Hollow vs Solid Expansion
We have two copper spheres of the same radius; one is hollow (vacuum inside) and the other is solid. Initially, both spheres are in a vacuum at the same temperature. Then they are heated up to the same temperature which causes them both to expand.
Which sphere has the greater final radius?
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3 solutions
This problem seems ambiguous, because the exact meaning of "expands more" isn't defined. If the question is strictly asking about proportional change in volume for the copper material, then yes they expand by the same amount. But in order for that to occur, I think the radius of the solid sphere and the outer radius of the spherical shell will no longer be equal after the expansion, because the spherical shell will expand both inward and outward, while the solid sphere will only expand outward. If that's the case, then while the proportional change in the volume of copper in both objects will be equal, the solid sphere will now have a larger radius, and thus could in a sense be considered to have expanded more. Am I wrong about that?
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The inner radius of the hollow does not decrease; it increases! On heating, the hollow sphere scales up uniformly, so both the inner and the outer radii of the hollow sphere increase, and that too by the same fraction. Note that the fractional change in volume would be the same for both spheres, and the final radius would also be the same for both spheres.
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Ah okay. That makes sense. I had to look at the algebra around the spherical shell volume increase for a bit to see that you will indeed get the same proportional volume increase in the spherical shell if you increase both the inner and outer radii by the cube root of the factor by which the volume is meant to increase. Thanks. This was very helpful. 😊
Shouldnt the Vo in the hollow sphere, be the volume of the sphere minus the volume of the vacuum inside it?
Can't the hollow sphere expand inwards ? I think so and I think that would apparently lessen it's total expansion than the solid one which cannot expand inwards as there's no space.
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But expanding inwards means the inner shell is contracting as the radius decreases. This means that the atoms on the inner surface are getting closer together, which isn't expanding. They need to expand too, which means that the inner radius, or hollow sphere inside, is also increasing.
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Provided that it heats up uniformly, there's going to be a pressure from the inside of the solid sphere as those atoms also try to displace their neighbors. If there were two hollow spheres at standard temperature, with one hollow sphere also containing 1ATM of air, and the other with a vacuum, the diameter of each would vary if either the temperature changed or the external pressure changed, if for no other reason than the pressure internally would be different. That is a force which cannot be ignored. It may not be significant, it may even make sense to model them the same in a certain range of values, but they musn't be the same. At 1084 °C they've both melted and this has become a different question, but at 1082 °C they would have very different internal pressures just before melting point.
Please everyone read fully:
These conclusions are wrong. Just because you looked up a formula in a book, it doesn't mean that you understand it's applications or that the formula accurately applies to real world problems. The mistake you are making is assuming that a hollow sphere has the same proportionality constant as a solid sphere just because they are made of the same material. Also, the problem is missing a key piece of information that will change the answer.
This problem should be reworded to include whether or not the inside of the hollow sphere contains gasses or is a vacuum as this does indeed affect the answer.
Gasses expand much larger than solids do. As the temperature changes, any gas inside the hollow sphere will exert an additional outward force that isn't accounted for in the given formula, unless the proportionality constant is correctly changed, but it would indeed cause the hollow sphere to expand larger than the solid sphere or even explode.
However, if the inside of the sphere is a vacuum, the exact opposite will occur. When a solid, sealed object expands with a vacuum inside of it, there will be a force resisting expansion applied on the sphere by the vacuum, which again can only be adjusted for by changing the proportionality constant. When the hollow sphere expands, it will create negative pressure that exerts an inward force on the hollow sphere that the solid sphere will not experience, and therefore the solid sphere will expand more.
You were correct when you said the expansion only relies on the temperature change, initial volume, and the nature of the material, but you were incorrect in assuming the nature of a solid sphere is the same as the nature of a hollow sphere, whether it has gasses or a vacuum within.
