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Question
The initial pressure and volume of a given mass of a gas (Cp/Cv = γ) are p0 and V0. The gas can exchange heat with the surrounding. (a) It is slowly compressed to a volume V0/2 and then suddenly compressed to V0/4. Find the final pressure. (b) If the gas is suddenly compressed from the volume V0 to V0/2 and then slowly compressed to V0/4, what will be the final pressure?
Solution
Given:
For the gas, `("C"_"p")/"C"_"v" = gamma`
Initial pressure of the gas = P0
Initial volume of the gas = V0
(a)
(i) As the gas is slowly compressed, its temperature will remain constant.
For isothermal compression,
P1V1 = P2V2
So, P0V0 =P2 `"V"_0/2 =>"P"_2 = 2"P"_0`
(ii) Sudden compression means that the gas could not get sufficient time to exchange heat with its surroundings. So, it is an adiabatiac compression.
So, for adiabatic compression,
P1V1γ = P2V2γ Or
`2"P"_0 ("V"_0/2)^gamma = "P"_2 ("V"_0/4)^ gamma`
`=> "P"_2 = "V"_0^gamma/2^ gamma xx 2"P"_0 xx 4^gamma/"V"_0^gamma`
= 2γ × 2P0 = P02γ+1
(b)
(i) Adiabatic compression:
P1V1γ = P2V2γ
`"P"_0"V"_0 ^ gamma = "P'"("V"_0/2)^ gamma`
⇒ P' = P02γ
(ii) Isothermal compression:
P1V1 = P2V2
`2 ^ gamma "P"_0 xx "V"_0/2 = "P''" ("V_0/2)`
P" = P02γ × 2
⇒ P" = P02γ+1
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