First slide

Surface tension and surface energy

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

Moderate

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

$P c^{3} - 4 a^{2} - 4 b^{2}$
D

$P c^{2} - 2 a^{2} - 3 b^{2}$

Solution

Assuming isothermal conditions,
$\begin{array}{l} (P + \frac{4 σ}{a}) (\frac{4}{3} π a^{3}) + (P + \frac{4 σ}{b}) (\frac{4}{3} π b^{3}) \\ = (P + \frac{4 σ}{c}) (\frac{4}{3} π c^{3}) \\ or P [a^{3} + b^{3} - c^{3}] = 4 σ [c^{2} - a^{2} - b^{2}] \\ or σ = \frac{P (c^{3} - a^{3} - b^{3})}{4 (a^{2} + b^{2} - c^{2})} \end{array}$

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