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Specific Heat Capacities of Gases — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsSpecific Heat Capacities of Gases5 Marks
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 $V_0.$
  1. 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.
  2. 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.
✓

Answer

Initial pressure $= P_0$
Initial Volume $= V_0$
$\gamma=\frac{\text{C}_\text{P}}{\text{C}_\text{V}}$
  1. 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 $=2^\frac{(\gamma+1)}{\gamma}\text{V}_0$
  1. 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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