Show that for an isotheremal process, work `W=muRTlog_(e)((V_(2))/(V_(1)))` where symbols have their usual meaning.

Explanation

For an isothermal process, PV=constant.
At any intermediate stage with pressure P and change in volume from. V to V+dV,
Work done dW=PdV. However for the entire process `W=underset(V_(1))overset(V_(1))intPdV`
Where `PV=muRT" "i.e,P=(muRT)/(V)`
Hence, `W=underset(V_(1))overset(V_(1))int(muRT)/(V)dV" "i.e,W=muRTlog_(e)((V_(2))/(V_(1)))`
For `V_(2)gtV_(1),Wgt0and"for"V_(2)ltV_(1),Wlt0`
In an isothermal expansion the gas absorbs heat and does work while in an isothermal compression, work is done on the gas by the environment and heat is released.