I'm studying Kardar's "Statistical mechanics of particles" book and tackled a problem. After solving it, I checked Kardar's solution and found that he has different approach. I'm interested in the community's opinion on the correctness of my solution compared to Kardar's approach.
The problem asks to estimate length scale when surface tension become comparable with gravitational forces, given surface tension $S = 7 \times 10^{-2} \, \text{N/m}$ for water at room temperature. Kardar approaches this by comparing the energies associated with surface tension and gravity: $$E_{\text{surface}} = E_{\text{gravity}}$$ For a spherical bubble we have: $$S \cdot 4\pi R^2=mgR$$ Meanwhile, I examined the balance between the additional pressure inside a water bubble, due to surface tension, and the pressure at the center of the bubble resulting from gravitational forces: $$\frac{2S}{R} = \rho \cdot g \cdot R$$
I'm curious, is the pressure-based method valid, and how does it conceptually compare to the energy-based reasoning in terms of accuracy and physical insight?
