Excess pressure inside a soap bubble

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

Excess pressure inside a spherical soap bubble is 4S/R .

How to deal with the cylindrical shape ? Why does the question specify it to be "long" ?

 

[Charles Link]

To achieve equilibrium, I think you need to have the work done by the gas if the bubble expands be equal to the energy needed to expand the cylindrical bubble because of surface tension. And in the case of a bubble, there is an inside surface and an outside surface. Are you familiar with the equation $dW=S \, dA$ for surface tension? (I believe I have this correct, but I'm going to need to google it, because only on occasion have I worked a surface tension problem). $\\$ Edit: And yes, the cylindrical shape does get a different answer than the spherical shape, and I have also verified the calculation gets the correct answer for a sphere. You need to write $dW=(P_2-P_1)A \,dr=S \, dA'$, where $A=(2 \pi r) \, L$ and $A'=2 \, ( 2 \pi r) \, L$. Does that seem logical? $\\$ Using the same type of calculation, you can also derive the result for a spherical bubble that you mentioned, where $A= 4 \pi r^2$, and $A'=2 (4 \pi r^2)$. Bubbles are normally spherical, because the spherical shape has minimum surface area for a given volume. Thereby, the potential energy from the surface tension for a given volume is minimized.

 

[Chestermiller]

If you slit the cylinder in half lengthwise and do a force balance on one of the halves, you have the following:

Pressure force on the half cylinder = $\Delta P (2RL)$

Surface force on the half cylinder = $4SL$

 

[haruspex]

The premiss for the question is quite non-physical. There is no way the bubble would even approximate a cylinder. The pressures would have to be non-uniform. So while it is possible to obtain an answer, it is rather meaningless.
I think it might have made sense if it had been specified as very short, so just a circular band between parallel plates.

 

[Jahnavi]

You need to write $dW=(P_2-P_1)A \,dr=S \, dA'$, where $A=(2 \pi r) \, L$ and $A'=2 \, ( 2 \pi r) \, L$. Does that seem logical? $\\$

What is A' and dA' ?

 

[Jahnavi]

If you slit the cylinder in half lengthwise and do a force balance on one of the halves, you have the following:

Pressure force on the half cylinder = $\Delta P (2RL)$

Surface force on the half cylinder = $4SL$

Nice

Why does cylinder have to be "long" ?

 

[Charles Link]

What is A' and dA' ?

$A$ is the surface area of the sphere or cylinder. $A'$ is the surface area that forms the surface tension. For a spherical droplet $A =A'$, but for a soap bubble with two surfaces,(inside and outside), and also for this cylindrical bubble that has two surfaces, $A'=2 A$.

 

[Jahnavi]

OK .

But how is A and A' appearing in the same expression i.e in dW = P2 - P1 = Adr = SdA' ?

 

[Charles Link]

$dU=S \, dA'$ is the potential energy obtained when the surface area $A'$ is increased by $dA'$. Meanwhile $dW= (P_2-P_1)A \, dr$ is the work available (to do work on increasing the surface area $A'$), as the bubble is expanded with $r$ increasing by $dr$. When the conditions are such that these two quantities (differentials) are equal, you will have equilibrium. (This is similar to setting forces equal, where $-\nabla U=\vec{F}$ etc.). (When $dW >dU$ the bubble will continue to expand, etc., until $dW=dU$).$\\$ You could also think of it as being a system that consists of two springs: $dW=F_1 \, dr$, (where $F_1=(P_2-P_1)A$), describes one spring system created by the excess pressure of the gas inside, and it is counterbalanced by what is essentially an elastic type material that has a potential energy $U$ that obeys $dU=S \, dA'$ The problem is to balance these two elastic forces. $\\$ @Chestermiller has an alternative and clever approach to set these forces equal, since the surface tension force is really at right angles to the pressure force on the walls of the bubble.

 

[Jahnavi]

In the expression for dW , A would be the outer surface area of the bubble , the one that is exposed to atmospheric pressure . Right ?

 

[Charles Link]

In the expression for dW , A would be the outer surface area of the bubble , the one that is exposed to atmospheric pressure . Right ?

Yes.

 

[Jahnavi]

Why do you think the cylinder has to be "very long " ?

 

[Charles Link]

Why do you think the cylinder has to be "very long " ?

They were just trying to try to avoid the effects of putting end faces on it. It really doesn't need to be long. It can be a cylindrical bubble contained between two metal discs as @haruspex points out above. I agree with his input that otherwise, a cylindrical bubble is very hypothetical and simply does not occur in the real world. In many ways, these equations could be describing a cylindrical (elongated) balloon though, so it can be worthwhile to perform the computation as it may have other real world applications. (The tension in a balloon will increase with increasing dimensions, so I am only making a loose comparison here).

 

[phinds]

I've seen soap bubbles somewhat more cylindrical than this and longer as well. Granted they are hardly perfect cylinders, but they are close (except being clearly rounded on the ends. They don't STAY that way for long but they do exist.

 

[Jahnavi]

They were just trying to try to avoid the effects of putting end faces on it. It really doesn't need to be long. It can be a cylindrical bubble contained between two metal discs as @haruspex points out above. I agree with his input that otherwise, a cylindrical bubble is very hypothetical and simply does not occur in the real world. In many ways, these equations could be describing a cylindrical (elongated) balloon though, so it can be worthwhile to perform the computation as it may have other real world applications. (The tension in a balloon will increase with increasing dimensions, so I am only making a loose comparison here).

OK .

Thanks @Charles Link and @Chestermiller .

 

[Charles Link]

It could also be worth mentioning that here we are writing equations to try to describe a bubble in equilibrium or near equilibrium. As @phinds has shown, other bubble forms are possible, and our equilibrium equations could be good approximations for these cases as well.