4°C
...

After that the volume starts increasing (while the density decreases) as shown in
Fig
...
2
...
10
...
As the pond cools, the colder, denser
water at the surface initially sinks to the bottom
...
The temperating of surface water
keeps on decreasing and freezes ultimately at 0°C
...
If this had not happened fish and all the marine
life would not have survived
...
1
...
This expansion is very large
as compared to solids and liquids
...
Hence we have to
consider either expansion of the gas with temperature at constant pressure or
the increase in its pressure at constant volume
...
2 KINETIC THEORY OF GASES
You now know that matter is composed of very large number of atoms and
molecules
...
Kinetic theory of gases attempts to relate the
macroscopic or bulk properties such as pressure, volume and temperature of an
ideal gas with its microscopic properties such as speed and mass of its individual
molecules
...
(A gas whose
molecules can be treated as point masses and there is no intermolecular force
between them is said to be ideal
...

10
...
1 Assumptions of Kinetic Theory of Gases
Clark Maxwell in 1860 showed that the observed properties of a gas can be
explained on the basis of certain assumptions about the nature of molecules, their
motion and interaction between them
...

We now state these
...
The intermolecular forces between
them are negligible
...

These collisions are perfectly elastic
...

Notes

(iv) Between collisions, molecules move in straight lines with uniform velocities
...

(vi) Distribution of molecules is uniform throughout the container
...
Moreover, since a molecule moving in space will have
velocity components along x, y and z–directions, in view of assumption (vi)it is
enough for us to consider the motion only along one dimension, say x-axis
...

10
...
Note that if there were N (= 6 ×1026 molecules m–3), instead of considering
3N paths, the assumptions have reduced the roblem to only one molecule in one
dimension
...
Its x,
y and z components are u, v and w, respectively
...
On striking the wall, this molecule will rebound in the
opposite direction with the same speed u, since the collision has been assumed to
be perfectly elastic (Assumption ii)
...
Hence, the change in momentum of a molecule is
mu – (–mu) = 2mu
If the molecule travels from face LMNO to the face ABCD with speed u along x–
axis and rebounds back without striking any other molecule on the way, it covers
a distance 2l in time 2l/u
...

y

According to Newton’s second law of motion, the
rate of change of momentum is equal to the
impressed force
...
10
...
Since there are N
molecules of the gas, the total rate of change of momentum or the total force
exerted on the wall ABCD due to the impact of all the N molecules moving along
x-axis with speeds, u1, u2,
...
+ u N2 )
l

We know that pressure is force per unit area
...
+ u N2
(
l
P =
l2

)

m
( u 12 + u 22 +
...
1)

–
If u 2 represents the mean value of the squares of all the speed components along
x-axis, we can write
2
2
2
2
–
u 2 = u 1 + u 2 + u 3 +
...
+ u N2

Substituting this result in Eqn
...
1), we get

Nmu 2
l3

P =

(10
...
This relation
also holds for the mean square values, i
...

–
–2
c–2 = u 2 + v 2 + w

Since the molecular distribution has been assumed to be isotropic, there is no
preferential motion along any one edge of the cube
...
(10
...
Hence,
we get
1
1
Nm c–2 = M c–2
(10
...
e
...
e
...


PV =

Notes

Eqn (10
...
4)

If we denote the ratio N/V by number density n, Eqn
...
3) can also be expressed
as
1
3

P = m n c2

(10
...
(10
...
Instead of a cube
we could have taken any other container
...


(ii) We ignored the intermolecular collisions but these would not have affected
the result, because, the average momentum of the molecules on striking the
walls is unchanged by their collision; same is the cose when they collide
with each other
...

This is illustrated by the following example
...

Then their mean speed is
1+ 2 + 3 + 4 + 5
= 3 units
5
Its square is 9 (nine)
...

Example 10
...

Solution : Change in momentum 2m u = 2 × (5 × 10–26 kg) × (500 m s–1)
= 5 × 10–23 kg m s–1
...
Hence
2 × 10 –2 m
Time = 500 ms –1 = 4 × 10–4 s

∴ Rate of change of momentum =

5 × 10−23 kg ms −1
= 1
...
25 × 10–19 × 1022 = 416
...
2 × 10–4 N m–2

INTEXT QUESTIONS 10
...
(i) A gas fills a container of any size but a liquid does not
...
Why?
2
...
How is pressure related to density of molecules?
286

PHYSICS

MODULE - 3

Kinetic Theory of Gases

Thermal Physics

4
...
Define coefficient of cubical expansion
...
A steel wire has a length of 2 m at 20°C
...
01 m at 120°C
...


