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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, 400 cm3 and 300 K respectively. The raito of the specific heat capacities of the gas is Cp/Cv = 1.5. Find the pressure and the temperature of the gas if it is (a) suddenly compressed (b) slowly compressed to 100 cm3.
Sol. P base 1 = 100 KPa = 10^5 Pa, V base 1 = 400 cm^3 = 400 * 10^-6 m^3, T base 1 = 300 k, γ = C base P/C base V = 1.5 (a) Suddenly compressed to V base 2 = 100 cm^3 P base 1 V base 1^γ = P base 2 V base 2^γ ⇒ 10^5 (400)^1.5 = P base 2 * (100)^1.5 ⇒ P base 2 = 10^5 * (4)^1.5 = 800 KPa (b) T base 1V base 1^γ-1 = T base 2V base 2^ γ-1 ⇒ 300 * (400)^1.5 – 1 = T base 2 * (100)^1.5-1 ⇒ T base 2 = 300*20/20 = 600 K (b) Even if the container is slowly compressed the walls are adiabatic so heat transferred is 0. Thus the values remain, P base 2 = 800 KPa, T base 2 = 600 K.
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