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That's what I said lmao need a combination of the proper equation with intuition about the scenario, the equation only works for solids with similar chemical composition
There is 3rd casein your setup that is if it is in a vacuum as well as a vacuum inside the hollow sphere in this case there would be no external or internal pressures and the volume of copper could expand both internally and externally the volume of the two spheres would expand proportionally but since the volume of the hollow sphere is made of the internal radius subtracted from the external radius and can expand inwards it would not achieve a final external radius as large as the solid sphere that can only expand externally.
The question specifically states that there is a vacuum inside the hollow sphere
You assume the proportionality constant, gamma, of the volume expansion formula is equal for both spheres, yet the materials of the spheres are different. The only equal expansions are the copper shell and the corresponding shell of the filled sphere. We must then consider the gas inside the shell versus the copper inside the filled sphere. It might not be that simple because gas might not push very forcefully on the shell, I'm not sure whether the copper inside or the gas inside will push harder. What I am sure of is that they will not push equally. Hope this doesn't upset anyone, but I think the answer key is wrong. :#
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This is exactly right.
How is copper different from copper?
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air is different than copper, the parts of the sphere that are identical expand the same, but the insides are different so they expand differently. Please read the whole comment before you retort, I had already addressed your question before you asked it
But the hollow sphere have air inside it
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The problem itself speaking in vaccum.
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Assumption is the mother of all.... So would we need to assume that the spheres remained in the vacuum, or could we assume otherwise? The ambiguous use of the word "initially" in a vacuum may imply that, when heated, they were removed from the vacuum. If they were removed, which would expand further?
How much are we heating this copper? If we get close to the melting point of 1085 degrees C, we can expect some size increase of the hollow sphere (due to grain boundary diffusion?). This is similar to heating a wire to close to melting point while under tension.
I don't like the question, because I don't have enough theoretical background information to answer it.
I disagree, I believe that the solid sphere will reach a slightly bigger diameter, even though locally the expansion is the same, the hollow sphere doesn't have internal resistance, so will expand towards the inside a little, instead of solid sphere needs to expands just towards the outside
Think of atoms as laying in a grid (actually a 3D-lattice).
When you heat them up, the (average) distance between atoms is increased, so the whole grid is scaled.
This means the sphere actually gets to the same outer radius at the end.
This solutions presume a perfect, near crystalline, lattice. The greater answer would be "how was this hollow sphere created so perfectly and how could NASA buy one for further material testing?"
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well, the other solutions assume the volume growth is linear with temperature, which is also false... We all need to make some assumption in order to give a solution (the question does not state clear assumptions). The greater comment would be "how hollow is the hollow sphere?"
what about the Thermal expansion of the air inside the hollow sphere, wont it generate additional expansion because of the increased preassure inside the hollow sphere.
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the thermal expansion of air is less than that of copper, so it would actually slow it down (but I doubt the effect would be significant, unless the sphere is so thin that it would be affected by pressure). IMO, there are important assumptions missing from the text of the question, just report this as a problem.
there is no air in a vacuum
Thermal expansion works by uniformly "scaling up" an object by factor 1 + α Δ T where α is coefficient of (linear) thermal expansion and Δ T is change in temperature. This means that distance between any two points of the object is multiplied by this factor. As a result the object is bigger but not deformed in any way. Since the two copper spheres had the same radii before heating up, they will have the same radii after it.
Why doesn't the hollow sphere "expand inwards"?
It may help to think about what happens to the copper on microscopic scale. When it is heated up the distances between adjacent atoms increase by the factor 1 + α Δ T . Now imagine a ring of equidistant atoms on the inner surface of the hollow sphere. Let's denote the distance between adjacent atoms a 0 . After heating the interatomic distances increase so the distance between adjacent atoms on the ring should become a 0 ( 1 + α Δ T ) . But this can only happen if radius of the ring also increased by 1 + α Δ T . This must be true for any such ring, so the hollow must have expanded.
Another way to look at this would be to imagine a smaller solid sphere that fits perfectly into the hollow one so together they form a solid sphere. Now heat up this 2-part solid sphere (hollow sphere + smaller sphere inside). It will uniformly expand just as a proper solid sphere would, the small sphere still perfectly fits into the hollow one. And since the smaller sphere has expanded the hollow must have expanded as well.
