An ideal gas `((C_(p))/(C_(v))=gamma)`has initial volume `V_(0)` is kept in a vessel. It undergoes a change and follows the following relation `P = kV^(2)` (where `P` is pressure, and `V` is volume) find the change in internal energy of the gas if its final pressure is `P_(0)`:

Answer

`(k)/(gamma-1) (((P_(0))/(k))^(3//2)-V_(0)^(3))`

Explanation

Initial volume of gas is `V_(i) = V_(0)`
final pressure of gas is `P_(f) = P_(0)`
Initial pressure of gas is `P_(i) = kV_(0)^(2)`
Change internal energy `DeltaU = nC_(v)DeltaT`
`DeltaU = (nR)/(gamma-1)DeltaT = (P_(f)V_(f)-P_(i)V_(i))/(gamma-1)`
final volume of gas is `V_(f) = ((P_(0))/(k))^((1)/(2))`
`DeltaU = (P_(0)((P_(0))/(k))^((1)/(2))-kV_(0)^(2)V_(0))/(gamma-1) =(k)/(gamma-1) [((P_(0))/(k))^((3)/(2))-V_(0)^(3)]`