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20

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?

Given C_{P}/C_{V} = γ

Let P1=P_{0} be the Initial Pressure, V1=V_{0} be the Initial Volume

(a)Since the volume is slowly compressed, temperature remains constant I.e. Isothermal compression. Let P2 and V2=V_{0}/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=2P_{0}

(b) Since the volume is suddenly compressed, I.e. it is an adiabatic process Let P2 and V2=V_{0}/2 be the pressure and volume after sudden compression, then

PV^{γ} = const.

I.e.

Substituting value of P1=P_{0}

Now, since the volume is slowly compressed, temperature remains constant I.e. Isothermal compression. Let P3 and V3=V_{0}/4 be the pressure and volume after slow compression.

∴