The equation which establishes the relation between kinetic energy per unit volume and gas pressure is derived based on the Kinetic Theory of Gases.

Kinetic energy is denoted by the following equation:

K.E = ½.mv2

Where, K.E = kinetic energy of gas molecules

m = mass of gas molecules

= root mean square speed of gas molecules

Since density = mass/volume

mass = density x volume

Mass = v

Putting the value of mass (m) in the equation of kinetic energy we get:

K.E = ½.Vv2

This is the kinetic energy of gas.

Kinetic energy of gas per unit volume = K.E / volume (v)

Kinetic energy of gas per unit volume = ½.v2      ………….. (i)

According to the Kinetic Theory of Gases, gas pressure is represented by the following equation:

  P =  ⅓ .v2         ………….. (ii)

Where, P = gas pressure

= density of the gas molecules

= root mean square speed of gas molecules

The relation between kinetic energy per unit volume and gas pressure is obtained by dividing equation (i) by equation (ii):

 K.E/P  = ½ .2/ ⅓ .2

K.E/P = 3/2

 K.E = 3/2 x p

Hence, the kinetic energy of gas per unit volume is related to pressure of gas by the following equation:

 K.E = 3/2.P

(where, K.E = kinetic energy of gas per unit volume; P = pressure of gas)