Why does the cavity in a lead sphere increase on heating?

[hale2bopp]

There is a spherical lead ball with a spherical cavity. It is heated. The size of the cavity increases. Why does it do so?
Also, does it matter what happens to the cavity if it is not present at the centre, but rather a little removed from the centre? Will it remain spherical?

 

[AliAhmed]

- In general, metals expand upon heating. For the moment, let's assume the following:

1) The lead sphere contains no impurities or vacancies;
2) It is heated uniformly;
3) The rate of expansion due to the heat on the metal is proportional to the diameter of the sphere. Since there is a cavity, let's consider the problem as two independently expanding spheres (i.e. the inner sphere and the outer sphere each expand independently);
4) There may be air or some other ideal gas within the cavity (I doubt it would be vacuum within the cavity since if it were, the lead sphere would probably collapse since lead is quite soft).

- The expansion will thus be caused by both the thermal expansion of the lead itself and the increasing pressure of the gas within the cavity. The thermal expansion arises due to the stretching of bonds within the lead (the higher the temperature, the more energy is supplied to the atoms and their bonds, the larger their equilibrium bond length).

- The thermal expansion of each sphere (the inner and outer ones) may be computed by the equation:
ΔD/D = αΔT; α for lead is about 2.9*10^-5K^-1
i.e. The change in diameter of the sphere divided by the initial diameter of the sphere is proportional to the change in temperature by the constant α (called the coefficient of linear thermal expansion).

- The pressure within the cavity (if it may be treated as an ideal gas) is given by:
PV = nRT; R = 8.31 J/(mol*K)
- In the above equation, P is the pressure within the cavity, V is the volume of the cavity, n is the number of moles of gas within the cavity, and T is the temperature. This pressure will act as a stress on the sphere and thus the sphere will stretch in proportion to its elastic modulus (which will also change with temperature).

- If there were impurities and vacancies within the lead, the heat might not distribute uniformly and the expansion might not be very predictable.

- I apologize if my reply is too technical (or not technical enough), but I will be more than glad to explain any concept more clearly.

 

[marcusl]

I think Mr. Ahmed's explanation misses the point. If you heat a flat washer, the hole expands and obviously air pressure plays no role whatsoever. You can understand it from geometry and linear coefficient of thermal expansion.

Imagine heating a hoop of infinitely thin wire. The wire expands along its length, increasing the hoop's diameter. A physical hoop has finite thickness, so place another hoop inside of the first, and another inside of that, etc. All of them expand outwards with heat. Now rotate through pi and you have a sphere whose cavity expands upon warming. No pressure required.

 

[hale2bopp]

First off, I want to thank you for giving me a detailed answer.
I do have a few questions.
You assumed that the two cavities are expanding independently. The solid sphere expands as ΔD/D=αΔt , where t is temperature. The inner cavity contains some gas which expands and exerts some pressure on the walls of the cavity and also makes it expand.
The coefficient of linear expansion will definitely increase the outer diameter of the sphere; I am not very clear why (except for the pressure exerted by the gas expansion) the inner diameter should increase.
Why doesn't the sphere expand both ways-inside the cavity and out?
One very frequent answer which was given was that since all linear dimensions increase, even the diameter of the inner sphere should increase.
Is it because, as you said, when we give heat to a body, the molecules start vibrating more and their bond length increases, and if we expect that the cavity decreases, the bonds which are at the surface of the cavity would have reduced bond lengths?

 

[hale2bopp]

Imagine heating a hoop of infinitely thin wire. The wire expands along its length, increasing the hoop's diameter. A physical hoop has finite thickness, so place another hoop inside of the first, and another inside of that, etc. All of them expand outwards with heat. Now rotate through pi and you have a sphere whose cavity expands upon warming. No pressure required.

That is the question. Sure, the hoop has a finite thickness; but, that doesn't stop objects from expanding. After all, the ring is not being pulled out, it stretched out-it is being expanded. It's volume is increasing. So I don't see why it can't expand both ways, to some limit?

 

[marcusl]

The hole expands just as if it were made of the same material as the shell. See, e.g.,
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thexp2.html
and
http://physics.bu.edu/~duffy/py105/Temperature.html

 

[AliAhmed]

Firstly, I want to apologize for being unclear.

I never meant to say that the pressure is what is causing the expansion phenomenon, but rather that the pressure within the cavity will itself cause some expansion (though it will likely be small compared to the actual thermal expansion of the metal itself).

The pressure is not the driving force for the thermal expansion, it is temperature and material properties that are responsible. Again, when temperature increases the entire sphere will expand (both the inner and outer diameters will increase). In fact, the outer diameter will increase by a larger amount than the inner diameter (if you consider the thermal expansion equation).

All in all, what I meant to say is that pressure is one factor causing expansion while the thermal properties of the material are another factor (most likely the larger contributor to the expansion).

Again, if I am unclear I will be glad to explain. Also, I apologize if I didn't quite answer your question(s) (I am new to the forum and stiff familiarizing myself with the works).

 

[A.T.]

It's volume is increasing. So I don't see why it can't expand both ways, to some limit?

It's volume is increasing uniformly. The only deformation that achieves that is uniform scaling. And that increases all lengths uniformly, including the inner diameter.

It is a rigid body, not a sack of gas. So increase in volume is not the only boundary condition. It also tries to preserve it's shape. Any deformation other than uniform scaling would change the shape (length ratios) and create internal stresses.

 

[hale2bopp]

Thank you for all your help.
marcusl, thank you for the links. They were very helpful.
So, in conclusion-the inner radius increases to avoid stresses associated with it reduces, and because otherwise the expansion will not be uniform. Is that right?

 

[marcusl]

Thank you for all your help.
marcusl, thank you for the links. They were very helpful.
So, in conclusion-the inner radius increases to avoid stresses associated with it reduces, and because otherwise the expansion will not be uniform. Is that right?

I'm not sure that the material "knows" to avoid stresses and to keep the expansion uniform, so that explanation is a bit anthropomorphic. The coefficient of linear expansion is isotropic in the ideal case, so the material expands the same percentage in all directions. If you work out the geometry, the spherical shell must therefore expand uniformly.

 

[Drakkith]

Personally I prefer to imagine the molecules joined together. If a chain of them are linked together in a circle, and the heat causes them to get further apart, then the chain MUST get larger in diameter, otherwise the molecules would be getting closer together, not further away.