10
...
(10
...
3 J mol–1 K–1
...
It denotes the number of atoms or molecules
n
in one mole of a substance
...
023×1023 per gram mole
...
Therefore, we can write
2

1 –
3 ⎛ R ⎞
3
m c2 = ⎜ N ⎟ T =
kT
2
2 ⎝ A⎠
2
R
k = N
A

where

(10
...
7)

is Boltzmamn constant
...
38 × 10–23 J K–1
...
8)

287

MODULE - 3

Kinetic Theory of Gases

Thermal Physics

Hence, kinetic energy of a gram mole of a gas is

3
RT
2

This relationship tells us that the kinetic energy of a molecule depends only on
the absolute temperature T of the gas and it is quite independent of its mass
...

Notes

Clearly, at T = 0, the gas has no kinetic energy
...
According to modern concepts, the energy of the system of
electrons is not zero even at the absolute zero
...

From Eqn
...
5), we can write the expression for the square root of c–2 , called
root mean square speed :

3kT
3RT
=
m
M
This expression shows that at any temperature T, the crms is inversely proportional
to the square root of molar mass
...
For example, the molar mass of oxygen is 16
times the molar mass of hydrogen
...
It is for this reason
that lighter gases are in the above part of our atmosphere
...

crms =

c2 =

10
...
(11
...
Thus, both M and c–2 on the right hand side of Eqn
...
3) are
constant
...
9)

This is Boyle’s law, which states that at constant temperature, the pressure of a
given mass of a gas is inversely proportional to the volume of the gas
...
(10
...
e, V ∝ c–2 , if M and P do not vary or M and P are constant
...
10)
This is Charle’s law : The volume of a given mass of a gas at constant pressure
is directly proportional to temperature
...

Using a vacuum pump designed by Robert Hook, he
demonstrated that sound does not travel in vacuum
...

A founding fellow of Royal Society of London, Robert Boyle remained a
bachalor throughout his life to pursue his scientific interests
...

(iii) Gay Lussac’s Law – According to kinetic theory of gases, for an ideal gas
1 M –
c2
3 V
For a given mass (M constant) and at constant volume (V constant),
P ∝ c–2

P =

But
∴

c–2 ∝ T

P∝T

(10
...
It states that the pressure of a given mass of a gas is
directly proportional to its absolute temperature T, if its volume remains
constant
...
Then from Eqn
...
3), we recall that
P1 V1 =

PHYSICS

1
–
m1 N1 c12
3

289

MODULE - 3

Kinetic Theory of Gases

Thermal Physics

and

P2 V2 =

1
–
m2 N2 c22
3

If their pressure and volume are the same, we can write
P1V2 = P2V2
Notes

Hence

1
1
–
–
m1 N1 c12 = m2 N2 c22
3
3

Since the temperature is constant, their kinetic energies will be the same, i
...

1
1
–
–
m1 c12 = m2 c22
2
2

Using this result in the above expression, we get N1 = N2
...
12)

That is, equal volume of ideal gases under the same conditions of temperature
and pressure contain equal number of molecules
...

(v) Dalton’s Law of Partial Pressure
Suppose we have a number of gases or vapours, which do not react chemically
...
and mean square speeds c12 , c22 , c32
...

We put these gases in the same enclosure
...

Then the resultant pressure P will be given by
P =

1 –2
1 –
1 –
ρ1 c1 + ρ2 c22 + ρ3 c32 +
...
signify individual (or partial) pressures of different
3 1 1 3 2 2 3 3 3
gases or vapours
...


(10
...
This is Dalton’s law of partial pressures
...
This is known as Graham’s law
of diffusion
...
From Eqn
...
4) we recall that

Thermal Physics

3P
c–2 = ρ
—

or

c 2 = crms =

3P
ρ

Notes

That is, the root mean square velocities of the molecules of two gases of densities
ρ 1 and ρ 2 respectively at a pressure P are given by
(crms)1 =

3P
ρ1

and

(crms)2=

3P
ρ
2

Thus,

ρ2
ρ1

Rate of diffusion of one gas
(crms )1
=
Rate of diffusion of other gas
(crms ) 2 =

(10
...

Example 10
...
Take m(H2) as 3
...
38 × 10–23 J mol–1 K–1
Solution : We know that

crms

=

3kT
=
m

3 × (1
...
347 × 10 –27 kg

= 1927 m s–1
Example 10
...
T
...
, pressure being constant (STP = Standard
temperature and pressure)
...
(10
...
T
...
be c0

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