Question

Two cylinders A and B fitted with the pistons contain an equal amount of an ideal diatomic gas at $300K$ . Piston A is free to move and piston B is fixed. The same amount of heat is given to the gases in two cylinders. Temperature of the gas in cylinder A increases by $30K$ . Then, increase in temperature of the gas in cylinder B is:
( $\gamma = 1.4$ for diatomic gas)
(A) $36K$
(B) $54K$
(C) $42K$
(D) $24K$

Answer 1

Given that the two cylinders A and B are fitted with pistons. Also they have diatomic gas. Piston A is free to move which means pressure remains constant and piston B is fixed which means that the volume inside that cylinder remains constant. So using the ratio of the specific heat we can find the increase in temperature.

Complete answer:
Piston A is free to move. That means that the pressure inside that cylinder will be constant, which means the process is isobaric.
Thus for given heat to the system is
$\therefore \Delta {Q_A} = n{C_p}\Delta {T_A} \ldots \ldots \left( 1 \right)$
Piston B is fixed. This implies that the volume inside the cylinder is constant. This means that the process is isochoric.
Thus for the given heat to the system is
$\therefore \Delta {Q_B} = n{C_v}\Delta {T_B} \ldots \ldots \left( 2 \right)$
Given in the question is that the same heat is given to both the cylinders. Therefore,
$\Delta {Q_A} = \Delta {Q_B}$ ……..(3)
Therefore equating (1) and (2)
$n{C_p}\Delta {T_A} = n{C_v}\Delta {T_B}$ ……… (4)
$\Delta {T_B} = \left( {\dfrac{{{C_p}}}{{{C_v}}}} \right)\Delta {T_A}$ ………. (5)
The ratio of the specific heat is given as,
$\gamma = \left( {\dfrac{{{C_p}}}{{{C_v}}}} \right) = 1.4$ for diatomic gases.
Also, the increase in temperature for cylinder A is given as
$\Delta {T_A} = 30K$
Substituting both the values in the equation (5),
$\Delta {T_B} = 1.4 \times 30K$
$\Delta {T_B} = 42K$
Therefore the correct option is C.

Note:
Note that the specific heat capacity of a body is the amount of heat required to raise the temperature of the body of unit mass from ${0^0}C$ to ${1^0}C$ . It is represented by C and its units are usually calories or $Joule/KgK$ . The specific heat capacity of water at room temperature is higher than most other materials. We use this property of water in our body to maintain constant body temperature.