The magnitude of work done by a gas is a reversible expansion is greater than that in an irreversible expansion provided the initial and final state are identical in both processes-explain.

Explanation

Suppose, an ideal gas expands isothermally from its initial state `P_(1) V_(1)` to final state `P_(2)V_(2)`.

A reversible process consists of an infinite number of small steps, and at each step, the driving from is infinitesimally greater than the opposing force. suppose, in a reversible expansion of an ideal gas, the external pressure of the gas at a step of the process is decreased by a very small amount dP, causing the increase in volume of the gas by dV. the work done by the gas will be, dw=`-(P_(ex)-dP)dV=-PdV` [`becauseP_(ex)-dP=P` (Pressure of the gas) and `dPxxdV` is very small as in the P-V diagram the point, m and n are very close and pressure in these two points are not much different]. in the above diagram, PdV is givenn by the area of the strip, mnqp. total work done in reversible process will be equal to the sum of the areas of all such strips. thus, in such a process, work done,
`-w=underset(V_(1))overset(V_(2))intPdV=`area ABNM
Absolute value of work done, `|w|=` area ABNM
Now, in isothermal irreversible expansion, suppose, the gas is expanded from volume `V_(1)` to `V_(2)` against an external pressure which is equal to the final pressure of the gas. in this process work done,
`w=underset(V_(1))overset(V_(2))intP_(2)dV=-P_(2)underset(V_(1))overset(V_(2))intdV` [`P_(2)=`constant during expansion]
Absolute value of work done, `|w|=(V_(2)-V_(1))`=area. ABNE. From the graphs, it is clear than work done is reversible isothermal expansion is greater than that in irreversible isothermal expansion, since the arae of ABNM is greater than that of ABNE. this also indicates that work involved in a process depends upon the path (or nature) o the process.