Derive an expression for maximum work in isothermal reversible expansion of two moles of an ideal gas.
Solution
 Consider the expansion of ‘n’ moles of an ideal gas enclosed in a cylinder fitted with a weightless and
frictionless piston at a constant temperature T.  Let the pressure of the gas be P which is equal to the external pressure.
 Let the external pressure be reduced by an infinitesimally small amount dp so that the new external pressure becomes (Pdp).
 Let the corresponding small increase in volume by dv.
 Therefore amount of work done in the expansion of the gas, dw =  [(p dp)dv] = [pdv dp.dv] ..(i)
 Since dp and dv are very small, their product (dp.dv) will be much smaller and can be neglected ∴ dw = pdv ...(ii)

During expansion of the gas, if the initial volume V_{1} changes to volume V_{2} then total amount of work alone (Wmax) can be obtained by integrating the work, dw between the two limits, V_{1} and V_{2}.
`therefore "W"_"max" = int_("v"_2)^("v"_1)  dw`
`= int_("v"_1)^("v"_2)  pdv` ...(iii)
But pv = nRT (Ideal gas Equation for n moles of a gas)
`therefore "p" = "nRT"/"v"`
Substitute pressure value in equation (iii)
`therefore "W" _"max" = int_("v"_1)^("v"_2)  ("nRT")/"v" "dv"`
`= "nRT" int_("v"_1)^("v"_2) "dv"/"v"` (n,R,T are constatnt)
`= "nRT" ["log v"]_("v"_1)^("v"_2)`
`= "nRT" "ln" "v"_2/"v"_1`
`therefore "W"_"max" = 2.303 "nRT" "log" "v"_2/"v"_1`
(In = 2.303 log_{10}) ...(iv)