Why 2S/r for a drop but 4S/r for a soap bubble, and which way does mercury go in a capillary?

7 answers & replies

• [+][–]Meera Patel11thSample· 1mo ago

  A soap bubble is a thin film of liquid with air inside and air outside, so it has two liquid–air surfaces, the outer one and the inner one. Each surface adds an excess pressure of $2S/r$ (radius almost the same), so the total is $4S/r$.

  Why a drop has $2S/r$: with one surface. Derivation: if the radius increases by $dr$, the work done by the excess pressure is $\Delta P \cdot 4\pi r^2\,dr$, and this equals the increase of surface energy $S\times 8\pi r\,dr$. So $\Delta P = 2S/r$. For a bubble, the area has two surfaces, so you multiply by 2 and get $4S/r$. An air bubble inside a liquid has just one surface, so $2S/r$.

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  • [+][–]Diya Menon11thSample· 1mo agoAuthor

    ahhh two surfaces = inner and outer skin of the film. like a samosa wrapper that's very thin but still has two faces 😂 thanks Meera!

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• [+][–]Yash Tiwari11thSample· 1mo ago

  bhai wetting ka scene hai. Contact angle chhota (acute) ho to liquid glass ko geela karta hai aur upar chadhta hai, jaise paani. Contact angle bada (obtuse) ho to liquid glass ko geela nahi karta, neeche jaata hai, jaise mercury. Real life examples: tel lamp ki batti mein upar chadhta hai, blotting paper se syahi chus leta hai, aur ped ke jad se paani upar jaata hai, capillarity ki wajah se (kuch had tak).

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• [+][–]Kavya NaikDropperSample· 1mo ago

  Diya, your staircase guess is the opposite of what happens. The formula says $h \propto 1/r$. Thinner tube, higher rise. Surface tension acts around the circumference ($2\pi r$) to pull the liquid up, and what it has to lift is the column of weight $\pi r^2 h\rho g$. The pull goes with $r$, the weight with $r^2$, so a wider tube has comparatively too much liquid to lift.

  For mercury, it isn't just the density. The contact angle between mercury and glass is obtuse (roughly 140°), so $\cos\theta$ is negative and $h$ comes out negative: mercury is depressed, with a convex meniscus. Water on clean glass has $\theta$ close to 0°, so $\cos\theta \approx 1$ and the meniscus is concave.

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  • [+][–]Diya Menon11thSample· 1mo agoAuthor

    wait so thin tube = higher rise, my staircase idea was totally backwards 😂 thanks Kavya didi

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• [+][–]Aman YadavDropperSample· 1mo ago

  Bubbles joined by a tube. Excess pressure is $4S/r$, so the smaller bubble has the higher pressure. Air flows from the smaller to the larger one. The small one shrinks further, the large one grows. Counter-intuitive, I know, it's a standard MCQ trap.

  Different from a balloon, where the rubber gets stiffer as it stretches. A soap film's surface tension is constant, it doesn't depend on the area.

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• [+][–]Ishita GoyalDropperSample· 1mo ago

  A quick numerical for feel. Soap bubble, $r = 1$ cm, $S = 0.03$ N/m: $\Delta P = 4S/r = 4\times0.03/0.01 = 12$ Pa. Tiny.

  Water in a tube of radius 0.5 mm, $S = 0.073$ N/m, $\theta \approx 0$:

  $h = \frac{2S\cos\theta}{r\rho g} = \frac{2\times0.073}{0.5\times10^{-3}\times1000\times9.8} \approx 0.03\ \text{m} = 3\ \text{cm}$

  It's why the narrow capillaries in plants can pull water up a bit, while a normal glass doesn't climb anywhere.

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