Savchenko Problem 5.5.34

The key insight is that surface tension creates an extra pressure inside each bubble—and when two bubbles merge, that pressure difference must be conserved through energy balance. A soap bubble has two surfaces (inner and outer), so the pressure excess is $\Delta P = \frac{4\sigma}{R}$ for radius $R$. When the bubbles combine, use the fact that the total volume is conserved and the final pressure relates to the initial pressures through the Young-Laplace equation to solve for $\sigma$.

The key insight is that surface tension creates an extra pressure inside each bubble—and when two bubbles merge, that pressure difference must be conserved through energy balance. A soap bubble has two surfaces (inner and outer), so the pressure excess is $\Delta P = \frac{4\sigma}{R}$ for radius $R$. When the bubbles combine, use the fact that the total volume is conserved and the final pressure relates to the initial pressures through the Young-Laplace equation to solve for $\sigma$.