Question

Derive relation between kinetic energy and pressure

Answer 1

The average kinetic energy of a gas particle is directly proportional to the temperature. An increase in temperature increases the speed in which the gas molecules move. All gases at a given temperature have the same average kinetic energy.
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Answer 2

Pressure, a macroscopic property, can be related to the average (translational) kinetic energy per molecule which is a microscopic property by P=nm¯¯¯¯¯v23 P = nm v 2 ¯ 3 . Since the assumption is that the particles move in random directions, the average value of velocity squared along each direction must be same.