Question 93: Thermodynamics & KTG

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 $27 \mathrm{P} / 8$. If initial volume of each compartment was 9 litres then the final volume in right-hand side compartment is $\_\_\_\_$ litres. (for this ideal gas $\mathrm{C}_{\mathrm{P}} / \mathrm{C}_{\mathrm{V}}=1.5$ )

Accepted Answer

In left compartment heat is added. The gas expands, increasing its pressure and volume.
In right compartment heat is not added because the wall is adiabatic, but the piston moves toward it because the pressure on the left increases. Since the walls and the piston are adiabatic, the gas in the right compartment undergoes an adiabatic compression.
The piston stops moving when the pressure in both compartments is equal. Therefore, the final pressure in the right compartment $\left(\mathrm{P}_{\mathrm{R}}ight)$ will also be $\frac{27 \mathrm{P}}{8}$.
For an adiabatic process, the relationship between pressure $(\mathrm{P})$ and volume $(\mathrm{V})$ is:
$$ \mathrm{P} \mathrm{~V}^\gamma=	ext { constant } $$
Where $\gamma=\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}=1.5$.
$$ P_1 V_1^\gamma=P_2 V_2^\gamma $$
$$\Rightarrow $$ $$ \frac{V_2}{V_1}=\left(\frac{P_1}{P_2}ight)^{1 / \gamma} $$
The initial pressure in right compartment is $\mathrm{P}_1=\mathrm{P}$ and its initial volume is $\mathrm{V}_1=9$ litres
The final pressure in right compartment is $\mathrm{P}_2=\frac{27 \mathrm{P}}{8}$
Substituting these values:
$$ \frac{V_{	ext {final }}}{9}=\left(\frac{P}{27 P / 8}ight)^{1 / 1.5} $$
$$\Rightarrow $$ $\frac{\mathrm{V}_{	ext {final }}}{9}=\left(\frac{8}{27}ight)^{2 / 3}$
$$\Rightarrow $$ $\frac{V_{	ext {final }}}{9}=\frac{4}{9}$
$$\Rightarrow $$ $$ V_{	ext {final }}=4 	ext { litres } $$
Therefore, the final volume in the right-hand side compartment is 4 litres.
Hence, the correct option is (B).

Answer

3

Answer

4

Answer

14

Answer

9

Thermodynamics & KTG

Choose the correct answer: