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Calculate the work involved in expansion and compression process. |
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Answer» Solution :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, theexternal 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. |
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