A gas is enclosed in a cylindrical can fitted with a piston. The walls of the can and the piston are adiabatic. The initial pressure , volume and temperature of the gas are `100 kPa, 400cm^(3)` and `300K` respectively. The ratio of the specific heat capacities of the gas is `(C_p / C_v = 1.5)` Find the pressure and the temperature of the gas if it is (a) suddenly compressed (b) slowly compressed to `100 cm^(3)`.

Explanation

`P_1=100 Kpa` ,
`V_1 = 400 cm^3`,
`=400xx10^(-6)m^3`,
`T_1=300K`,
`gamma=(Cp)/(Cv) = 1.5`
(a) Suddenly compressed to `V_2 = 100 cm^3`
`P_1V_1^gamma = P_2V_2^gamma`
implies 10^5 xx (400)^1.5 = P_2(100)^1.5`
`P_2 = 10^(5)(4)^1.5 = 800 KPa`
`T_1V^(gamma-1)=T_2V_2^(gamma-1)`
implies `300xx(400)^(1.5-1) = T_2(100)^(1.5-1)`
implies `300 xx (400)^0.5 = T_2 (100)^0.5`
`T_2 =600K`.
(b) Even if the container is slowly compressed the walls are adiabatic so heat transferred is zero.
Thus the values remain,
`P_2=800 KPa`
`T_2 =600 K`