Question
A cylinder with adiabatic walls is closed at both ends and is divided into two compartments by a frictionless adiabatic piston. Ideal gas is filled in both (left and right) the compartments at same $P,V$, T. Heating is started from left side until pressure changes to $\frac{27P}{8}$. If initial volume of each compartment was 9 litres then the final volume in right-hand side compartment is $\_\_\_\_$ litres. (for this ideal gas $C_{p}/C_{v} = 1.5$ )
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Solution
Sol. [Key Concept: The right compartment is compressed adiabatically, so $P_{i}V_{i}^{\gamma} = P_{f}V_{f}^{\gamma}$ with $P_{f} = \frac{27P}{8}$ and $\gamma = 1.5$; solve for $V_{f}$.]
$P_{\text{Left}\text{~}} = P_{\text{Right}\text{~}}$ (final)
for right chamber
$${P_{i}V_{i}^{\gamma} = P_{f}V_{f}^{\gamma} }{P(9)^{3/2} = \frac{275}{8} \times V_{f}^{3/2} }{27 = \frac{27}{8}{\text{ }V}_{f}^{3/2} }$$$\therefore V_{f} = 4$ litres
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