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Thermodynamics

JEE Main level, single correct, medium

57 of 190

Question 57 of 190

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

Show answer and solution

Answer: (B) $4$

The right compartment is enclosed by adiabatic walls and an adiabatic piston, and no heat is supplied to it. Therefore, the gas in the right compartment undergoes an adiabatic compression.

Since the piston is frictionless, the pressure in the right compartment is always equal to the pressure in the left compartment. The final pressure in both compartments is

P_f = \dfrac{27P}{8}

.

For the adiabatic process in the right compartment, the equation is:

P_i V_i^\gamma = P_f V_f^\gamma

Given

P_i = P

V_i = 9

P_f = \dfrac{27P}{8}

, and

\gamma = \dfrac{C_P}{C_V} = 1.5 = \dfrac{3}{2}

. Substituting these values:

P(9)^{3/2} = \left(\dfrac{27P}{8}\right) V_f^{3/2}

Since

9^{3/2} = (3^2)^{3/2} = 3^3 = 27

, the equation becomes:

27P = \dfrac{27P}{8} V_f^{3/2}

Dividing both sides by

27P

1 = \dfrac{1}{8} V_f^{3/2}

V_f^{3/2} = 8

Taking the power of

2/3

on both sides:

V_f = (8)^{2/3} = (2^3)^{2/3} = 2^2 = 4

litres.

Answer:

4

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