Or imagine building a house out of many identical bricks. And after that building its exact brick-for-brick copy out of larger bricks. Will the second house have smaller or bigger rooms?
I can't post, but would like to comment... I read "initially both are in a vacuum" and assuming they stay in the vacuum, the expansion would be identical. But with the linguistic use of "initially", it lead me to assume that when they were heated that they were removed from the vacuum, and if so, the air in the hollow copper sphere would expand like a balloon being blown up, more than the solid copper sphere, hence I got it wrong. Assumption is the mother of all.... So would we need to assume that they remained in the vacuum, or could we assume otherwise?
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How would the air get into the hollow sphere? Furthermore, whether the spheres are in a vacuum and what (if anything) is inside the hollow one are independent concepts.
i got it wrong, but i thought to myself that since the same amount of temperature is received by the spheres and since the hollow one is less dense ( contains less copper) so less copper will receive more temperature, so it will expand more. what is wrong with this reasoning?
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They do not receive temperature. They are heated up to a certain temperature and to get there need different amounts of energy. So in the end the temperature difference for both sphere is the same, but the energy difference isn't.
ps. I also got the answer wrong, since I forgot about the vacuum and thought brought the gas expansion of the hollow sphere's content in account as well.
Me too thought in dis way. Got it wrong
The "smaller solid sphere inside a larger hollow one" analogy seems flawed to me, as it already assumes the question that a solid sphere has the same outward expansion as a hollow one. In a solid sphere, all the expansion must be in the outward direction, as there is no inner space at all. The solid ball's expansion will necessarily fill all the available space, and therefore "fit perfectly," but will the overall outer diameter of the pair be the same as a hollow ball alone?
Really cool intuitions for thinking about the problem. Thanks!
My intuition, obviously wrong, was this: The circumference of the sphere must dilate faster than the radius: dS = d(2 pi R ). Thus, the surface if the sphere would be under tension, and resist the radial expansion...
The hollow sphere has less mass so it will get hotter once they both have the same properties, right? Wouldn't it be a case on which the hollow one could expand more?
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This was my conclusion. The solid one wouldn’t heat up as much as the hollow one, so the particle expansion so to speak would be greater for the hotter object; the hollow sphere. Maybe by “heat” they presumed both were heated to a constant temperature (as impossible as that may be in reality). But I still think we were right for what we’d actually see in the physical world.
I thought solid would expand more, since it can only expand outwards, the inside of it is already filled. It simply has no space on the inside. While the hollow copper sphere would expand its atoms in both ways, inside and outside. Thus, a solid one would have a greater radius than the hollow one. Bearing in mind, it's in vacuum and no external or internal pressures are present, the hollow sphere, to my mind, would expand on the inwards too.
Someone could comment on this?
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You are considering only expansion along a radial axis. The heat will force the atoms apart in tangential directions as well. If the atoms expanded inwards, then those lining the inside of the hollow sphere would become more tightly packed together, and therefore it wouldn't reach an equilibrium.
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Thank you Stewart. I was grappling with the same idea as Andrejs, and your explanation, plus Karel's example above about the brick house, helped this make sense to me.
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@Amber Baker – I had the same thought as Andrejs. Stewert's response makes sense, but what if there is more then enough room on the inside for a disproportional expansion, so that the inner expansion of the atoms use disproportional more inner space, and the distance between the inner atoms is still evenly distributed? (They would get less tightly packed because they use more space).
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@Glenn Mungra – Correction: the sphere would become smaller on the inside, so some of the atoms atoms would have to drop out from the area of a larger sphere into the area of a smaller sphere on the inside. If there is enough room for inward expansion, the pressure and heat on the inside would not increase.
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@Glenn Mungra – If I look at it the other way around: if there is no room for inward expansion the outward expansion would greater. More inside atoms would be pushed out into a sphere with a greater circumference in order to maintain the same relative distance to each other.
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@Glenn Mungra – So in that case the solid sphere would expand more.
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@Glenn Mungra – But I don't know if there is a vacuum inside the hollow sphere or if the heating will expand the gasses inside the hollow sphere. That would change the inner expansion more or less.
