Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

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
...
4 : Calculate the average kinetic energy of a gas at 300 K
...
38 × 10–23 JK–1
...
38 × 10–23 J K–1 and T= 300 K, we get
∴

E

=

3
(1
...
21 × 10–21 J

INTEXT QUESTIONS 10
...
Five gas molecules chosen at random are found to have speeds 500 m s–1,
600 m s–1, 700 m s–1, 800 m s–1, and 900 m s–1
...

2
...
When we blow air in a balloon, its volume increases and the pressure inside
is also more than when air was not blown in
...
4
...

Suppose you are driving along a road and several other roads are emanating
from it towards left and right
...
Now, say the

292

PHYSICS

MODULE - 3

Kinetic Theory of Gases

road has a flyover at some point and you take the flyover route
...
You can move only along the flyover and we say that your degree
of freedom is ‘1’
...
10
...
A string is tied in a taut manner from one end A to other
end B between two opposite walls of a room
...
Then
its degree of freedom is ‘1’
...
10
...
Now, it can move along x or
y direction independently
...
And if the ant
has wings so that it can fly
...

A monatomic molecule is a single point in space and like the winged ant in the
above example has 3 degrees of freedom which are all translational
...
Hence a diatomic
molecule has (3 + 2 = 5) degrees of freedom: three translational and two
rotational
...
5 THE LAW OF EQUIPARTITION OF ENERGY
We now know that kinetic energy of a molecule of a gas is given by

1 —2 3
m c = kT
...
e
...
That is to say, for a molecule all the three directions
are equivalent :
u = v = w

and
Since

1 –
–
–
–
u 2 = v2 = w 2 = c 2
3

c2 = u2 + –v 2 + w2
–2
c–2 = u–2 + –v 2 + w

PHYSICS

293

MODULE - 3

Kinetic Theory of Gases

Thermal Physics

Multiplying throughout by

1
m, where m is the mass of a molecule, we have
2

1 –
1
1 –
2
m u 2 = m –v 2 = m w
2
2
2

1 –
m u 2 = E = total mean kinetic energy of a molecule along x–axis
...
But the total mean kinetic energy of a molecule is k T
...
This is the law of equipartition of energy and was
deduced by Ludwing Boltzmann
...


system is equally divided among all its degrees of freedom and it is equal to

So far we have been considering only translational motion
...
Hence, for one molecule of a monoatomic gas,
total energy
3
kT
(10
...

Such a molecule can rotate about any one of the three mutually perpendicular
axes
...
It means that rotational
1
1
energy consists of two terms such as I ω2y and I ω2z
...
Thus, for a diatomic gas molecule, both rotational
and translational motion are present but it has 5 degrees of freedom
...
16)

PHYSICS

MODULE - 3

Kinetic Theory of Gases

Thermal Physics

Ludwing Boltzmann
(1844 – 1906)
Born and brought up in Vienna (Austria), Boltzmann completed
his doctorate under the supervision of Josef Stefan in 1866
...
A very
emotional person, he tried to commit suicide twice in his life
and succeeded in his second attempt
...


Notes

He is famous for his contributions to kinetic theory of gases, statistical
mechanics and thermodynamics
...


10
...
For example, the volume or the pressure may be kept
constant or both may be allowed to vary in some arbitrary manner
...
Hence, we say that a gas has two different
heat capacities
...
Thus

Heat capacity =

Specific heat capacity, c =

heat capacity
m

(10
...
(10
...
17) may be combined to get
ΔQ
c = m ΔT

(10
...

The SI unit of specific heat capacity is kilo calories per kilogram per kelvin (kcal
kg–1K–1)
...
For example the specific
heat capacity of water is
1 kilo cal kg–1 K–1 = 4
...

PHYSICS

295

MODULE - 3
Thermal Physics

Kinetic Theory of Gases

The above definition of specific heat capacity holds good for solids and liquids
but not for gases, because it can vary with external conditions
...

Consequently, we define two specific heat capacities :
(i)

Notes

Specific heat at constant volume, denoted as cV
...

(a) The specific heat capacity of a gas at constant volume (cv) is defined as
the amount of heat required to raise the temperature of unit mass of a gas
through 1K, when its volume is kept constant :

⎛ ΔQ ⎞
⎟
cv = ⎜
⎝ ΔT ⎠V

(10
...

⎛ ΔQ ⎞
cp = ⎜
⎟
⎝ ΔT ⎠ P

(10
...

We know that when pressure is kept constant, the volume of the gas increases
...

This means the specific heat capacity of a gas at constant pressure is greater than
its specific heat capacity at constant volume by an amount which is thermal
equivalent of the work done in expending the gas against external pressure
...
21)

10
...
10
...
Since the gas has been assumed to be
ideal (perfect), there is no intermolecular force between its molecules
...


296

PHYSICS

MODULE - 3

Kinetic Theory of Gases

Thermal Physics

A
P
V1

V2

Notes

Fig
...
5 : Gas heated at constant pressure

Let P be the external pressure and A be the cross sectional area of the piston
...
Now suppose that the gas is heated at constant
pressure by 1K and as a result, the piston moves outward through a distance x, as
shown in Fig
...
5
...
Therefore, the work W done by the gas in pushing the piston through
a distance x, against external pressure P is given by
W =P×A×x
= P × (Increase in volume)
= P (V2 – V1)
We know from Eqn
...
22) that cp – cv = Work done (W) against the external
pressure in raising the temperature of 1 mol of a gas through 1 K, i
...

cp – cv = P (V2 – V1)

(10
...
e
...
23)

PV2 = R (T + 1)

(10
...
(10
...
(10
...
25)

Hence from Eqns
...
19) and (10
...
26)

where R is in J mol–1 K–1
Converting joules into calories, we can write
c p – cv =

R
J

(10
...
18 cal is the mechanical equivalent of heat
...
5 : Calculate the value of cp and cv for a monoatomic, diatomic and
triatomic gas molecules
...
e
...

