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:

Let the surface tension of the soap film be $$T$$. Because a soap bubble has two liquid surfaces, the excess pressure inside a bubble of radius $$r$$ in vacuum is given by the formula

$$P = \dfrac{4T}{r}$$

Here the outside pressure is zero (perfect vacuum), so the whole pressure inside the bubble is only this excess pressure.

For the first bubble of radius $$r_1$$ we therefore have

$$P_1 = \dfrac{4T}{r_1}$$

For the second bubble of radius $$r_2$$ we have in the same way

$$P_2 = \dfrac{4T}{r_2}$$

The volumes of the two initial bubbles are, respectively,

$$V_1 = \dfrac{4}{3}\pi r_1^{3}, \qquad V_2 = \dfrac{4}{3}\pi r_2^{3}$$

When the two bubbles coalesce isothermally, the gas contained in them is conserved. At constant temperature the ideal-gas relation $$PV = nRT$$ tells us that the product $$PV$$ is directly proportional to the number of moles of gas. Hence the total $$PV$$ before coalescence equals the final $$PV$$ after coalescence.

So we write

$$P_1 V_1 + P_2 V_2 = P_f V_f$$

where $$P_f$$ and $$V_f$$ are the pressure and volume of the single final bubble. Let the radius of this final bubble be $$R$$. Then

$$P_f = \dfrac{4T}{R}, \qquad V_f = \dfrac{4}{3}\pi R^{3}$$

Substituting all these expressions, we get

$$\dfrac{4T}{r_1}\left(\dfrac{4}{3}\pi r_1^{3}\right)\;+\;\dfrac{4T}{r_2}\left(\dfrac{4}{3}\pi r_2^{3}\right) \;=\;\dfrac{4T}{R}\left(\dfrac{4}{3}\pi R^{3}\right)$$

Every term on both sides contains the common factor $$\dfrac{4T}{\phantom{2}}\times\dfrac{4}{3}\pi$$, so we cancel it out completely. After cancelling we are left with

$$\dfrac{r_1^{3}}{r_1}\;+\;\dfrac{r_2^{3}}{r_2}\;=\;\dfrac{R^{3}}{R}$$

which simplifies step by step as follows:

$$r_1^{2} + r_2^{2} = R^{2}$$

Now taking the square root of both sides (and keeping only the positive root because radius is positive) we find

$$R = \sqrt{\,r_1^{2} + r_2^{2}\,}$$

This expression exactly matches Option C.

Hence, the correct answer is Option C.