As far as I remember, copper belongs to metals. Metals form a crystal lattice, whereas atoms are ubiqutously spaced relative to each other, whether radial or tangential axis. Well, at least there is a rigid structure and no matter how we move an atom in a crystal lattice in a 3D space, it will try to exert force to a neighbouring atom.
In a crystal lattice, each copper atom when pushed by another atom, whatever the spacing - horizontal, vertical or diagonal, will try to repel or move the next standing one to a certain degree, depending on the chemical material it is made of. In other words, it will move the other atom as much as it has been moved with a slight attenuation force due to friction and resistance as we go further along.
Solid sphere. In a solid, the space is already taken at max. When pushed to max density on the inside, the only way it can move, would be outwards. While the most outer layer will heat up, it will expand and free up space to the internal layers, which will begin to heat up. Since the crystal lattice is arranged in such a way, that it not only repels the atoms, when they are pushed towards each other, but also binds and holds them together, when stretched apart from each other. And since heating up is a timely process, not a spontaneous one, the heat transfer will continue outwards, dragging more atoms out.
The hollow one. It has free space on the inside and since there is no pressure in a vacuum, it will expand isometrically (by the way, it doesn't state if the hollow space is vacuum or has air). The atoms that are sitting outwards would expand by the same token as for a solid, to my mind. The difference for a hollow sphere is, that it will expand both ways, first outwards as temperature first will heat up the outer layers first, and then, the closer we get to heating the internal layers on the inside, the atoms would expand on the inside too because there is untaken free space on the inside and less forces are required to push other atoms inside, than to push other atoms on the outside. Here, the atoms will still be bound together and try to hold on to each other as they expand until a critical temperature point or melting point is reached. Copper also belongs to thermal conductive materials, i.e. easily transfers heat (reason why pipes, heat exhangers and ventilation systems are usually made out of copper). So, if we apply heat, copper will try to distribute heat evenly throughout all of its layers first before moving any atoms, whether in or out. But as more and more heat would be applied it would expand both on the inside and outside.
Conslusion. Since a hollow sphere would "lose" some of its atoms travelling to the inside, it would expand less than a solid one. Well, that's how I see the story about two brothers. Might be equal actually.
The values I am going to use are for illustration purposes and are not indicative of actual possibilities but I believe the general idea is sound.
I do not agree with your premise that just because the distance between atoms increases that the radius must also increase.
Imagine this scenario you have a very thin ring just 2 atoms thick but of a very large outer radius say 2 miles, the inner radius is 2 miles - 2x thickness of atom. When this is resting the atoms align in a crystal lattice such that they are equidistant from two others ,',',',',',',',',',' like this. you could draw an imaginary line between them to make a circle that would be 1/2 the distance between all of them of a radius 2 miles - 1 atom thick. Now heat is applied so that the atoms get excited and double the distance between them. They will expand and influence each other to separate but they will retain the crystal lattice of equi-distance from the other two like this , ' , ' , ' , '. But the imaginary line you put between them is still in the same location directly between the two atoms as they have uniformly expanded away from one another so that now the outer radius is 2M + 1atom the original separation the imaginary circle remains at 2 miles - 1atom now the inner radius obviously cannot get bigger inwards but what determines the shift it is equally likely that an atom will expands outwards as inwards as the space is taken up so until enough get forced inwards to where there is more influence going out the ring could expand in any direction some in some out. I do not see how it would be restricted to only one direction.
Take it one step further and say the ring is 1 atom thick when heat is applied which direction do they go???
When both spheres are heated they will undergo the cubical expansion(Volume Expansion) and increase in the volume or shape is determined by the relation Δ V = γ V o Δ T c c c c where γ is proportionality constant The above relation shows that the increase in the volume of sphere depends upon the nature of material, initial volume V o and increase(change) in the temperature but doesn't depend upon the whether sphere is solid or hollow. All the factors responsible for increase in volume for both shperes are same which shows that both will equally expand by same amount on heating irrespective of theirs shapes.