Temperature a la Galileo

There's famously dozens of ways to measure the height of a building with a barometer. If you're sufficiently clever, you can think of many, many more ways to measure temperature with just about anything.

One of the most visually impressive ways to measure temperature is the Galilean thermometer, which is also sometimes called the Galileo thermometer. It consists of little fluid-filled glass bulbs immersed in another liquid. The suspending liquid is itself encased in a long glass tube. Hanging from each of the glass bulbs is a marker with a temperature label. Look at the temperature on the highest bulb which has sunk to the bottom and the temperature on the lowest bulb which has floated to the top and in between those numbers is the temperature. As the temperature rises, one bulb after another on the top will gradually sink to the bottom. As the temperature falls, one bulb after another on the bottom will rise to the top. In this way the temperature is reflected in which bulbs are floating and which are not. The tube itself is narrow enough to preserve the numerical order of the temperature markers, and the bulbs are too large to slide past each other.

To figure out how it works, let's examine the forces on a given bulb. There two of them. First there's the upward force of buoyancy. Second, there's the downward force of the bulb's weight.

According to Archimedes' principle, the buoyant force is equal to the weight of the liquid displaced. The mass is going to be equal to the volume of the fluid times the density of the fluid, and multiply that by g to get the weight. Let's call this force with the letter u to remind us that it's the upward force on a bulb:

The downward force from the weight of the bulb itself is found in pretty much the same way. The weight is the mass of the bulb times g, but let's write the mass in terms of the density of the bulb itself, as the density of the bulb times the volume of the bulb:

And that's the upward force with the letter u. Subtract the downward force from the upward force and you'll get the total force on the bulb:

If the fluid density is greater than the bulb density the result will be a positive number, the force will be up, and the bulb will float. The the fluid density is less than the bulb density the result will be a negative number, the force will be down, and the bulb will sink.

Now you know that in general objects expand when heated and contract when cooled. The fluid itself is some hydrocarbon and its density changes in a comparatively large way with temperature. The fluid surrounding the bulbs will contract when it gets colder which means its density must increase. Denser fluid increases the upward force on each bulb. The bulb doesn't expand or contract very much at all because it is made of glass. Glass changes volume very little with temperature and so the density of each bulb is essentially constant with respect to the density of the fluid, and so the designers don't have to worry about the expansion of the glass bulb offsetting the expansion of the liquid. Pyrex glass, for instance, expands something like five hundred times less than ethanol per degree. This means the maker of these thermometers can carefully adjust the density of each bulb so that it will just exactly have the same density as the fluid when the fluid is at the particular temperature labeled on the bulb.

I have one of these on my desk, given to me by my lovely girlfriend for our anniversary. They're an elegant and useful demonstration of a rather interesting physical principle. And they're only about twenty or thirty bucks
depending on which size you're looking at.

Much more interesting than a boring old digital thermometer, I think.