Question 144 of 168

Two spherical soap bubbles of radii ${r}_{1}$ and ${r}_{2}$ in vacuum combine under isothermal conditions. The resulting bubble has a radius equal to:

A$\dfrac{{r}_{1}{r}_{2}}{{r}_{1}+{r}_{2}}$B$\sqrt{{r}_{1}{r}_{2}}$C$\sqrt{{r}_{1}^{2}+{r}_{2}^{2}}$D$\dfrac{{r}_{1}+{r}_{2}}{2}$

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Answer: (C) $\sqrt{{r}_{1}^{2}+{r}_{2}^{2}}$

Number of moles is conserved.
${n}_{1}+{n}_{2}={n}_{3}$
${P}_{1}{V}_{1}+{P}_{2}{V}_{2}={P}_{3}V$
$\dfrac{4S}{{r}_{1}}\left(\dfrac{4}{3}\pi {r}_{1}^{3}\right)+\dfrac{4S}{{r}_{2}}\left(\dfrac{4}{3}\pi {r}_{2}^{3}\right)=\dfrac{4S}{{r}_{3}}\left(\dfrac{4}{3}\pi {r}_{3}^{3}\right)$
${r}_{1}^{2}+{r}_{2}^{2}={r}_{3}^{2}$
${r}_{3}=\sqrt{{r}_{1}^{2}+{r}_{2}^{2}}$

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