KINETIC THEORY OF GASES

1. Kinetic Theory of Gases
Physical properties of gases
1. Compressible
2. Viscous
3. No fixed volume
4. Lesser density

2. Kinetic Theory of Gases
Boyles law
At contant temperature,volume of a gas is inversely proportional to pressure.

3. Kinetic Theory of Gases
Charles law
At constant pressure volume of a gas directly proportional to temperature.

4. Kinetic Theory of Gases
Concept of perfect gas
1. Always obeys Boyles and Charles law
2. Pressure Coefficient = Volume coefficient
3. Molecules are infinitesimally small
4. No force of attraction between molecules

5. Kinetic Theory of Gases
Equation of a Perfect gas

6. Kinetic Theory of Gases
Concept of pressure or law of pressure
At constant volume , pressure of a gas is directly proportional to the temperature.

7. Kinetic Theory of Gases
Work done in compressing a gas

8. Kinetic Theory of Gases
Assumptions of kinetic theory
1. Molecules of a gas are rigid and very small.
2. There is no force of attraction among molecules they moves in linear path with a constant velocity.
3. Collision between two molecules is completely elastic ie kinetic energy is conserved.
4. Time taken in collision of two molecules is negligible as compare to time taken to cover distance by
molecules between two consecutive collisions.
5. Since molecules moves in a vessel in possible direction therefore molecules are equally distributed
throughout the vessel.
6. Since velocity of molecules are high and mass is small that’s why there is no effect of gravitation on its
motion.

9. Kinetic Theory of Gases
Mean free path
Relaxation time
Avogadros number
Absolute zero temperature

10. Kinetic Theory of Gases
Kinetic Interpretation of Temperature
The temperature of a body is the measure of the average kinetic energy of a body.
the temperature of a body always depends upon its average kinetic energy and since
the average kinetic energy can have a minimum possible value of zero, therefore an
object cannot be cooled below a certain minimum value, this value is known as
absolute zero. This is the lowest possible temperature in our universe and no object
could be cooled to this temperature, it is equivalent to -273 degree Celsius or 0 Kelvin.

11. Kinetic Theory of Gases
RMS Speed of Gas Molecules
Average velocity of gas molecules

12. Kinetic Theory of Gases
Degree of Freedom
The number of independent ways in which a molecule of gas can move is called the degree of freedom.
Types of degree of freedom
(a) Translational degree of freedom
the ability of gas molecules to move freely in space.
A molecule may move in the x, y, and z directions of a Cartesian coordinate system.

13. Kinetic Theory of Gases

14. Kinetic Theory of Gases
(b) Rotational degree of freedom
the number of unique ways the molecule may rotate in space about its center of mass
with a change in the molecule’s orientation.
A monatomic gaseous molecule such as a noble gas possesses no rotational degrees of
freedom, as the center of mass sits directly on the atom and no rotation which creates
change is possible.

15. Kinetic Theory of Gases
diatomic molecule lying along the Y-axis can undergo rotation about the
mutually perpendicular X-axis and Z-axis passing through its centre of gravity,
This shows that the linear molecule has two rotational degrees of freedom.

16. Kinetic Theory of Gases
non-linear molecules have three rotational degrees of freedom.

17. Kinetic Theory of Gases
(c) Vibrational degree of freedom
the number of unique ways the atoms within the molecule may move relative to
one another, such as in bond stretches or bends.
As already mentioned, monoatomic atoms possess only a translational degree of freedom.
A diatomic molecule has only one vibrational degree of freedom During the vibrational motion the
bonds of the molecules behave like a spring and the molecule exhibits simple harmonic motion.
A polyatomic molecule containing N atoms has 3N degrees of freedom.

18. Kinetic Theory of Gases
Degree of freedom of monoatomic gas
Since a monatomic molecule consists of only a single atom of point mass it has three
degrees of freedom of translatory motion along the three coordinate axes x, y and z.
Examples: Molecules of Inert gases like helium(He), Neon(Ne), Argon(Ar), etc.

19. Kinetic Theory of Gases
Degree of freedom of diatomic molecule
The diatomic molecule can rotate about any axis at right angles to its own axis. Hence it has
two rotational degrees of freedom, in addition, it has three translational degrees of freedom
along the three axes.
A diatomic molecule shows one vibrational degree of freedom. So, a diatomic molecule has a
total of six degrees of freedom at high temperatures.
At room temperature the total degree of freedom of a diatomic molecule is five because
vibrational motion is not contributed. Examples: molecules of O2, N2, CO, Cl2, etc.

