Question

Two soap bubbles of radius R1 and R2 combine to form a single soap bubble isothermally. What is the radius of single soap bubble?​

Answer 1

Answer:

$\boxed{\mathfrak{R = \sqrt{ {R_1}^{2} + {R_2}^{2}}}}$

Explanation:

Radius of two soap bubbles is given as $\sf R_1$ & $\sf R_2$.

Let the radius of soap bubble after combining both bubble to form one soap bubble be R.

Excess pressure inside soap bubble = $\sf \dfrac{4T}{R}$

T → Surface Tension

As the soap bubble are combined isothermally. i.e.

PV = constant

$\sf \implies P_1 V_1 + P_2 V_2 = PV \\ \\ \sf \implies \dfrac{ \cancel{4T}}{R_1} \times \cancel{\dfrac{4}{3} \pi} {R_1}^{3} + \dfrac{ \cancel{4T}}{R_2} \times \cancel{\dfrac{4}{3} \pi} {R_2}^{3} = \dfrac{ \cancel{4T}}{R} \times \cancel{\dfrac{4}{3} \pi} {R}^{3} \\ \\ \sf \implies \dfrac{ {R_1}^{3}}{ R_1} + \dfrac{ {R_2}^{3}}{ R_2} = \dfrac{ {R}^{3} }{ R} \\ \\ \sf \implies {R}^{2} = {R_1}^{2} + {R_2}^{2} \\ \\ \sf \implies R = \sqrt{ {R_1}^{2} + {R_2}^{2}}$