A soap bubble of radius r exists inside another soap bubble of radius R > r, at atmospheric pressure P₀. For surface tension T and T/(RP₀), T/(rP₀) ≪ 1, the change in radius of inner bubble after outer bubble bursts is Δr = 4Tr/(nP₀R). Find n. [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

A soap bubble of radius r exists inside another soap bubble of radius R > r, at atmospheric pressure P₀. For surface tension T and T/(RP₀), T/(rP₀) ≪ 1, the change in radius of inner bubble after outer bubble bursts is Δr = 4Tr/(nP₀R). Find n. [See Image.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Initial pressure inside inner bubble is approximately P₀ + 4T/R + 4T/r. After outer bubble bursts, pressure is approximately P₀ + 4T/(r+Δr). Step 2 — Assuming the gas in the smaller bubble is isothermal, PV = constant and small changes satisfy ΔV/V ≈ −ΔP/P₀. Step 3 — Outer-shell pressure decreases by 4T/R. Therefore 3Δr/r ≈ 4T/(P₀R). Step 4 — Δr ≈ (4Tr)/(3P₀R). Compare with (4Tr)/(nP₀R): n = 3.