Derive the equation for `'W_("rev")'` in isothermal reversible process.

Explanation

Work done during the expansion of a gas against external pressure can be expressed as
`W=P_("ext")(-DeltaV)=-P_("ext")(V_(f)-V_(i))`
In isothermal reversible process the above equation can be written as
`W_("rev")=-underset(V_i)overset(V_f)intP_("ext")dV=underset(V_i)overset(V_f)int(P_("in")pmdP)dV`
Since `d_(p)xxd_(v)` is very small it can be neglected and we can write the equation as
`W_("rev")=underset(V_f)overset(V_i)P_("in")dV`
Now the pressure of the gas `P_("in")` can be written as P.
This can be obtained from ideal gas equation.
`P=(nRT)/(v)`
Therefore at constant temperature (isothermal process)
`W_("rev")=underset(V_f)overset(V_i)intnRT(dV)/(V)=-"nRTIn"(V_f)/(V_i)`
`W_("rev")=-2.303 nRT log""(V_f)/(V_i)`