The kinetic gas equation is given as:

PV = 1/3 mnc^{2}

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 = c_{1} + c_{2} + … + c_{n} /n

The average kinetic energy per molecule is = average K.E per mole / N_{A}

\(\overline{K.E}\)= 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}.\(\overline{K.E}\)= 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:

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

**1. Deriving Boyle’s law:**

From kinetic theory

PV = 1/3 mnc^{2}

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 mnc^{2}

At constant pressure and amount,

V ∝ c^{2}

∵ c^{2} ∝ 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 [m_{1}c_{1}^{2}] = kT = ½ [m_{2}c_{2}^{2}] and so [m_{1}c_{1}^{2}] = [m_{2}c_{2}^{2}]

But P_{1}V_{1} = 1/3 [m_{1}n_{1}c_{1}^{2}] and P_{2}V_{2} = 1/3 [m_{2}n_{2}c_{2}^{2}]

Now if P_{1} = P_{2} and V_{1} = V_{2}, [m_{1}n_{1}c_{1}^{2}] = [m_{2}n_{2}c_{2}^{2}]

Therefore: n_{1} = n_{2}** … **Avogadro’s law

**5. Deriving Dalton’s law of partial pressures:**

For a mixture of gases:

PV = 1/3 ([m_{1}n_{1}c_{1}^{2}] + 1/3 [m_{2}n_{2}c_{2}^{2}] + …)

P = 1/3 ([m_{1}n_{1}c_{1}^{2}]/V + 1/3 [m_{2}n_{2}c_{2}^{2}]/V + …) = P_{1} + P_{2} + … Where P_{1}, P_{2} … 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 = P_{1} + P_{2} + …