Mean Kinetic Energy and Pressure of Gas

Mean Kinetic Energy of Gas
The mean kinetic energy per molecule of a gas is given by the equation
E = (3/2)kT
where E is the mean kinetic energy per molecule, k is Boltzmann's constant, and T is the absolute temperature of the gas.

Pressure of Gas
The pressure of a gas is related to the mean kinetic energy per unit volume of the gas by the equation
P = (2/3)E/V
where P is the pressure, E is the mean kinetic energy per unit volume, and V is the volume of the gas.

Derivation of Pressure-Energy Relation
The pressure of a gas can be derived from the ideal gas law, which relates the pressure, volume, and temperature of a gas:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the absolute temperature of the gas.

The number of moles of gas can be expressed in terms of the mass of the gas and its molecular weight:
n = m/M
where m is the mass of the gas and M is its molecular weight.

Substituting this expression for n into the ideal gas law and rearranging, we get:
P = (m/M)V(RT/m)

The term RT/m is equal to the mean kinetic energy per molecule of the gas, which we can express in terms of the mean kinetic energy per unit volume by dividing by the volume:
E = (3/2)kT/N = (3/2)kT/V * (1/N)
where N is the number of molecules in the gas.

Substituting this expression for E into the equation for pressure, we get:
P = (2/3)E/V

Answer
Therefore, the correct answer is option 'D', i.e., the pressure of a gas in terms of its mean kinetic energy per unit volume E is equal to 2E/3.