(i) We know that for monoatomic gas, total energy =
∴
Hence

monoatomic gas cV =

3
3
3
R (T + 1) – R T = R
...

2
2

5
RT
2

5
5
5
R (T + 1) – R R T = R
2
2
2

5
7
R + R = R
...


cp = cV + R =

INTEXT QUESTIONS 10
...
What is the total energy of a nitrogen molecule?
2
...
3J mol–1 K–1)
...
Why do gases have two types of specific heat capacities?
Brownian Motion and Mean Free Path
Scottish botanist Robert Brown, while observing the pollen grains of a flower
suspended in water, under his microscope, found that the pollen grains were
tumbling and tossing and moving about in a zigzag random fashion
...
But when motion
of pollens of dead plants and particles of mica and stone were seen to exhibit
298

PHYSICS

MODULE - 3

Kinetic Theory of Gases

Thermal Physics

the same behaviour, it became clear that the motion of the particles, now
called Brownian motion, was caused by unbalanced forces due to impacts of
water molecules
...
The Brownian displacement was found to depend on
...

Notes
(ii) The Brownian motion also increases with the increase in the temperature
and decreases with the viscosity of the medium
...

The average distance between two successive collisions of the molecules is
called mean free path
...


WHAT YOU HAVE LEARNT
z

The specific heat of a substance is defined as the amount of heat required to
raise the temperature of its unit mass through 1°C or 1 K
...


z

Kinetic theory relates macroscopic properties to microscopic properties of
individual molecules
...


z

Kinetic energy of a molecule depends on the absolute temperature T and is
independent of its mass
...


z

Gas law can be derived on the basis of kinetic theory
...


z

Depending on whether the volume or the pressure is kept constant, the amount
of heat required to raise the temperature of unit mass of a gas by 1ºC is
different
...


cp – cV =

Notes

z

z

The law of equipartition of the energy states that the total kinetic energy of
a dynamical system is distributed equally among all its degrees of freedom
and it is equal to

z

1
k T per degree of freedom
...

2

TERMINAL EXERCISE
1
...
What will be the velocity and kinetic energy of the molecules of a substance
at absolute zero temperature?
3
...
What should be the ratio of the average velocities of hydrogen molecules
(molecular mass = 2) and that of oxygen molecules (molecular mass = 32) in
a mixture of two gases to have the same kinetic energy per molecule?
5
...
5, 1 and 2 km s–1 respectively, calculate
the ratio between their root mean square and average speeds
...
Explain what is meant by the root-mean square velocity of the molecules of
a gas
...

7
...

ii) At what temperature does the average energy have half this value?
300

PHYSICS

Kinetic Theory of Gases

8
...
0 Pa
and at a temperature of 27 0C
...


MODULE - 3
Thermal Physics

( R= 8
...
Mass of 1 mole of hydrogen
molecule = 20 × 10–3 kg mol–1)
...
A closed container contains hydrogen which exerts pressure of 20
...


Notes

(a) At what temperature will it exert pressure of 180 mm Hg?
(b) If the root-mean square velocity of the hydrogen molecules at 10
...
State the assumptions of kinetic theory of gases
...
Find an expression for the pressure of a gas
...
Deduce Boyle’s law and Charle’s law from kinetic the theory of gases
...
What is the interpretation of temperature on the basis of kinetic theory of
gases?
...
What is Avagardo’s law? How can it be deduced from kinetic theory of gases
15
...
09
kg m–3)
...
Calculate the pressure in mm of mercury exerted by hydrogen gas if the
number of molecules per m3 is 6
...
90 × 10 m s–1
...
02 × 1023 and molecular
weight of hydrogen = 2
...

17
...
Derive the relationship
between cp and cV
...
Define specific heat of gases at constant volume
...
Calculate cP and cV for argon
...
3 J mol–1 K–1
...
1
1
...

(ii) Because the molecules in a solid are closely packed
...

PHYSICS

301

MODULE - 3
Thermal Physics

Kinetic Theory of Gases

2
...

3
...
The specific heat of a substance is the amount of heat required to raise the
temperature of its unit mass through 1°C or 1K
...
The coefficient of cubical expansion is defined as the change in volume per
unit original volume per degree rise in temperature
...
0
...
2
1
...
The resultant pressure of the mixture will be the sum of the pressure of gases
1 and 2 respectively i
...
P = P1 + P2
...
Boyle’s law is not applicable
...
3
1
...

2

PHYSICS

MODULE - 3

Kinetic Theory of Gases

Thermal Physics

2
...
3 J mol–1 K–1 = 20
...


cp = cV + R = 29
...

Notes
Answers to Terminal Problem
2
...
becomes 4 times, doubles, becomes 4 time
...
4 : 1
5
...
6
...
12 × 1020, 7
...
2634ºC, 2560 m s–1
15
...
3
...
12
...
75 J mol–1 K–1

---