A closed hollow insulated cylinder is filled with gas at $0^{o} C$ and also contains an insulated piston of negligible weight and negligible thickness at the middle point. The gas on one side is heated to $100^{o} C$ . If the piston moves through 5 cm, the length of the hollow cylinder is

13.65 cm

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27.3 cm

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38.6 cm

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64.6 cm

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Solution

The correct option is D
64.6 cm

Let L be the length (in cm) of the follow cylinder and r its radius. Since the mass of the gas remains unchanged and the pressures of the gas in both sides are equal, we have, from Charles' law,
$\frac{V_{1}}{T_{1}} = \frac{V_{2}}{T_{2}} . . . . . . (1) G i v e n V_{1} = (\frac{L}{2} - 5) π r^{2}, V_{2} = (\frac{L}{2} + 5) π r^{2}, T_{1} = 0^{o} C = 273 K a n d T_{2} = 100^{o} C = 373 K .$ Using these values in (1), we get
$\frac{\frac{L}{2} - 5}{273} = \frac{\frac{L}{2} + 5}{373}$
which gives L = 64.6 cm.

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