Two spherical soap bubbles of radii r _ 1 and r _ 2 in vacuum...

Number of moles is conserved.

${n}_{1} + {n}_{2} = {n}_{3}$

${P}_{1}{V}_{1} + {P}_{2}{V}_{2} = {P}_{3}V$

$\dfrac{4S}{{r}_{1}}\left(\dfrac{4}{3}\pi{r}_{1}^{3}\right) + \dfrac{4S}{{r}_{2}}\left(\dfrac{4}{3}\pi{r}_{2}^{3}\right) = \dfrac{4S}{{r}_{3}}\left(\dfrac{4}{3}\pi{r}_{3}^{3}\right)$

${r}_{1}^{2} + {r}_{2}^{2} = {r}_{3}^{2}$

${r}_{3} = \sqrt{{r}_{1}^{2} + {r}_{2}^{2}}$