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

The problem necessitates an analysis of the correlation between piston displacement and their contained volumes. Given identical pressure in both cylinders, the work performed by the gas during piston displacement is equivalent due to congruent alterations in internal energy. The gas work formula is \( W = P \Delta V \), where \( P \) denotes pressure and \( \Delta V \) represents the volume change. With constant pressure for gases A and B, the work done is expressed as \( W_A = P \times \pi r_A^2 \times 16 \) and \( W_B = P \times \pi r_B^2 \times 9 \). As the internal energy change is uniform, \( W_A = W_B \), leading to the equation \( P \pi r_A^2 \times 16 = P \pi r_B^2 \times 9 \). Eliminating common terms and simplifying yields \( r_A^2 \times 16 = r_B^2 \times 9 \). Dividing both sides by 9 produces \( \frac{r_A^2}{r_B^2} = \frac{9}{16} \). Taking the square root of both sides results in \( \frac{r_A}{r_B} = \frac{3}{4} \). Consequently, the ratio \( r_A / r_B \) equals \(\frac{3}{4}\), aligning with the provided correct choice.