Question

The initial pressure and volume of a given mass of a gas $(\frac{C_p}{C_v}=\gamma )$ are $P_0$ and $V_0$ The gas can exchnage heat with the surrounding,

(a) It is slowely comproessed to a volume $\frac{V_0}{2}$ and then suddenly compressed to $\frac{V_0}{4}$. Find the final presure.

(b) If the gas is suddenly compressed from the volume $V_0$ to $\frac{V_0}{2}$ and then slowly compressed to $\frac{V_0}{4}$, what will be the final pressure ?

Answer 1

Given, $\frac{C_p}{C_v}=\gamma$

$P_0$ ( Initial pressure), $V_0$ (Initial volume)

(a) (i) Isothermal compression,

$P_1{V}_1=P_2{V}_2$

or $P_0{V}_0=P_2\frac{V_0}{2}\Rightarrow P_2=2P_0$

(ii) Adiabatic compression

$P_1{V}_1\gamma =P_2{V}_2\gamma$

or $2P_0{\left(\frac{V_0}{2}\right)}^{\gamma }=P_2{\left(\frac{V_0}{4}\right)}^{\gamma }$

$\Rightarrow P_2=\frac{V_0^{\gamma }}{2^{\gamma }}\times 2P_0\times \frac{4^{\gamma }}{V_0^{\gamma }}$

= $2\gamma \times 2P_0=P_0{2}^{\gamma +1}$

(b) (i) Adiabatic compression

$P_1{V}_1\gamma =P_2{V}_2^{\gamma }$

or $P_0{V}_0^{\gamma }=P{V}_1^{\gamma }$

$\Rightarrow P^{\prime }=P_{0}2\gamma$

(ii) Isothermal compression

$P_1{V}_1=P_2{V}_2$

$2\gamma P_0\times \frac{V_0}{2}=P"\left(\frac{V_o}{2}\right)$

$P"=P_{0}2\gamma \times 2$

$\Rightarrow =P_0{2}^{\gamma +1}$