Conider a given sample of an ideal gas `(C_p / C_v = gamma )` having initial pressure `(p_0) and volume (V_0)` . (a) The gas is isothermally taken to a pressure `(P_0 / 2 )` and from there adiabatically ta a pressure `(p_0 / 2)` and from there isothermally to a pressure `(p_0 / 4 )` . find the volume.

Explanation

(a) Given that
`P_1 =P_0`, `V_1=V_0`
For isothermal process,
`P_2= ((P_0V_0)/(P_0))/((P_0)/(2)) = 2V_0`
For adiabatic process `P_3 = P_0/4 , V_3 =?`
`P_2V_2^(gamma)=P_3V_3^(gamma)`
implies `((V_3)/(V_2))^gamma = P_3V_3^gamma`
implies `((V3)/(V2))^gamma = ((P_2)/(P_3))`
implies ` ((V_3)/(V_2))^gamma = ((P_0//2)/(P_0//4))=2`
implies `(V_3)/(V_2) = 1/2^gamma`
:. `V_3 =V_2^(2^((1)/gamma)) =2V_0^(2^((1)/gamma))`
`= 2 ^((gamma+1)/(gamma))V_0`
(b) `P_1V_1^gamma = P_2V_2^gamma`
or `((V_2)/(V_1))=((P_1)/(P_2))^((1)/(gamma))`
or `V_2 = V_0^(2^(1/gamma))`
Again isothermal process,
`P_2V_2 = P_3V_3`
` :. `V_3 = (P_2V_2)/(P_3) = 2.2^((1)/(gamma))V_0` ,
`=2^((gamma+1)/(gamma))V_0`