Calculate the work involved in expansion and compression process.

Explanation

Work involved in expansion :
(i) For understanding pressure- volume work, let us consider a cylinder which contains 'n' moles of an ideal gas fitted with a frictionless piston of cross sectional area A. The total volume of the gas inside is `V_(i)` and pressure of the gas inside is `P_("int")`.
(ii) If the external pressure `P_("ext")` is greater than `P_("int")` , the piston moves inward till the pressure inside becomes equal to `P_("ext")` . Let this change be achieved in a single step and the final volume of `V_(f)`
(iii) In this case, the work is done on the system (+w) . It can be calculated as follows
`w=-F.Deltax" "...(1)`
(iv) where dx is the distance moved by the piston during the compression and F is the force acting on the gas.
`F=P_("ext")A" "...(2)`
Substituting (2) in (1)
`w=-P_("ext").A.Deltax`
`A.Deltax= " change volume "=V_(f)-V_(i)`
`w=-P_("ext").(V_(f)-V_(i))" "...(3)`
`w=-P_("ext").(-DeltaV)" "...(4)`
`=P_("ext").DeltaV`

(vi) Since work is done on the system , it is a positive quantity.
(vi) If the pressure is not constant , but changes during the process such that the pressure of the gas , then , at each stage of compression , the volume decreases by an infinitesimal amount, dV. In such a case we can calculate the work done on the gas by the relation.
`W_("rew")int_(V_(i))^(V_(f))P_("int")dV`
(vii) In a compression process, `P_("ext")` the external pressure is always greater than the pressure of the system .
i.e. `P_("ext")=(P_("int"+dP))`.
(viii) In a expansion process, the external pressure is always less than the pressure of the system .
i.e. `P_("ext")=(P_("int"-dP))`.

(ix) When pressure is not constant and changes in infinitesimally small steps (reversible condition) during compression from ... The P-V plot looks like in image work done on the gas is respected by the shaded area.
(x) `P_("ext")=(P_("int")+dP)` Such processes are called reversible processes. For a compression process work can related to internal pressure of the system under reversible conditions by writing equation
`W_("rev")=int_(V_(i))^(v_(f)) =P_("int")dV`
For the the system with ideal gas
`P_("int")V=nRT`
`P_("int")=(nRT)/(V)`
`W_("rev")=int_(V_(i))^(V^(f))(nRT)/(V)dV` `W_("rev")=-nRTint_(V_(i))^(V^(f))((dV)/V)`
`W_("rev")=-nRTIn((V_(f))/(V_(i)))`
`W_("rev")=-2.303nRTlog((V_(f))/(V_(i)))`
(xi) If `V_(f)gtV_(i)` (expansion) ,the sign of work done by the process is negative
(xii) If `V_(f)ltV_(i)` (compression) the sign of work done on the process is positive.