Deduce an expression for work done in an adiabatic process.

Explanation

Consider one mole of a gas in insultaing cylinder with piston of cross sectional area A. Let the volume of gas be V, exerting pressure P. If the piston moves through a distance dx, then work done is given by :
`dW = f.dx = Padx = PdV`.
Now, the total work done, when the gas expands from volume `V_(1)` to `V_(2)` is given by
`W = int dW = underset(V_(1))overset(V_(2))int PdV`
or `W = underset(V_(1))overset(V_(2))int (k)/(V^(gamma)) dV` {`because` In adiabatic process `PV^(gamma) = k` or `P = (k)/(V^(gamma)`}
or `W = k underset(V_(1))overset(V_(2))int V^(-gamma)dV` or `W = (1)/(1-gamma) [kV_(1)^(1-gamma) - kV_(1)^(1-gamma)]`
But `P_(1)V_(1)^(gamma) = P_(2)V_(2)^(gamma) = k rArr W = (1)/(1-gamma)[P_(2)V_(2)^(gamma)V_(2)^(1-gamma) - P_(1)V_(1)^(gamma)V_(1)^(1-gamma)]`
or `W = (1)/(gamma - 1)[P_(1)V_(1) - P_(2)V_(2)]`.