Question

Two soap bubbles of radii $a$ and $b$ combine to form a single bubble of radius $c$. If $P$ is the external pressure, then the surface tension of the soap solution is

• A. $\frac{P\left(c^3+a^3+b^3\right)}{4\left(a^2+b^2-c^2\right)}$
• B. $\frac{P\left(c^3-a^3-b^3\right)}{4\left(a^2+b^2-c^2\right)}$
• C. $P{c}^3-4a^2-4b^2$
• D. $P{c}^2-2a^2-3b^2$

Answer 1

Answer: (B)

Assuming isothermal conditions,
$\left(P+\frac{4\sigma }{a}\right)\left(\frac{4}{3}\pi a^3\right)+\left(P+\frac{4\sigma }{b}\right)\left(\frac{4}{3}\pi b^3\right)$
$=\left(P+\frac{4\sigma }{c}\right)\left(\frac{4}{3}\pi c^3\right)$
or $P\left[a^3+b^3-c^3\right]=4\sigma \left[c^2-a^2-b^2\right]$
or $\sigma =\frac{P\left(c^3-a^3-b^3\right)}{4\left(a^2+b^2-c^2\right)}$