Consider a given sample of an ideal gas ( C_P C_V = ) having initial pressure P_0 and volume V_0. 1. The gas is isothermally taken to a pressure P_0 2 and from there, adiabatically to a pressure P_0 4 . Find the final volume. 2. The gas is brought back to its initial state. It is adiabatically taken to a pressure P_0 2 and from there, isothermally to a pressure P_0 4 . Find the final volume.

Initial pressure = P_0 Initial Volume = V_0 = C_P C_V 1. Isothermally to pressure = P_0 2 P_0V_0= P_0 2 V_1 _1=2V_0 Adiabetically to pressure P_0 4 P_0 2 (V_1)^ = P_0 4 (V_2)^ P_0 2 (2V_0)^ = P_0 4 (V_2)^ 2^ +1 V_0^ =V_2^ _2=2^( +1) V_0 Final Volume =2^( +1) V_0 2. Adiabetically to pressure P_0 4 to P_0 P_0 (2^ +1 V_0^ )= P_0 2 (V')^ Isothermal to pressure P_0 4 P_0 2 2^1 V_0= P_0 4 V'' ''=2^( +1) V_0 Final Volume =2^( +1) V_0

Consider a given sample of an ideal gas ( C_P C_V = ) having init…

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Question

Consider a given sample of an ideal gas $\Big(\frac{\text{C}_\text{P}}{\text{C}_\text{V}}=\gamma\Big)$ having initial pressure $P_0$ and volume $V_0.$

The gas is isothermally taken to a pressure $\frac{\text{P}_0}{2}$ and from there, adiabatically to a pressure $\frac{\text{P}_0}{4}.$ Find the final volume.
The gas is brought back to its initial state. It is adiabatically taken to a pressure $\frac{\text{P}_0}{2}$ and from there, isothermally to a pressure $\frac{\text{P}_0}{4}.$ Find the final volume.

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Answer

Initial pressure $= P_0$
Initial Volume $= V_0$
$\gamma=\frac{\text{C}_\text{P}}{\text{C}_\text{V}}$

Isothermally to pressure $=\frac{\text{P}_0}{2}$

$\text{P}_0\text{V}_0=\frac{\text{P}_0}{2}\text{V}_1$
$\Rightarrow\text{V}_1=2\text{V}_0$
Adiabetically to pressure $\frac{\text{P}_0}{4}$
$\frac{\text{P}_0}{2}(\text{V}_1)^\gamma=\frac{\text{P}_0}{4}(\text{V}_2)^\gamma$
$\frac{\text{P}_0}{2}(2\text{V}_0)^\gamma=\frac{\text{P}_0}{4}(\text{V}_2)^\gamma$
$\Rightarrow2^{\gamma+1}\text{V}_0^\gamma=\text{V}_2^\gamma$
$\Rightarrow\text{V}_2=2^\frac{(\gamma+1)}{\gamma}\text{V}_0$
$\therefore$ Final Volume $=2^\frac{(\gamma+1)}{\gamma}\text{V}_0$

Adiabetically to pressure $\frac{\text{P}_0}{4}$ to $P_0$

$\text{P}_0\times\big(2^{\gamma+1}\text{V}_0^\gamma\big)=\frac{\text{P}_0}{2}\times(\text{V}')^\gamma$
Isothermal to pressure $\frac{\text{P}_0}{4}$
$\frac{\text{P}_0}{2}\times2^\frac{1}{\gamma}\text{V}_0=\frac{\text{P}_0}{4}\text{V}''$
$\Rightarrow\text{V}''=2^\frac{(\gamma+1)}{\gamma}\text{V}_0$
$\therefore$ Final Volume $=2^\frac{(\gamma+1)}{\gamma}\text{V}_0$

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