# Kinetic Gas Equation of an Ideal Gas

The kinetic gas equation is given as:

PV = 1/3 mnc2

P = pressure

V = volume

M = mass of molecule

n = no. of molecules present in the given amount of gas

c = root mean square speed.

Kinetic energy for one mole of gas is given as c = c1 + c2 + … + cn /n

The average kinetic energy per molecule is = average K.E per mole / NA

$$\overline{K.E}$$= 3/2 RT/NA = 3/2 kT

Where k = R/NA and is known as the Boltzmann constant.

The total kinetic energy for 1 mol of the gas is

Etotal = NA.$$\overline{K.E}$$= 3/2 RT

Deduction of gas laws from kinetic gas equation:

The ideal gas laws can be derived from the kinetic theory of gases which is based on the following two important assumptions:

1. The volume occupied by the molecules is negligible in comparison to the total volume of the gas
2. The molecules exert no forces of attraction upon one another.

1. Deriving Boyle’s law:

From kinetic theory

PV = 1/3 mnc2

At constant temperature and at fixed amount of gas, c is constant.

Therefore, PV = constant … Boyle’s law

2. Deriving Charle’s law:

PV = 1/3 mnc2

At constant pressure and amount,

V ∝ c2

∵ c2 ∝ T

∴ V ∝ T … Charle’s law

3. Deriving Gay Lussac’s law:

PV = 1/3 mnc²

At constant volume and amount of gas

P ∝ c²

∵ c² ∝ T

∴ P ∝ T

Deriving Avogadro’s Maxwell showed that the average kinetic energies of molecules are equal at the same temperature, that is:

½ [m1c12] = kT = ½ [m2c22] and so [m1c12] = [m2c22]

But P1V1 = 1/3 [m1n1c12] and P2V2 = 1/3 [m2n2c22]

Now if P1 = P2 and V1 = V2, [m1n1c12] = [m2n2c22]