Work done by a system under isothermal change from a volume `V_(1)` to `V_(2)` for a gas which obeys Vander Waal's equation `(V-betan)(P+(alphan^(2))/(V))nRT`

Options

A. `nRT log_(c)((V_(2)-nbeta)/(V_(1)-nbeta))+alphan^(2)((V_(1) -V_(2))/(V_(1)V_(2)))`
B. `nRT log_(10)((V_(2)-alphabeta)/(V_(1)-alphabeta))+alphan^(2)((V_(1) -V_(2))/(V_(1)V_(2)))`
C. `nRT log_(e)((V_(2)-nalpha)/(V_(1)-nalpha))+betan^(2)((V_(1) -V_(2))/(V_(1)V_(2)))`
D. `nRT log_(e)((V_(1)-nbeta)/(V_(2)-nbeta))+alphan^(2)((V_(1) V_(2))/(V_(1)-V_(2)))`

Answer

A

Explanation

According to given Vander Waal's equation
`P = (nRT)/(V-nbeta) - (alphan^(2))/(V^(2))`
work done , `W = int_(V_(1))^(V_(2)) PdV`
`= nRT int_(v_(1))^(v_(2)) (dV)/(V-nbeta) - alphan^(2) int_(v_(1))^(v_(2)) (dV)/(V^(2))`
` = nRT [log_(e) (V-nbeta)]_(v_(1))^(v_(2)) + alphan^(2) [(1)/(V)]_(v_(1))^(v_(2))`
`= nRT log_(e)'(V_(2) - nbeta)/(V_(1) - nbeta) + alphan^(2) ((V_(1) - V_(2))/(V_(1) V_(2)))`