Derive an expression for the work done in adiabatic process.

Consider a unit mole of gas contained in a perfectly non-conducting cylinder provided with a non-conducting and frictionless piston.
Let C_v be the specific heat of gas at constant volume.
Let at any instant, when the pressure of gas is P, the gas be compressed by small volume dV.
Then work done on the gas is, 
                          dW = PdV 
Total work done on gas to compress from volume V₁ to V₂ is given by 
                        ...(1) 
According to first law of thermodynamics, 
               
For adiabatic process,   
∴           PdV = -dU =  
∴    
where,
T₁ is the temperature of gas when volume is V₁ and T₂ when volume is V₂. 
Thus, work done is given by, 
      
        
         
Therefore,
 
The above expression gives us the amount of work done in adiabatic process.