Relation between pressure (P) and energy (E) for a unit volume of gas :

CONCEPT:

The pressure exerted by a gas:

• According to kinetic theory, the molecules of a gas are in a state of continuous random motion.
  • They collide with one another and also with the walls of the vessel.
  • Whenever a molecule collides with the wall, it returns with a changed momentum, and an equal momentum is transferred to the wall (conservation of momentum).
• According to Newton's second law of motion, the rate of transfer of momentum to the wall is equal to the force exerted on the wall.
  • Since a large number of molecules collide with the wall, a steady force is exerted on the wall.
  • The force exerted per unit area of the wall is the pressure of the gas.
  • Hence a gas exerts pressure due to the continuous collisions of its molecules with the walls of the vessel.

EXPLANATION:

• From kinetic theory of gases, the pressure P exerted by an ideal gas of density ρ and rms velocity of its gas molecules C is given by

• Mean kinetic energy of translation per unit volume of the gas is

On dividing equations (1) and (2), we get

The above equation can be written as