**Deduction of the Gas Laws from the Kinetic Equation**

All the gas laws may be deduced from equation: **PV = 1/3 mnc ^{2}** … … (1)

This is the kinetic equation for gases. Although the above deduction has been made for a cubical vessel the equation is valid for a vessel of any shape as the vessel can be divided into a large number of small cubes for each of which this equation is valid.

**(a) Boyle’s law:** Temperature is a measure of the kinetic energy of the molecules. The higher the temperature higher is the kinetic energy of the molecules Equation (1) may be written in the form.

**PV = (2/3 x 1/2) mnc ^{2} = 2/3 x total Kinetic Energy **… … (2)

Since the kinetic energy of a moving body is – ½ mc^{2}. For a given quantity of gas, when the temperature is constant, the kinetic energy is constant. Hence,

PV = constant (at constant T) which is Boyle’s law.

**Charles’ or Gay-Lussac’s law:** The total kinetic energy of the molecules is proportional to the absolute temperature. i.e K.E. = const x T (hence from equation 2)

**PV = const x T**

At constant volume for a given quantity of gas P ∞ T. This is Charles’ or Charles’ or Gay-Lussac’s law.

**Avogadro’s law:** Consider that two gases are at the same pressure, P, and contained in vessels of equal volume, V. Both the vessels are at the same temperature. For the first gas, from the kinetic equation (1)

**PV= 1/3 n _{1}m_{1}c_{1}^{2}**

Where n_{1} is the number of molecules, m_{1}, is the mass of each molecule and c_{1} is the r.m.s. velocity. Similarly, for the second gas;

**PV = 1/3 n _{2}m_{2}c_{2}^{2}**

Where n_{2}, m_{2} and c_{2} have the same significance as for the first gas. Since P, V and T of the two gases are same, according to equation (1)

**1/3 n _{1}m_{1}c_{1}^{2} = 1/3 n_{2}m_{2}c_{2}^{2}** … … … (3)

At the same temperature the Kinetic Energy of the molecules of both the gases are the same; that is,

**1/2 m _{1}c_{1}^{2} = 1/2 m_{2}c_{2}^{2}** … … … (4)

If equation (3) is divided by equation (4) One obtains; **n _{1} = n_{2}**

That is, the number of molecules in equal volumes of the two gases under the same conditions of T and P is the same. This is Avogadro’s hypothesis or Avogadro’s law.