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Surface Tension

Two soap bubb...

Question

Two soap bubbles of radii $R_{1}$ and $R_{2}$ kept in atmosphere are combined isothermally to form a big bubble of radius $R$ . The expression for surface tension will be

A

\frac{P_{0} (R^{3} + R_{1}^{3} - R_{2}^{3})}{4 (R^{2} - R_{1}^{2} - R_{2}^{2})}

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B

\frac{P_{0} (R_{1}^{3} + R_{2}^{3} + R^{3})}{4 (R^{2} - R_{1}^{2} - R_{2}^{2})}

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C

P_{0} (R_{1}^{3} + R_{2}^{3} - R^{3})

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D

4 P_{0} (R_{1}^{3} + R_{2}^{3} - R^{3})

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Solution

The correct option is A $\frac{P_{0} (R^{3} + R_{1}^{3} - R_{2}^{3})}{4 (R^{2} - R_{1}^{2} - R_{2}^{2})}$
$\begin{matrix} A c c o r d i n g t o q u e s t i o n . . . . . . . . . . . . . . . . P . V = n R T \Rightarrow P_{1} V_{1} + P_{2} V_{2} = P V b u t, \Rightarrow P_{i n} - P_{e x t} = \frac{4 T}{r}, (P_{e x t} = P) \Rightarrow (P + \frac{4 T}{R_{1}}) \frac{4}{3} π R_{1}^{3} + (P + \frac{4 T}{R_{2}}) \frac{4}{3} π R_{2}^{3} = (P + \frac{4 T}{R}) \frac{4}{3} π R^{3} \Rightarrow P (R_{1}^{3} + R_{2}^{3} - R^{3}) = 4 T (R^{2} - R_{1}^{2} - R_{2}^{2}) ∴ T = \frac{P_{0} (R^{3} + {R_{1}}^{3} - {R_{2}}^{3})}{4 (R^{2} - {R_{1}}^{2} - {R_{2}}^{2})} \end{matrix}$ so the correct option is A

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