# Graphically show the total work done in an expansion when the state of an ideal gas is changed reversibly and isothermally from to $(p_{f} ,V_{f})$. With the help of a pV plot compare the work done in the above case with that carried out against a constant external pressure $P_{f}$.

Let us assume that a cylinder has 1-mole gas without any weight or friction having area of cross-section A. Hence, the total volume is depicted by Vi and the initial pressure is assumed to be p.

Let us assume that pext is the external pressure.

Now, if pext> p, the piston moves down until pext = p,

Hence, the final volume is Vf and the distance moved by piston be ?l

Thus, ?V = ?l × A (eq-1)

?V = Vf - Vi

We know that, force = pressure ×area

Therefore, F = pext× A (eq-2)

If w is work done on the system

W = force × displacement

= pext × A ×?l

From eq-1

W = pext×(-?V)

W = -pext?V

W = -pext( vf – vi)

If vf> vi work is done by the system and w is negative. If vf< vi work is on the system and w is positive

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