Question

Two spherical soap bubbles collapses. If $V$ is the consequent change in volume of the contained air and $S$ is the change in the total surface area and $T$ is the surface tension of the soap solution, then if relation between $P_0,V,S$ and $T$ are $\lambda P_{0}V+4ST=0$, then find $\lambda$? (If $P_0$ is atmospheric pressure): Assume temperature of the air remain same in all the bubbles.

Answer 1

${\left(△P\right)}_{soapbubble}=\frac{4T}{R}$

Let radii of two spherical bubbles be $r_1\&r_2$ respectively.

Now, as two collapse to form a single.

then,

$\Rightarrow (P+\frac{4T}{r_1})\frac{4}{3}\pi r_1^3+(P+\frac{4T}{r_2})\left(\frac{4}{3}\pi r_2^3\right)=(P+\frac{4T}{R})\frac{4}{3}\pi R^3$

$\Rightarrow PV+\frac{4T}{3}(4\pi r_1^2+4\pi r_2^2-4\pi R^2)=0$

$\Rightarrow 3PV=4TS=0$

Hence, the answer is $3PV=4TS=0.$