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, 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 cm³.

Answer 1

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,
P₁V₁^γ = P₂V₂^γ

⇒ 10⁵ × (400)^1.5 = P₂ (100)^1.5

⇒ P₂ =10⁵ (4)^1.5 = 800 kPa

Also,
T₁V^γ−1 = T₂V₂^γ−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