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Kinetic Theory of an Ideal Gas

The behaviour of gases can be studied in terms of the macroscopic variables as pressure, volume and temperature. But the same properties can be expressed in in terms of the microscopic variables like root mean square velocity and the average kinetic energy of the gas molecules. Kinetic theory of gases deals with the behaviour of gases in terms of molecular (atomic) motion.

Kinetic theory was developed by Robert Boyle, Daniel Bernoulli, James Joule, Rudolph Clausius, Clerk Maxwell and others. Bernoulli is considered as the father of the kinetic theory, because, for the first time he explained Boyle’s law using molecular motion.
When the kinetic theory of gases was developed there was no direct experimental evidence. But at present, we have a number of evidences in favour of this theory. They include the Brownian motion, diffusion, evaporation of a liquid etc.

Mean Speed and r.m.s Speed

Molecules of a gas move in all directions with all possible speeds. Let v1,v2,v3 … be the speeds of the different molecules of the gas. The mean or average speed is given by
V =
Where N is the total number of molecules and VN is the speed of the Nth gas molecule.

The root mean square speed of a gas molecule is the square root of the mean of the squares of the speeds of the individual molecules. The r.m.s. speed is given by.
Vrms = =
It can be shown that, the average speed = 1.6 and the root mean square speed
Vrms = 1.7, where k is the Boltzmann’s constant, T is the temperature of the gas and m mass of each molecule of the gas.

Derivation of Expression for the Pressure Exerted by a Gas using Kinetic Theory of Gases
A number of assumptions are made to develop the kinetic theory of gases, regarding the nature of gas molecules and the motion of the molecules Clausius made the following postulates.

Postulates of Kinetic Theory

  1. The molecules of a gas are considered to be rigid, elastic solid spheres, identical in all respects.
  2. They are considered to be point-masses and of negligible size, as compared with their distance apart.
  3. They are in a state of continuous random motion, moving in all directions, with all possible velocities.
  4. The molecules in motion collide with one another and with the walls of the containing vessel. During collision, their velocities are changed in magnitude and direction.
  5. The collision are perfectly elastic and there are no forces of attraction or repulsion between the molecules, which means the energy of the gas is all kinetic and is proportional to the absolute temperature.
  6. The time spent in a collision is negligible, as compared, with that during which the molecules are moving independently.
  7. Between collisions the molecules move in a straight line with uniform velocity. The distance between two collisions is called the free path of the molecule. The average distance traveled by a molecule between successive collisions is called the mean free path.

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