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 Surface Tension of the soap solution is:

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

Answer 1

Answer: (B)

$\frac{P(c^3-a^3-b^3)}{4(a^2+b^2-c^2)}$

The ideal gas equation is:

$P.V=nRT$

For an isolated system:

$\Rightarrow P_1{V}_1+P_2{V}_2=PV$

But according to the definition of surface tension:

$Pin-Pext=\frac{4T}{r}$, ($P_{exr}=P$)

Substitute the value of pressure and volume:

$(P+\frac{4T}{a})\frac{4}{3}\pi a^3+(P+\frac{4T}{b})\frac{4}{3}\pi b^3=(P+\frac{4T}{c})\frac{4}{3}\pi c^3$

$\Rightarrow P(a^3+b^3-c^3)=4T(c^2-a^2-b^2)$

$\Rightarrow T=\frac{P(c^3-a^3-b^3)}{4(a^2+b^2-c^2)}$