July Dec Revision - Kinetic theory of gases

Any sample of gas is made of molecules.

The observed behaviour of gas results from the behaviour of its large number of molecules.

Kinetic theory of gases attempts to develop a model of the molecular behaviour which should result in the observed behaviour an ideal gas.


Assumptions of kinetic theory of gases

1. All gases are made of molecules moving randomly in all directions
2. The size of 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 collision (no atraction force or repulsion force).
4. All collisions between two molecules or between a molecule and a wall are perfectly elastic. Also the time spent during a collision is negligibly small.
5. the molecules obey Newton's laws of motion.
6. When a gas is left for sufficient time in a closed container, it comes to a steady state. The density and the distribution of molecules with different velocities are independent of position, direction and time.

The assumptions are close to the real situations at low densities.
The molecular size is roughly 100 times smaller than the average separation between the molecules at 0.1 atm and room temperature.

The real molecules do exert electric forces on each other but these forces can be neglected as the average separation between molecules is large as compared to their size.


Pressure of an ideal gas
p = (1/3)ρ*Avg(v²) ........ (1)

where
ρ = density of gas = mass per unit area
Avg(v²) = average of the speeds of molecules squared

pV = (1/3)M*Avg(v²) ....... (2)

M = Mass of gas in the closed container

pV = (1/3)nm*Avg(v²) ........ (3)
n = number of molecules of gas in the container
m = mass of each molecule


RMS Speed: The square root of mean square speed is called root-mean-square speed or rms speed.
It is denoted by the symbol v_rms

Avg(v²) = (v_rms)²

The equation (1) can be written as

p = (1/3)ρ*(v_rms)²

Then
v_rms) = √[3p/ρ] = √[3pV/M]


Total translational kinetic energy of all the molecules of the gas is

K = Σ (1/2mv² = (1/2)M(v_rms)² ... (4)

The average kinetic energy of a molecule = (1/2)m(v_rms)²

Then from equation (2)
K = (3/2)pV

according to the kinetic theory of gases, the internal energy of an ideal gas is the same as the total translational kinetic energy of its molecules.

For different kinds of gases, it is not the rms speed but average kinetic energy of individual molecules that has a fixed value at a given temperature.

The heavier molecules move with smaller rms speed and the lighter molecules move with larger rms speed.

All gas laws can be deduced from kinetic theory of gases.

Ideal gas equation

pV = nRT

R = universal gas constant = 8.314 J/mol-L

The average speed of molecules is somewhat less than the rms speed.

Average speed = (Σv)/n = √[8kT/πm]