Laplace pressure in a bubble: derivation using forces

By considering the balance of forces on a surface area element, we derive expressions for the excess pressure inside a spherical bubble in terms of the surface tension. We consider two different cases - a gas bubble surrounded by liquid, and a bubble surrounded by a thin liquid film floating in a gas.
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About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#physics #mathematics #pressure #bubble #equilibrium #forces #resolving #sphericalcoordinates #polarcoordinates #surfacearea #surfacetension #arclength #infinitesimal #balance #bubbles #laplacepressure #physicsproblems #maths #math #science #education

[@paxshild4924]

Dr. Ben, thanks for taking my suggestion and making a video about it. Your expertise really came through, and I appreciate you considering viewer recommendations. Keep up the great work!

[@DrBenYelverton]

There's another one coming soon which is actually closer to your suggestion!

[@franzmaina3080]

Love this one, nice work. We can arrive at this also using thermodynamics

[@geoffcheckan7240]

Been trying to find a really sound derivation of this and this is the only one I’ve located that really goes through it rigorously without making leaps in logic that obscure the explanation, great work!

[@DrBenYelverton]

Thanks for saying so, glad this was helpful. There are quicker ways to do this but I wanted to show that it still works out if you consider all the details!

[@orenpedatzur6070]

Thank you! This is the infinitesimal derivation I was looking for.

[@marcovillalobos5177]

really good, as usual!

[@marcovillalobos5177]

you revisit really classical and important problems from alternative points of view, and the sense of completeness at the end of the video is priceless... if you write someday a physics book, I'll buy it 100%

[@marcovillalobos5177]

The math and the intuition go together in a really good proportion

[@DrBenYelverton]

Thanks for your kind words and I'm glad you're still enjoying the videos! Never thought about writing a book but it's a nice idea, definitely something to consider.

[@kennarnett8220]

Fantastic! What graphics/interface software/hardware did you use for this presentation?

[@DrBenYelverton]

Thanks! The software is Xournal++ and the hardware is a One by Wacom graphics tablet.

[@MathwithMing]

Great problem as usual! I'll have to work it through the details myself later though because right now I'm confused about how surface tension is defined. Btw do you have any reference for such topic (namely surface tension) and perhaps this very problem?

[@DrBenYelverton]

Imagine changing the side lengths of our surface area element - intuitively, you can see that the longer the sides get, the greater the pulling force on each side will be. That's because the origin of this tension is ultimately the attractive force between the fluid molecules, and a longer side will have more molecules pulling on it. The only sensible way to quantify this tension is therefore as a force per unit length, which is exactly the definition of surface tension. No particular references come to mind but once you've understood the above definition hopefully things will fall into place!

[@MathwithMing]

@@DrBenYelverton Many thanks! Here is what I understood. Consider a surface element of the interface of two contacting mediums (e.g. air and water). The surface tension experienced by a `perimeter element' (for the lack of a better term) of that surface element is tangent to interface and is in the outward normal direction of the surface element. The magnitude of that force is proportional to the length of the perimeter element, and the proportionality constant is denoted by gamma, which, presumably, depends on the two contacting medium. If you could point out any mistake in my understanding that would be greatly appreciated! Thank you again for your work!

[@DrBenYelverton]

That sounds good to me! Just one comment, I think "the outward normal direction of the surface element" would be clearer as "the direction perpendicular to the perimeter element", since the normal direction to the surface element would really be perpendicular to the interface. I understood what you meant, but since surface elements are technically 2D they only have normals pointing "up" and "down", not to the sides!

[@MathwithMing]

@@DrBenYelverton Oops I meant to say the direction “is tangent to the interface and is in the outward normal direction of the *perimeter* element”! Thanks, I think I got it!

[@DrBenYelverton]

Makes sense!

[@lordmandarin7212]

Thank you! Could you also make a video in which you derive a formula for the stress acting on a balloon such that the express only depends on the thickness, the radius of the sphere and the material properties? If this is possible, of course

[@DrBenYelverton]

Perhaps this is what you're looking for? kzbin.info/www/bejne/aHOcipmXZpqohsk

[@lordmandarin7212]

@DrBenYelverton Unfortunately, no. Let me explain the problem: starting from 0 km, the balloon ascend until it reaches the desired height. Calling dp = helium press - air press, which of the following scenarios happens? 1. dp = 0 ----> dp increases as the balloon expands, until it bursts 2. dp >0 -----> ? Then, considering the fact that the thickness reduces, how do you compute the stress? In your expression for the stress, sigma = E*(r -r0)/r0, I don't see the initial thickness and the change in the pressure variation

[@GaneshKumar-wb7bz]

Sir please solve jee advanced physics problem and their concept