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If the water were uniform temperature, it would have uniform density, so a bubble should either be all the way at the top (if it's lighter than water) or all the way at the bottom (if heavier). But in reality you don't see this neat separation. Sometimes you see bubbles hovering in the middle.

Is this because the water temperature is not uniform? Does this mean that with time, if the room temperature stays constant, all bubbles will neatly separate along the top and bottom?

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I think there's a temperature gradient in the water. The other candidate is that the density of water changes due to its compressibility under pressure. Let's examine the pressure effect first. From an estimate its size, we can see whether it's a significant factor compared to temperature gradients.

The bulk modulus of water is about $2*10^9 Pa$ (Wikipedia source). The pressure in a column of water is $\rho g h$ with $\rho$ the density, $g$ gravitational acceleration, and $h$ the height of the column. For a column of water half a meter tall, this comes to about $5*10^3 Pa$. This means the water at the density of water at the bottom is slightly greater, and the size of the effect is about $2*10^{-6} \rho$

Compare this to the effect of temperature. Water's density varies by about $7*10^{-5} \rho$ for a $1 C$ change in temperature at room temperature (Wikipedia source). That means that the density gradient due to pressure and compressibility is equivalent to a temperature difference of about $.03 C$.

$.03 C$ is tiny - only .01% of the temperature. There must be temperature fluctuations larger than .01% in a thermometer that's just sitting in your room, so the baubles float mid-way due to a temperature gradient in the water, as you originally suggested.

As for whether the baubles will separate - yes, I think they will, but I don't have a Galilean thermometer to test this on. If you've got one and some good ideas on temperature control, make the experiment!

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Although I could be wrong, I think the reason is that the density of any fluid, including water, increases very slightly with depth.

There is a well-known relation between pressure and depth,

$$\Delta p = -\rho g \Delta y$$

which says that at the difference in pressure between any two points in the fluid is proportional to the difference in height between those two points. Now, we usually think of a liquid as being incompressible, so that its volume remains the same at any pressure, but in reality this is not true. Liquids do undergo a very slight decrease in volume as the pressure rises. So since the water pressure at the bottom of the tube is higher, the volume of a given amount of water will be slightly smaller at the bottom of the tube than at the top, and therefore the density at the bottom will be a little higher.

If the density of one of the glass bulbs just happens to fall in the range between the density of water at the top and the density of water at the bottom of the tube, then the bulb will come to equilibrium in the middle, at the point where its own density happens to be equal to that of the surrounding water.

Incidentally, Wikipedia reports that the liquid used in a Galilean thermometer is usually not water, but something else that has a more drastic variation of density with temperature. But the argument I've outlined above applies equally well to any normal liquid.

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""Wikipedia reports that the liquid used in a Galilean thermometer is usually not water, but something else that has a more drastic variation of density with temperature."" Wiki assumes (not repots) that reaseon is the dgree of expansion. Hydrocarbons of the category mentioned by Wiki have about 0.09 % volume expansion per degree, Water at 20 °C has 0.02 %/°C . This is less but would be enough nevertheless. Main reason against water is simple: algae and other "mould"! It is not easy to prevent some "life" in water over many years. –  Georg Mar 31 '11 at 21:14
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