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 + … + c_n /n 
The average kinetic energy per molecule is = average K.E per mole / N_A 

= 3/2 RT/N_A = 3/2 kT 

Where k = R/N_A and is known as the Boltzmann constant.
The total kinetic energy for 1 mol of the gas is 

E_total = N_A.= 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:
i.            The volume occupied by the molecules is negligible in comparison to the total volume of the gas
ii.            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

4.      Deriving Avogadro’s law: 

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]
Therefore: n₁ = n₂ …… Avogadro’s law

5.      Deriving Dalton's law of partial pressures:

For a mixture of gases:
PV = 1/3 ([m₁n₁c₁²] + 1/3 [m₂n₂c₂²] + …………)
P = 1/3 ([m₁n₁c₁²]/V + 1/3 [m₂n₂c₂²]/V + …………) = P₁ + P₂ + …..Where P₁, P₂ ... are the partial pressures of the gases, and this is Dalton's law (the sum of the partial pressures of all the gases occupying a given volume is equal to the total pressure).
P = P₁ + P₂ +……