Question

The initial pressure and volume of a given mass of a gas (Cp/Cv = 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

Final pressure -

(a)  When the volume is compressed to to  $\frac{V_0}{2}$  the condition is $\mathrm{P}_{2}=2 \mathrm{P}_{0}$ and $\frac{V_0}{2}$ to $\frac{V_0}{4}$ is $P_2=\mathrm{P}_{0} 2^{\mathrm{y}+1}$

(b) The condition when the volume compressed from $V_0 \text {to} \frac{V_0}{2}$ is $\mathrm{P}^{\prime}=\mathrm{P}_{0} 2^{\gamma}$ and from$\frac{V_0}{2}$ to  $\frac{V_0}{4}$  is $\mathrm{P}^{n}=\mathrm{P}_{0} 2^{\gamma+1}$

Explanation:

Given Data

In case of  gas,

$\gamma=\frac{c_{p}}{c_{v}}$

$\text {Initial gas pressure} = P_0$

$\text {Initial gas volume }=\mathrm{V}_{0}$

(a)  Slowly compressed to a volume $\frac{V_0}{2}$ and then slowly compressed to $\frac{V_0}{4}$

(i) As the gas is compressed gradually the temperature should stay constant.  

For compression by isothermal,

$\mathrm{P}_{1} \mathrm{V}_{1}=\mathrm{P}_{2} \mathrm{V}_{2}$

$\mathrm{P}_{0} \mathrm{V}_{0}=\mathrm{P}_{2} \frac{\mathrm{V}_{0}}{2}$

$\mathrm{P}_{2}=2 \mathrm{P}_{0}$

(ii) Sudden compression means that the gas has not been able to get enough time to exchange heat with its atmosphere. So, it's a compression of adiabatic.  

So, to compact adiabatically,

$\mathrm{P}_{1} \mathrm{V}_{1}^{\gamma}=\mathrm{P}_{2} \mathrm{V}_{2}^{\gamma}$

$2 \mathrm{P}_{0}\left(\frac{\mathrm{V}_{0}}{2}\right)^{\gamma}=\mathrm{P}_{2}\left(\frac{\mathrm{V}_{0}}{4}\right)^{\gamma}$

$\mathrm{P}_{2}=\frac{\mathrm{V}_{0}^{\gamma}}{2} \times 2 \mathrm{P}_{0} \times \frac{4^{\gamma}}{\mathrm{V}_{0}^{\gamma}}$

    $=2^{\gamma} \times 2 \mathrm{P}_{0}$

   $=\mathrm{P}_{0} 2^{\mathrm{y}+1}$

$P_2=\mathrm{P}_{0} 2^{\mathrm{y}+1}$

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

(i) Compression with adiabatic effect:

$\mathrm{P}_{1} \mathrm{V}_{1}^{\gamma}=\mathrm{P}_{2} \mathrm{V}_{2}^{\gamma}$

$\mathrm{P}_{0} \mathrm{V}_{0}^{\gamma}=\mathrm{P}^{\prime}\left(\frac{\mathrm{V}_{0}}{2}\right)^{\gamma}$

$\mathrm{P}^{\prime}=\mathrm{P}_{0} 2^{\gamma}$

(ii) Compression with Isothermal effect:

$\mathrm{P}_{1} \mathrm{V}_{1}=\mathrm{P}_{2} \mathrm{V}_{2}$

$\mathrm{P}_{0} 2^{\mathrm{Y}} \times \frac{\mathrm{V}_{0}}{2}=\mathrm{P}^{\prime \prime}\left(\frac{\mathrm{V}_{0}}{2}\right)$

$\mathrm{P}^{\prime \prime}=\mathrm{P}_{0} 2^{y} \times 2$

$\mathrm{P}^{n}=\mathrm{P}_{0} 2^{\gamma+1}$

Therefore the final pressure, when the volume is compressed to  $\frac{V_0}{2}$  the condition is $\mathrm{P}_{2}=2 \mathrm{P}_{0}$ and from $\frac{V_0}{2}$ to $\frac{V_0}{4}$ is $P_2=\mathrm{P}_{0} 2^{\mathrm{y}+1}$and the condition for the final pressure when the volume compressed from  $V_0 \text {to} \frac{V_0}{2}$ is $\mathrm{P}^{\prime}=\mathrm{P}_{0} 2^{\gamma}$ and from$\frac{V_0}{2}$ to  $\frac{V_0}{4}$  is $\mathrm{P}^{n}=\mathrm{P}_{0} 2^{\gamma+1}$ where $V_o$ and $P_0$ be the initial pressure.