Obtain an expression for the work done by an ideal gas during adiabatic change and explain .

Explanation

Work done during Adiabatic process : Consider n mole of perfect gas contained in a cylinder having insulating walls . When piston moves through a small distance dx , then small work (dW) will be done .

`thereforedW=(Pa)dx=PdV`,
where a is area of cross - section of the piston . Therefore , when the system goes from initial state A `(P_(1),V_(1))` to the final state `B(P_(2),V_(2))` the amount of work done ,
`W=int_(v_(1))^(v_(2))PdV" " ...(1)`
For an adiabatic change ,
`PV^(gamma)=K("a constant) or "P=KV^(-gamma)`
Substituting for P in equation (1) , the work done in an adiabatic process
`W_("adi")=int_(v_(1))^(v_(2))KV^(-gamma)dV=Kint_(v_(1))^(v_(2))V^(-gamma)dV`
`=K[(V^(1-gamma))/(1-gamma)]_(V_(1))^(V_(2))=(K)/(1-gamma)[V_(2)^(1-gamma)-V_(1)^(1-gamma)]`
`(1)/(gamma-1)[KV_(1)^(1-gamma)-V_(2)^(1-gamma)]`
Since , `K=P_(1)V_(1)^(gamma)=P_(2)V_(2)^(gamma)`
`W("adi")=(1)/(gamma-1)[P_(1)V_(1)^(gamma)V_(1)^(1-gamma)-P_(2)V_(2)^(gamma)V_(2)^(1-gamma)]`
`=(1)/(gamma-1)[P_(1)V_(1)-P_(2)V_(2)]`
`therefore` Work done in adiabatic process ,
`W=(1)/(gamma-1)[nRT_(1)-nRT_(2)]`
Since `P_(1)V_(1)=nRT_(1)andP_(2)V_(2)=nRT_(2)`
`=(nR)/(gamma-1)(T_(1)-T_(2))`