Calculate Final Volume of Compartment in Adiabatic Cylinder Heating

To find the final volume of the right-hand side compartment, we analyze the thermodynamic process undergone by the gas in that compartment.

Given parameters:

• Initial pressure in both compartments: $P_i = P$
• Initial volume of each compartment: $V_{R,i} = V_{L,i} = 9\text{ litres}$
• Ratio of specific heats: $\gamma = \frac{C_p}{C_v} = 1.5 = \frac{3}{2}$
• Final pressure in the cylinder (since the piston is frictionless, the pressures in both compartments equalize at equilibrium): $P_f = \frac{27}{8} P$

Since the piston is adiabatic and moves frictionlessly, no heat is exchanged with the gas in the right compartment, and work is done on it reversibly. Thus, the gas in the right-hand side compartment undergoes a reversible adiabatic compression.

For a reversible adiabatic process: $P_i V_{R,i}^\gamma = P_f V_{R,f}^\gamma$

Substituting the known values into the equation: $P \times (9)^{3/2} = \left(\frac{27}{8} P\right) \times V_{R,f}^{3/2}$

Dividing both sides by $P$: $(9)^{3/2} = \frac{27}{8} \times V_{R,f}^{3/2}$

Since $(9)^{3/2} = (3^2)^{3/2} = 3^3 = 27$: $27 = \frac{27}{8} \times V_{R,f}^{3/2}$

Canceling $27$ from both sides: $1 = \frac{1}{8} \times V_{R,f}^{3/2}$ $V_{R,f}^{3/2} = 8$

Taking the power of $\frac{2}{3}$ on both sides: $V_{R,f} = 8^{2/3} = \left(2^3\right)^{2/3} = 2^2 = 4\text{ litres}$

Thus, the final volume of the right-hand side compartment is $4\text{ litres}$.