Question Details

Two gases A and B are filled at the same pressure in separate cylinders with movable pistons of radius rA and rB, respectively. On supplying an equal amount of heat to both the systems reversibly under constant pressure, the pistons of gas A and B are displaced by 16 cm and 9 cm, respectively. If the change in their internal energy is the same, then the ratio rA/ rB is equal to

Options

4/3

3/4

2/√3

√3/2

Show Answer

Correct Answer :

Option B

3/4

Solution :

The correct option is 3/4.

To find the ratio of the radii of the two pistons, we can apply the first law of thermodynamics and the definition of work done in isobaric (constant pressure) processes.

According to the first law of thermodynamics, the heat supplied to a system ( $Q$ ) is equal to the change in its internal energy ( $Δ U$ ) plus the work done by the system ( $W$ ):

$Q = Δ U + W$

From the problem statement, we are given that:
1. The heat supplied to both systems is the same: $Q_{A} = Q_{B}$
2. The change in their internal energy is the same: $Δ U_{A} = Δ U_{B}$

Substituting these relations into the first law of thermodynamics, we get that the work done by both gases must also be equal:

$W_{A} = W_{B}$

The work done ( $W$ ) by a gas in a cylinder with a movable piston under constant pressure ( $P$ ) is given by:

$W = P \cdot Δ V$

where $Δ V$ is the change in volume. The change in volume can be expressed in terms of the cross-sectional area of the piston ( $A$ ) and the displacement of the piston ( $d$ ):

$Δ V = A \cdot d = π r^{2} d$

where $r$ is the radius of the piston. Therefore, the work done is:

$W = P \cdot π r^{2} d$

Now, equating the work done for both gases A and B:

$P \cdot π r_{A}^{2} \cdot d_{A} = P \cdot π r_{B}^{2} \cdot d_{B}$