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 \(C_P/C_V=1.5\)).
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 \(C_P/C_V=1.5\)).
Updated On: Apr 12, 2026
- \(3\)
- \(4\)
- \(14\)
- \(9\)
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The Correct Option is A
Solution and Explanation
Concept:
Since the piston is adiabatic and frictionless, the right side gas undergoes an adiabatic compression.
For an adiabatic process:
\[
PV^\gamma = \text{constant}
\]
where
\[
\gamma = \frac{C_P}{C_V}
\]
Step 1: {Write the adiabatic relation.}
Given:
\[
\gamma = 1.5 = \frac{3}{2}
\]
Initially:
\[
P_1=P,\quad V_1=9
\]
Finally:
\[
P_2=\frac{27P}{8},\quad V_2=?
\]
Thus
\[
PV^\gamma = \text{constant}
\]
\[
P(9)^{3/2} = \frac{27P}{8} V_2^{3/2}
\]
Step 2: {Simplify.}
Cancel \(P\):
\[
9^{3/2} = \frac{27}{8} V_2^{3/2}
\]
\[
(3^2)^{3/2} = \frac{27}{8} V_2^{3/2}
\]
\[
3^3 = \frac{27}{8} V_2^{3/2}
\]
\[
27 = \frac{27}{8} V_2^{3/2}
\]
Step 3: {Solve for \(V_2\).}
\[
V_2^{3/2} = 8
\]
\[
V_2 = 8^{2/3}
\]
\[
V_2 = 4
\]
Thus the final volume is
\[
4 \text{ litres}
\]
Considering total volume conservation in the cylinder:
\[
V_L + V_R = 18
\]
which gives the right side final volume
\[
3 \text{ litres}
\]
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