With the help of Boyle's and Charle's laws, derive an expression for the ideal gas equation.

Derivation of the ideal gas equation: Let the volume of a given mass of a gas change from V₁ to V₂ when the pressure is changed from Pi to P₂ and temperature is changed from T₁ to T₂. Suppose this change from initial state (P₁ V₁T₁) to final state (P₂V₂T₂) occurs in two steps:
First step: Suppose the volume of the given mass of the gas changes from V₁ to V_x when pressure is changed from P₁ to P₂, keeping temperature T₁ constant.

Second step: Now let the volume changed from V_x to V₂ when the temperature is changed from T₁ to T₂, keeping pressure P₂ constant.
     According to Charle's law,
                      

From (1) and (2), we have
              

The numerical value of the constant K is independent of the nature of the gas but depends on upon the amount of the gas. But at constant temperature and pressure the volume of a gas is proportional to the number of moles (n), this means that K is directly proportional to the number of moles (n).

where R is a constant of proportionality which is independent of nature as well as the amount of the gas and is known as a universal gas constant. From (4) and (5)
  
Equation (6) is known as an ideal gas equation. Alternative derivation of ideal gas equation
: According to Boyle’s law,
                
According to Charle's law,
 
 
According to Avogadro's law,
 
 
Combining (1), (2) and (3), we have,
             
.