The initial pressure and volume of a given mass of a gas `(C_p / C_v = gamma)` are `(p_0)` and `(V_0)`. The gas can exchange heat with surrounding. (a) It is slowly compressed to a volume `(V_0 / 2 )` and then suddenly compressed to `(V_0 /4)` . Find the final pressure. (b) If the gas is suddenly cpmpressed from the volume `(V_0 to V_0 / 2 )` and then slowly compressed to `(V_0 / 4 )`, What will be the final pressure?.

Explanation

Given , `(Cp)/(Cv) =gamma`
`P_0` (Initial pressure), `V_0` (Initial volume)
(a)(i) Isothermal compression,
`P_1V_1 = P_2V_2`
`P_0V_0 = P_2(V_0)/(2) implies P_2=2P_0`
(ii) Adiabatic compression
`P_1V_1^gamma=P_2V_2^gamma`
or `2P_0((V_0)/(2))^gamma = P_2((V_0)/4)^gamma`
implies `P_2= (V_(0^gamma))/(2^gamma) xx2P_0xx(4gamma)/V_(0^gamma)`
`2^gamma xx 2P_0 = P_(0^(2gamma+1))`
(b) (i) Adiabatic compression
`P_1V_1^gamma=P_2V_2^gamma`
or `P_0V_0^gamma = P^'V_1^gamma` implies `P^' = P_(0^(2gamma))`
(ii) Isothermal compression
or `P_1V_1 = P_2V_2`
`2^gammaP_0 xx (V_0)/(2) = P^''((V_0)/(2))`
`P^'' = P_0^(2gamma)xx2` ,
`=P_(0^(2gamma+1))`