Question

Two soap bubbles combine to form a single bubble. In this process, the change in volume and surface area are $V$ and $A$ respectively. If $P$ is the atmospheric pressure, and $T$ is the surface tension of the soap solution, then which of the following relation is true?

• A. 4 PV + 3 TA $=$ 0
• B. 3 PV - 4 TA $=$ 0
• C. 4 PV - 3 TA $=$ 0
• D. 3 PV + 4 TA $=$ 0

Answer 1

The correct option is D. 3 PV + 4 TA $=$ 0.

Let the radii of two soap bubbles are $a$ and $b$ and radius of single large bubble is $c$. Let $P_a{P}_b,Pc$ are the internal pressures of soap bubbles a,b,c and P is the atmospheric pressure.

$P_a=P+4T/r_a$ ,$P_b=P+4T/r_b$ ,$P_c=P+4T/r_c$

$V_a=\frac{4\pi r_a^3}{3}$, $V_b=\frac{4\pi r_b^3}{3}$, $V_c=\frac{4\pi r_c^3}{3}$

as temperature remains constant (atmospheric temperature)

$P_a{V}_a+P_b{V}_b=P_c{V}_c$...(1)

change in volume $V=V_c-V_b-V_a$....(2)

change in surface area $A=4\pi (r_c^2-r_a^2-r_b^2)$....(3)

from 1, 2 and 3 equation we get

$\frac{3PV}{4\pi }+\frac{TA}{\pi }=0$

$3PV+4TA=0$