Two gases $A$ and $B$ are filled at the same pressure in separate cylinders with movable pistons of radius $r_A$ and $r_B$, 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 $\frac{r_A}{r_B}$ is equal to

We are given two gases A and B that are initially at the same pressure \(P\) and are heated reversibly at constant pressure, with the same amount of heat \(Q\) supplied to each.

For a constant‑pressure process the first law gives

$Q=\Delta U+P\Delta V$

The problem states that the change in internal energy is the same for both gases, i.e.

$\Delta U\_\{A\}=\Delta U\_\{B\}$

Because the heat supplied \(Q\) is also equal for the two gases, we have

$\Delta U\_\{A\}+P\Delta V\_\{A\}=\Delta U\_\{B\}+P\Delta V\_\{B\}$

Substituting \(\Delta U_{A}=\Delta U_{B}\) cancels the internal‑energy terms, leaving

$P\Delta V\_\{A\}=P\Delta V\_\{B\}$

Since the pressure \(P\) is the same on both sides, the equality reduces to

$\Delta V\_\{A\}=\Delta V\_\{B\}$

The volume change produced by each piston is the cross‑sectional area of the piston multiplied by its displacement:

$\Delta V\_\{A\}=A\_\{A\}\backslash ,\Delta x\_\{A\}=\pi r\_\{A\}^\{2\}\backslash ,\Delta x\_\{A\}$

$\Delta V\_\{B\}=A\_\{B\}\backslash ,\Delta x\_\{B\}=\pi r\_\{B\}^\{2\}\backslash ,\Delta x\_\{B\}$

Given the displacements \(Δx_{A}=16\text{ cm}\) and \(Δx_{B}=9\text{ cm}\), and using the equality \(ΔV_{A}=ΔV_{B}\), we obtain

$\pi r\_\{A\}^\{2\}\backslash ,16=\pi r\_\{B\}^\{2\}\backslash ,9$

Cancel the common factor \(π\) and solve for the radius ratio:

$r\_\{A\}^\{2\}\backslash ,16=r\_\{B\}^\{2\}\backslash ,9$

$\backslash frac\{r\_\{A\}^\{2\}\}\{r\_\{B\}^\{2\}\}=\backslash frac\{9\}\{16\}$

Taking the positive square root (radii are positive):

$\backslash frac\{r\_\{A\}\}\{r\_\{B\}\}=\backslash sqrt\{\backslash frac\{9\}\{16\}\}=\backslash frac\{3\}\{4\}$

Therefore the required ratio of the piston radii is

$\backslash frac\{r\_\{A\}\}\{r\_\{B\}\}=\backslash frac\{3\}\{4\}$