Question

Two gases A and B are filled at the same pressure in separate cylinders with movable pistons of radii \(r_A\) and \(r_B\) respectively. On supplying an equal amount of heat to both the cylinders, their pressures remain constant and their pistons are displaced by 16 cm and 9 cm respectively. If the change in their internal energies is the same, then the ratio \(r_A / r_B\) is:

• A. \( \frac{4}{3} \)
• B. \( \frac{2}{\sqrt{3}} \)
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( \frac{3}{4} \)

Hint:

For an isobaric process, the work done is \(P \Delta V\). If the heat supplied and change in internal energy are the same for two processes, the work done must also be the same. Equate the work done for both gases using the given displacements and radii.

Answer 1

The Correct Option is D

To solve this problem, we need to analyze the relationship between the pistons' displacement and the volumes they encapsulate.
Given that both cylinders are filled at the same pressure, we'll use the fact that the work done by the gas during piston displacement must be equal, due to the same change in internal energy.
The work done by gas is given by \( W = P \Delta V \), where \( P \) is pressure, and \( \Delta V \) is the change in volume.
Given that the pressure is constant, for gas A and gas B:
\( W_A = P \times \pi r_A^2 \times 16 \) and \( W_B = P \times \pi r_B^2 \times 9 \).
Since the change in internal energy is the same, \( W_A = W_B \), thus:
\( P \pi r_A^2 \times 16 = P \pi r_B^2 \times 9 \).
Canceling out the common factors and simplifying gives:
\( r_A^2 \times 16 = r_B^2 \times 9 \).
By dividing both sides by 9, we get:
\( \frac{r_A^2}{r_B^2} = \frac{9}{16} \).
Taking the square root of both sides results in:
\( \frac{r_A}{r_B} = \frac{3}{4} \).
Therefore, the ratio \( r_A / r_B \) is \(\frac{3}{4}\), which matches the correct option.