The initial pressure and volume of a given mass of a gas (C P /C V = γ) are P 0 and V 0 . The gas can exchange heat with the surrounding. (a) It is slowly compressed to a volume V 0 /2 and then suddenly compressed to V 0 /4. Find the final pressure. (b) If the gas is suddenly compressed form the volume V 0 to V 0 /2 and then slowly compressed to V 0 /4, what will be the final pressure?

Question

Philoid · Accepted Answer

Given C_P/C_V = γ

Let P1=P₀ be the Initial Pressure, V1=V₀ be the Initial Volume

(a)Since the volume is slowly compressed, temperature remains constant I.e. Isothermal compression. Let P2 and V2=V₀/2 be the pressure and volume after slow compression

∴

When volume is suddenly compressed, it is an adiabatic process .Let P3 and V3 be the pressure and volume after sudden compression .i.e.

PV^γ = const.

I.e.

Substituting value of P2=2P₀

(b) Since the volume is suddenly compressed, I.e. it is an adiabatic process Let P2 and V2=V₀/2 be the pressure and volume after sudden compression, then

PV^γ = const.

I.e.

Substituting value of P1=P₀

Now, since the volume is slowly compressed, temperature remains constant I.e. Isothermal compression. Let P3 and V3=V₀/4 be the pressure and volume after slow compression.

∴