Two soap bubbles of radii $x$ and $y$ coalesce to constitute a bubble of radius $z$ . Then $z$ is equal to:

\sqrt{x^{2} + y^{2}}

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\sqrt{x + y}

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x + y

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\frac{x + y}{2}

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Solution

The correct option is A $\sqrt{x^{2} + y^{2}}$
Let $S$ be the surface tension of the soap bubble.
Excess pressure inside the soap bubble, $P = \frac{4 S}{r}$ where $r$ is the radius.
When the two bubbles coalesce to form another bubble, then there is no loss of air inside the bubble.
From: $P V = n R T$ , we get $n = \frac{P V}{R T}$
$\Rightarrow n_{1} + n_{2} = n$

$\Rightarrow \frac{P_{1} V_{1}}{R T} + \frac{P_{2} V_{2}}{R T} = \frac{P V}{R T}$

$\Rightarrow P_{1} V_{1} + P_{2} V_{2} = P V$ $(∵ T = c o n s t a n t)$

$\Rightarrow \frac{4 S}{x} \times \frac{4}{3} π x^{3} + \frac{4 S}{y} \times \frac{4}{3} π y^{3} = \frac{4 S}{z} \times \frac{4}{3} π z^{3}$

$\Rightarrow x^{2} + y^{2} = z^{2}$

$\Rightarrow z = \sqrt{x^{2} + y^{2}}$

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