20. Kinetic Theory of Gases
Degree of freedom of triatomic molecule
In the case of a triatomic molecule of linear type, the centre of mass lies at the
central atom.
It, therefore, behaves like a diatomic molecule with three degrees of freedom of
translation and two degrees of freedom of rotation, it has five degrees of
freedom as shown at room temperature.
At high temperatures, It shows four vibrational degrees of freedom. Hence, it
shows a total of nine degrees of freedom. Examples: molecules of CO2, CS2, etc.
At room temperature a triatomic nonlinear molecule possesses three degrees of
freedom of rotation in addition to three degrees of freedom of translation.
Hence it has six degrees of freedom.
At high temperatures, it shows a total of nine degrees of freedom. Examples :
molecules of H2O, SO2, etc.

21. Kinetic Theory of Gases
A polyatomic molecule containing N atoms has 3N degrees of freedom.
If we subtract the translational and rotational degree of freedom from the total degree of
freedom then find the total number of vibrational degrees of freedom of linear and
nonlinear molecules.
Degree of freedom Monatomic diatomic molecules polyatomic molecules
Translational 3 3 3
Rotational 0 2 3
Vibrational 0 3N – 5 3N – 6
Total 3 3N 3N

22. Kinetic Theory of Gases
Equipartition law of energy
For a system in equilibrium, there is an average energy of ½ kT or ½ RT per molecule associated with each
degree of freedom. (where k = Boltzmann constant and T is the temperature of the system).
This energy associated with each degree of freedom is in the form of kinetic energy and potential energy.
One translational degree of freedom = ½ kT or ½ RT
One rotational degree of freedom= ½ kT or ½ RT
One vibrational degree of freedom= kT or RT
Note: As regards the vibrational motion, two atoms oscillate against each other therefore both potential
and kinetic energy the energy of vibration involve two degrees of freedom, so that vibrational motion in a
molecule is associated with energy= 2 x ½ kT = kT
Total energy E= Etr + Erot + Evib + Eelc

23. Kinetic Theory of Gases
At room temperatures, the degrees of freedom need not include the vibrational modes.
For molecules to vibrate in their normal modes they require much higher energies which
is not possible at room temperature.
Energy contribution for linear molecules
Degree of freedom Translational Rotational Vibrational
Linear molecule 3 2 3N-5
Energy contribution At room temperature 3 x ½ kT 2 x ½ kT Inactive (no contribution)
Energy contribution At high temperature 3 x ½ kT 2 x ½ kT (3N-5) x kT

24. Kinetic Theory of Gases
Energy contribution for non-linear molecules
Degree of freedom Translational Rotational Vibrational
non-linear molecule 3 3 3N-6
Energy contribution At room temperature 3 x ½ kT 3 x ½ kT Inactive (no contribution)
Energy contribution At high temperature 3 x ½ kT 3 x ½ kT (3N-6) x kT

25. Kinetic Theory of Gases
Equipartition law of energy is used to calculate the value of CP − CV and the ratio
between them γ = CP / CV.
Here γ is called adiabatic exponent.
i) Monatomic molecule
Average kinetic energy of a molecule
Applications to specific heat capacities

26. Kinetic Theory of Gases
For one mole, the molar specific heat at constant volume

27. Kinetic Theory of Gases
ii) Diatomic molecule
Average kinetic energy of a diatomic molecule at low
temperature = 5/2kT
Total energy of one mole of gas
For one mole Specific heat at constant volume

28. Kinetic Theory of Gases
Energy of a diatomic molecule at high temperature is equal to 7/2RT
Note that the CV and CP are higher for diatomic molecules than the mono atomic molecules.
It implies that to increase the temperature of diatomic gas molecules by 1°C it require more
heat energy than monoatomic molecules.

29. Kinetic Theory of Gases
iii) Triatomic molecule
a) Linear molecule

30. Kinetic Theory of Gases
b) Non-linear molecule
Note that according to kinetic theory model of gases the specific heat capacity at constant
volume and constant pressure are independent of temperature. But in reality it is not sure.
The specific heat capacity varies with the temperature.

31. Kinetic Theory of Gases