Question: 41. 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 r_{A} / r_{B} is equal to
(1) \frac{4}{3}
(2) \frac{3}{4}
(3) \frac{2}{\sqrt{3}}
(4) \frac{\sqrt{3}}{2}
Answer: Option (2)
Explanation:
For each gas at constant pressure,
the first law of thermodynamics gives \Delta U = q - P \Delta V,
where \Delta U is change in internal energy, q is heat supplied,
P is pressure and \Delta V is change in volume.
It is given that equal heat is supplied to both gases, so q_{A} = q_{B},
and the change in internal energy is also same,
so \Delta U_{A} = \Delta U_{B}.
Using \Delta U = q - P \Delta V for each gas:
\Delta U_{A} = q_{A} - P \Delta V_{A} \Delta U_{B} = q_{B} - P \Delta V_{B}Since \Delta U_{A} = \Delta U_{B} and q_{A} = q_{B},
we get:
q_{A} - P \Delta V_{A} = q_{B} - P \Delta V_{B} \Rightarrow P \Delta V_{A} = P \Delta V_{B}The gases are at the same pressure,
so P is same for both.
Thus:
\Delta V_{A} = \Delta V_{B}The change in volume for a cylinder is given by
\Delta V = \text{area} \times \text{displacement} = \pi r^{2} h,
where r is radius and h is displacement of the piston.
For gas A: \Delta V_{A} = \pi r_{A}^{2} \times 16
For gas B: \Delta V_{B} = \pi r_{B}^{2} \times 9
Since \Delta V_{A} = \Delta V_{B}, we have:
\pi r_{A}^{2} \times 16 = \pi r_{B}^{2} \times 9Cancel \pi from both sides:
16 r_{A}^{2} = 9 r_{B}^{2}Rearrange to get the ratio of the squares:
\frac{r_{A}^{2}}{r_{B}^{2}} = \frac{9}{16}Taking square root on both sides:
\frac{r_{A}}{r_{B}} = \frac{3}{4