Question

The initial pressure and volume of a given mass of a gas (C_p/C_v = 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?

Answer 1

Given:
For the gas, $\frac{\mathit{C}\mathit{p}}{\mathit{C}\mathit{v}}\mathit{=}\gamma$
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 $\frac{{\mathrm{V}}_0}{2}$ ⇒ 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,
P₁V₁^γ = P₂V₂^γ
$$\begin{gathered} \text{Or}2P_0{\left(\frac{V_0}{2}\right)}^{\gamma }={\mathrm{P}}_2{\left(\frac{V_0}{4}\right)}^{\mathrm{\gamma }} \\ \Rightarrow P_2=\frac{{V_{\mathit{0}}}^{\gamma }}{2^{\mathrm{\gamma }}}\times 2P_0\times \frac{4^{\mathrm{\gamma }}}{{V_{\mathit{0}}}^{\gamma }} \end{gathered}$$
= 2^γ × 2P₀ = P₀2^γ+1

(b)
(i) Adiabatic compression:
P₁V₁^γ = P₂V₂^γ
P₀V₀^γ = P'${\left(\frac{V_0}{2}\right)}^{\gamma }$
⇒ P' = P₀2^γ

(ii) Isothermal compression:
P₁V₁ = P₂V₂
$2^{\gamma }{P}_0\times \frac{V_0}{2}=P"\left(\frac{V_0}{2}\right)$
P" = P₀2^γ × 2
⇒ P" = P₀2^γ+1