Question

Figure shows an adiabatic container which is divided into two equal parts by an adiabatic freely moving piston. Both sides contain same gas having \( \gamma = 1.5 \). If heat is supplied to left part such that its pressure becomes \( \frac{27P_0}{8} \), find volume of right part.

Hint:

In problems with adiabatic piston, always use \( PV^\gamma = \text{constant} \) for the side where no heat is supplied, and remember pressures become equal at equilibrium.

Answer 1

Step 1: Understand the process.
The container and piston are adiabatic, so no heat exchange occurs for the right part. Thus, the right side undergoes an adiabatic process.
Also, the piston is freely movable, so final pressures on both sides must be equal.
Final pressure on left side = final pressure on right side = \( \frac{27P_0}{8} \).

Step 2: Apply adiabatic relation for right side.
For adiabatic process:
\[ P V^\gamma = \text{constant} \] Initially for right side:
\[ P_0 V_0^\gamma \] Finally:
\[ \frac{27P_0}{8} \cdot V^\gamma \] So,
\[ P_0 V_0^\gamma = \frac{27P_0}{8} \cdot V^\gamma \]
Step 3: Simplify the equation.
Cancel \( P_0 \):
\[ V_0^\gamma = \frac{27}{8} V^\gamma \] \[ \left(\frac{V_0}{V}\right)^\gamma = \frac{27}{8} \]
Step 4: Substitute \( \gamma = 1.5 = \frac{3}{2} \).
\[ \left(\frac{V_0}{V}\right)^{3/2} = \frac{27}{8} \] Taking power \( \frac{2}{3} \) on both sides:
\[ \frac{V_0}{V} = \left(\frac{27}{8}\right)^{2/3} \] \[ \frac{V_0}{V} = \left(\frac{3^3}{2^3}\right)^{2/3} = \frac{3^2}{2^2} = \frac{9}{4} \]
Step 5: Final result.
\[ V = \frac{4}{9} V_0 \] Final Answer: \( \frac{4}{9}V_0 \)