Question

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

Answer 1

i)        Consider that one-gram molecule of a perfect gas is taken in a cylinder

ii)        Let the cylinder be placed on a source of heat at temperature T°A.

iii)        If the piston is now moved slowly outwards, the gas expands, does some work and tends to cool but it absorbs required amount of heat from the source to keep it at the same temperature.

iv)        The expansion is thus isothermal.

Here piston of area of cross-section A move through a small distance dx.

so that the gas expands by a small amount dV.

As the expansion is small, the pressure of the gas remains, practically, constant, say P.

Then the force acting on the piston, F = P×A

small work done by the gas,

dW = F dx

      = PA dx

dW = P dV        [Since, A dx = dV]

The net work done to expand the gas from a volume $V_{1}$ to volume $V_{2}$will be,

W = Area ABDC

Thus, area occupied below the curve in between AB and CD gives the required value of work done.

The isotherm PQ for the process is given in the figure.