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Question
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 ratio of the specific heat capacities of the gas, 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.
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Solution
Initial pressure of the gas, P1 = 100 kPa
Initial volume of the gas,V1 = 400 cm3 = 400 × 10−6 m3
Initial temperature of the gas, T1 = 300 K
`gamma =("C"_"p")/("C"_"v") =1.5`
(a) The gas is suddenly compressed to volume, V2 = 100 cm3 .
So, this is an adiabatic process.
For an adiabatic process,
P1V1γ = P2V2γ
⇒ 105 × (400)1.5 = P2 (100)1.5
⇒ P2 =105 (4)1.5 = 800 kPa
Also,
T1Vγ−1 = T2V2γ−1
⇒ 300 × (400)1.5−1 = T2 (100)1.5−1
⇒ 300 × (400)0.5 = T2 (100)0.5
⇒ T2 = 600 K
(b) If the container is slowly compressed, the heat transfer is zero, even thought the walls are adiabatic.
Thus, the values remain same. Thus,
P2 = 800 kPa
T2 = 600 K
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