## Wednesday, August 20, 2008

### Kinetic theory of gases - July-Dec 2008 revision

Kinetc theory of gases is also called kinetic molecular theory of gases.

The model takes into account molecular concept and the kinetic concept of gas molecules.

The theory was put forward by Bernoulli and was further developed and extended by Clausius, Maxwell, Boltzmann and others.

Postulates of the theory

1. All gases are made up of large number of minute particles called molecules.

2. The molecules are separated from one another by large distances.

3. The molecules are in a state of ceaseless and random motion in all directions. They keep colliding with other molecules and walls of the container and change their directions.

4. Molecular collissions are perfectly elastic (See physics for concept of elastic collisions)

5. There are no forces of interaction (attrative or repulsive) between molecules.

6. The pressure exerted by the gas is due to the collisions of the its molecules on the walls of the container per unit area.

7. Teh average kinetc engery of the gas molecules is directly proportional to the absolute temperature.

Kinetic gas equation

From the postulates of kinetic molecular theory, an equation was derived for the pressure of the gas. This equation is known as kinetic gas equation and is

pV = 1/3 mNu²

where

m = mass of a molecule
N = the number of molecules in the volume V
u = root mean square velocity of the molecules. u² is the mean square velocity of molecules. The velociytof each molecule is first squared and then its average is taken.

Averge Kinetic Energy of Molecules of a gas

The average translational kinetic energy of a molecule is

1/2 mu²

where

m = mass of a molecule

u = root mean square velocity of the molecules. u² is the mean square velocity of molecules. The velociytof each molecule is first squared and then its average is taken.

The total kinetic energy of the whole gas is

Ek = 1/2 mNu²

From the formulas for pV and Ek we can get

pV = 2/3 Ek

If we take one mole of gas and define Ek as total kinetic energy of one mole of gas

pV = RT = 2/3 Ek

=> Ek = 3/2RT for one mole of gas

For n moles of gas
Ek = 3/2 nRT.

If we want averge kinetic energy of one molecule, we divide Ek of one mole by Avogadros' number NA(6.022*1023).

averge kinetic energy of one molecule = (3/2) RT/NA = 3/2kbT

Where kb = R/NA is called Boltzmann constant.

Thus Ek α T

As Kinetic energy is proportional to u²

u² α T

u α √T

Molecular velocity of any gas is directly proportional to the square root of the absolute temperature.

This molecular motion is also referred to as thermal motion of the molecules. It will be zero when T = 0.

Past IIT JEE Questions

1. Helium atom is two times heavier than a hydrogen molecule. At 298 K, the average kinetic energy of a helium atom is

a. same as that of a hydrogen molecule
b. two times that of a hydrogen molecule
c. four times that of a hydrogen molecule
d. half that of a hydrogen molecule
(JEE 1982)

2. State whether the statement is True or False.

Kinetic energy of molecules is zero at 0°C. (JEE 1985)

Ans: False

The equation relating energy of molecules to temperature is in absolute temperature and not centigrade temperature.

3. the average velocity of an ideal gas molecule at 27°C is 0.3 m/s. the average speed at 927°C will be:

a. 0.6 m/s
b. 0.3 m/s
c. 0.9 m/s
d. 3.0 m/s
(JEE 1986)

4. Eight grams of oxygen and hydrogen at 27°C will have the total kinetic energy in the ratio of __________________.
(JEE 1989)

Reason: For a mixture of gases, in thermal equilibrium, average kinetic energy of all molecules is same.

(½)*m1* v1² = (1/2)*(m2*v2²

Oxygen’s molecular weight is 32 and hence 8 grams will have 8/32 = ¼ moles.
Hydrogen’s molecular weight is 2 and hence 8 grams will have 8/2 = 4 mols

Total kinetic energy of oxygen moleculues = (¼ )* (½)*m1* v1²
Total kinetic energy of hydrogen molecules = 4*(1/2)*(m2*v2²

The ratio will be (1/4)/4 = 1/16 as remaining terms or equal in both cases.