Deduce the expression for work done in an isothermal process.

Explanation

Consider one mole of a gas in conducting with frictionless piston of cross sectional area A. Let the volume of gas be V, exerting pressure P.
`therefore` Work done to move the piston though distance dx is
`dW = Fdx = PA.dx = PdV`.
Now tota work done, when 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 (RT)/(V)dV = RT underset(V_(1))overset(V_(2))int (dV)/(V) " "{because PV = RT "or " P = (RT)/(V)}`
or `W = RT log_(e)((V_(2))/(V_(1)))`
Hence, `W = 2.303 RT log_(10)((V_(2))/(V_(1))) " "(because log_(e) x = 2.303 log_(10)x}`