Why do objects float in substances more dense than themselves?

[kashiark]

I've been thinking about this forever, and I can't figure it out. Let's explore a simplified situation: 2 moles of neon and 2 moles of helium are the same temperature in a box that is 100% efficient at trapping heat and the gases. The helium will "float" to the top of the box; why? It's less massive, so each particle will have a greater velocity, but shouldn't it move randomly with probability ensuring that it moves equally in all directions? Am I missing something? Also, I'd appreciate if you would explain the more complicated scenario of a metal box with air in it that floats in water (I didn't want to say a boat because I didn't want you to think I was talking about a canoe or something like that.)

 

[espen180]

Think about it this way. Given a container with two fluids of different mass density or an object suspended in a fluid, what configuration of the fluids or fluid and object will minimize the total gravitational potential energy of the system?

Calculate the potential energy of the following configurations to see my point:
-Helium on bottom, Xenon on top
-Mixed
-Xenon on bottom, Helium on top

You will find that the third configuration has the least gravitational potential energy.
In short, the potential energy lost from having the Xenon sink to the bottom is greater that the gain of potential energy from having the helium ris to the top. The net change in potential energy is negative.

It is useful to remember that a freely moving system will always configure itself to minimize its potential energy.

 

[kashiark]

That makes sense. Thanks! I wasn't even thinking about it in terms of energy; I was stuck on kinetic molecular theory.

 

[the4thamigo_uk]

Although the energy explanation seems good, I find the macroscopic explanations always a bit dissatisfying though. How do we explain it microscopically then?

In a 50-50 (by moles) mixture of two ideal gases we have the same temperature and partial pressure and volume for each gas, and we know all particles do not interact. In the absence of gravity it will be a random mixture. In the presence of gravity we observe that the lighter gas will rise. Consider that the particles of the heavier gas have the same 'mean' kinetic energy as the particles of the lighter gas (by 3kT/2) but they must therefore travel at slower speeds on average. We know from basic projectile theory that for a particle projected at velocity v directly upwards in a gravitational field, that it will reach a maximum height of h = v^2/2g (from KE = PE). The maximum height therefore is dependent only on the velocities of the particles and therefore the lighter gas can travel higher relative to the heavier gas. This would seem to give the basis of a microscopic explanation.

I have a further question though? Do the two mixtures separate continuously or do they form a distinct boundary layer, a discontinuity. Can anyone answer this because my high school physics is a very distant memory. I know 3kt/2 is the 'mean' KE per particle, but we know that in reality there is a distribution that has a very long tail to allow particles of all energies. So we would probably see a continuously changing distribution as we move upwards I think. So can anyone calculate this distribution for two gases of different molecular masses?

 

[espen180]

I have a further question though? Do the two mixtures separate continuously or do they form a distinct boundary layer, a discontinuity. Can anyone answer this because my high school physics is a very distant memory. I know 3kt/2 is the 'mean' KE per particle, but we know that in reality there is a distribution that has a very long tail to allow particles of all energies. So we would probably see a continuously changing distribution as we move upwards I think. So can anyone calculate this distribution for two gases of different molecular masses?

I won't derive it mathematically, but I can try to give a rough explanation of the distribution. I hope my using energy arguments won't be a problem.

Like I said above, the system wants to minimize its potential energy. Therefore, if there was only one fluid in the container, the density of the molecules would be greatest at the bottom and decrease with increasing height. The same is the case with two fluids. At the bottom be have the largest xenon density. As we move upwards, the xenon density decreases, making room for helium. The helium will make its way into the regions of low xenon density in order to further dicrease the system's poential energy. So there is not a boundary where all the helium rests on all the xenon. I think the pressure gradient of the xenon should still be such that we don't get a steadily changing distribution, but we should have a volume occupied by xenon, a volume where we have an overlap of gases and a volume of helium.

 

[kashiark]

That makes sense; what's a microscopic explanation of a plastic boat floating the surface in a bathtub after you put it under water?

 

[sophiecentaur]

That makes sense. Thanks! I wasn't even thinking about it in terms of energy; I was stuck on kinetic molecular theory.

I always find that the 'energy' (scalar) argument is the most straightforward when applicable. It doesn't involve forces and directions (vectors) so fewer sums to do! It also is easy to apply 'feasibility' tests to the result - e.g. "you never get more energy out than you put in " and others.

 

[diazona]

That makes sense. Thanks! I wasn't even thinking about it in terms of energy; I was stuck on kinetic molecular theory.

Not to toot my own horn, so to speak - well, actually I guess I am tooting my own horn I'm reminded of something I wrote about this in connection with an episode of Mythbusters a few weeks ago. The connection is just that I used an energy argument to derive the expression for buoyant force. I don't know if you might find it interesting at all, but I thought I might mention it just in case... http://www.ellipsix.net/blog/post.78.html

 

[the4thamigo_uk]

That makes sense; what's a microscopic explanation of a plastic boat floating the surface in a bathtub after you put it under water?

Hmm. I can give you a macroscopic explanation that isn't dependent on 'things wanting to minimize their energy'. Energy is useful for calculations but I think sometimes it masks the physics.

Ok so for a cube underwater of side length L. We have hydrostatic pressure being the weight of the fluid above it (i.e. mgh), this is due to the density of the water . Therefore the pressure at the top of the cube is slightly less than the pressure at the bottom of the cube. Therefore there is a net force upwards due to pressure. On the other hand there is a force downward due to the force of gravity. If they are not in balance the cube will rise.

Net pressure upwards =
pressure(bottom) - pressure(top) * L^2 = density(water) * g * L^3

Gravitational force on cube =
mass(cube) * g = density(cube) * g * L^3

So since the g * L^3 cancel, the cube will rise if its density is less than the density of the surrounding water.

Also, as an afterthought, answer me this... what might happen to the cube's bouyancy if you heat the water to a higher temperature?


As for microscopic explanations, well since we arent dealing with gases anymore, the explanation is more complicated than for a gas...

 

[sophiecentaur]

"Hmm. I can give you a macroscopic explanation that isn't dependent on 'things wanting to minimize their energy'."
I should hope so too. It's never a good idea to say that objects "want" anything. But describing objects hitting something and all the implied Newtonian behaviour is really no more Physics than describing the situation energetically. The particles have Kinetic Energy and the floating object has Potential Energy. It just happens that the least energetic condition is with the less dense thing on top and the more dense thing underneath.
Of course, the mechanics of the situation can give a rosy glow of understanding but it is also possible to come to the wrong conclusion if you just use a particle description and happen to forget some vital factor. When you do, there may be no 'obvious' check.
I remember solving mechanics problems at School, using the principle of Virtual Work when it was very hard to solve them just using 'forces'.

 

[the4thamigo_uk]

Ooops I didnt mean to criticize the explanation. Of course it is equally valid explanation I agree with you sophie. They are different ways to look at the same thing and energy sometimes does simplify calculations. It just sometimes feels more intuitive to me, for some explanations, to imagine what is going on under the hood.

 

[sophiecentaur]

Not even the slightest hint of offense taken. There are always many ways of skinning a cat - as long as anthropomorphism isn't used as one of them! :)