20. The initial pressure and volume of a given mass of a gas \( \left(C_{p} / C_{V}=\gamma\right) \) 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 from the volume \( V_{0} \) to \( V_{0} / 2 \) and then slowly compressed to \( V_{0} / 4 \), what will be the final pressure?

Question

20. The initial pressure and volume of a given mass of a gas (C_p/C_v=gamma) are P₀ and V₀The 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?

Answer 1

Given,

For the gas, \(\frac{C_p}{C_v}=\gamma\)

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,

\(P_1V_1=P_2V_2\)

So, P₀V₀ = \(P_2 \frac{V_0}{2}\) ⇒ P₂ = 2P₀

(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,

\(P_1V_1^{\gamma} =P_2V_2 ^{\gamma}\) or