An ideal gas `(C_(p)//C_(v) =gamma)` having initial pressure `P_(0)` and volume `V_(0)` (a) The gas is taken isothermally to a pressure `2P_(0)` and then adiabatically to a pressure `4P_(0)`. Find the final volume. (b) The gas is brought back to its initial state. It is adiabatically taken to a pressure `2P_(0)` and then isothermally to a pressure `4P_(0)`. Find the final volume.

Answer

`(V_(0))/(2^((gamma+1)/(gamma)))` in each cases

Explanation

Initial pressure of an ideal gas `=P`
Initial volume of an idela gas `= V_(0)`
(a) For isothermal process `P_(2)V_(2) = P_(1)V_(1)`
`rArr V_(2) = (P_(1)V_(1))/(P_(2)) = (V_(0))/(2)`
For adiabatic process `P_(3)V_(3)^(gamma-1) =P_(2)V_(2)^(gamma)`
`rArr V_(3) = ((P_(2))/(P_(3)))^(1//gamma)V_(2) rArr V_(3) = (V_(0))/(2^((gamma+1)/(gamma)))`
(b) For adiabatic process
`P_(2)V_(2)^(gamma) = P_(1)V_(1)^(gamma) = V_(2) =((P_(1))/(P_(2)))^(1//gamma) V_(1)`
`rArr V_(2) = (V_(0))/(2^((1)/(gamma)))`
For isothermal process `P_(3)V_(3) = P_(2)V_(2)`
`rArr V_(3) = (P_(2)V_(2))/(P_(3)), V_(3)=(V_(0))/(2^((gamma+1)/(gamma)))`