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 r_A/r_B is equal to

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. The first law of thermodynamics states that the heat supplied to a system is equal to the sum of the change in its internal energy and the work done by the system:
$dQ=dU+dW$
where:
• $dQ$ is the heat supplied to the gas,
• $dU$ is the change in internal energy,
• $dW$ is the work done by the gas during expansion.

For cylinder A, we can write:
$d{Q}_A=d{U}_A+d{W}_A$
For cylinder B, we can write:
$d{Q}_B=d{U}_B+d{W}_B$

According to the problem description:
1. An equal amount of heat is supplied to both systems: $d{Q}_A=d{Q}_B$
2. The change in their internal energy is the same: $d{U}_A=d{U}_B$

Substituting these equalities into the first law of thermodynamics gives:
$d{W}_A=d{W}_B$

Under constant pressure, the work done $dW$ during gas expansion is given by:
$dW=P·dV$
where $P$ is the pressure and $dV$ is the change in volume. Since the pressure $P$ is the same in both cylinders, we have:
$P·d{V}_A=P·d{V}_B$
which simplifies to:
$d{V}_A=d{V}_B$

The change in volume in a cylindrical vessel of radius $r$ with a piston displacement of $x$ is:
$dV=\pi r^{2}x$

Substituting this expression for both gases A and B:
$\pi r_A^2{x}_A=\pi r_B^2{x}_B$

Dividing both sides by $\pi$ and rearranging the terms to find the ratio of the radii:
$\frac{r_A^2}{r_B^2}=\frac{x_B}{x_A}$
Taking the square root on both sides:
$\frac{r_A}{r_B}=\sqrt{\frac{x_B}{x_A}}$

Given the displacements of the pistons are:
• Displacement for gas A, $x_A=16\text{cm}$
• Displacement for gas B, $x_B=9\text{cm}$

Substitute these values into the ratio:
$\frac{r_A}{r_B}=\sqrt{\frac{9}{16}}=\frac{3}{4}$

Thus, the ratio of the radii $r_A/r_B$ is equal to 3/4.