Mercury does expand with temperature; moreover, it expands more rapidly with temperature than glass goes. That's why the column of mercury rises inside its glass container. While both materials expand as they get hotter, the mercury experiences a larger increase in volume and must flow up the narrow channel or "capillary" inside the glass to find room for itself. Mercury is essentially incompressible so that, as it expands, it pushes as hard as necessary on whatever contains it in order to obtain the space it needs. That's why a typical thermometer has an extra chamber at the top of its capillary. That chamber will receive the expanding mercury if it rises completely up the capillary so that the mercury won't pop the thermometer if it is overheated. In short, the force pushing mercury up the column can be enormous.
The force pushing mercury back down the column as it cools is tiny in comparison. Mercury certainly does contract when cooled, so that the manufacturer is telling you nonsense. But just because the mercury contracts as it cools doesn't mean that it will all flow back down the column. The mercury needs a push to propel it through its narrow channel.
Mercury is attracted only weakly to glass, so it doesn't really adhere to the walls of its channel. However, like all liquids, mercury has a viscosity, a syrupiness, and this viscosity slows its motion through any pipe. The narrower the pipe, the harder one has to push on a liquid to keep it flowing through that pipe. In fact, flow through a pipe typically scales as the 4th power of that pipe's radius, which is why even modest narrowing of arteries can dramatically impair blood flow in people. The capillaries used in fever thermometers are so narrow that mercury has tremendous trouble flowing through them. It takes big forces to push the mercury quickly through such a capillary.
During expansion, there is easily enough force to push the mercury up through the capillary. However, during contraction, the forces pushing the mercury back down through the capillary are too weak to keep the column together. That's because the only thing above the column of liquid mercury is a thin vapor of mercury gas and that vapor pushes on the liquid much too feebly to have a significant effect. And while gravity may also push down on the liquid if the thermometer is oriented properly, it doesn't push hard enough to help much.
The contracting column of mercury takes hours to drift downward, if it drifts downward at all. It often breaks up into sections, each of which drifts downward at its own rate. And, as two readers (Michael Hugh Knowles and Miodrag Darko Matovic) have both pointed out to me in recent days, there is a narrow constriction in the capillary near its base and the mercury column always breaks at that constriction during contraction. Since the top portion of the mercury column is left almost undisturbed when the column breaks at the constriction, it's easy to read the highest temperature reached by the thermometer.
Shaking the thermometer hard is what gets the mercury down and ultimately drives it through the constriction so that it rejoins into a single column. In effect, you are making the glass accelerate so fast that it leaves the mercury behind. The mercury isn't being pushed down to the bottom of the thermometer; instead, the glass is leaping upward and the mercury is lagging behind. The mercury drifts to the bottom of the thermometer because of its own inertia.
You're right that the glass tube acts as a magnifier for that thin column of mercury. Like a tall glass of water, it acts as a cylindrical lens that magnifies the narrow sliver of metal into a wide image.
You're right about the glass expanding along with the liquid inside it. But liquids normally expand more than solids as their temperatures increase. That's because the atoms and molecules in a liquid have more freedom to move around than those in a solid and they respond to increasing temperatures by forming less and less tightly packed arrangements. Since the liquid in a thermometer expands more than the glass container around it, the liquid level rises as the thermometer's temperature increases.
The term "digital display" usually refers to a system that reports the value of a physical quantity in numerical form. A digital watch display is a good example. The physical quantity it reports is time and it makes its report in the form of hours, minutes, and second—all in numerical form. In a digital watch, the display makes use of liquid crystals that are sensitive to electric fields. When you look at the display, you are actually looking through a layer of polarizing filter, some transparent electric wires, and a layer of liquid crystals. Liquid crystals are liquids that contain molecules that naturally orient themselves relative to one another. In the display, these liquid crystals adopt different orientations when they are exposed to electric fields than when they're not exposed to such fields. This electrically altered orientation affects their optical properties and causes them to appear dark when viewed through the polarizing filter. The watch can control the appearance of each segment of its digital display by the pattern of electric charge on its transparent wires. Since it takes very little energy to change the orientation of the liquid crystals, the watch uses almost no power for its display and can operate for years on a button battery.
A common liquid in glass thermometer takes advantage of the fact that liquids generally expand more than solids as their temperatures increase. The glass envelope of the thermometer contains a fine hollow capillary with a sealed reservoir at its base that's filled with a liquid such as alcohol or mercury. If both the liquid and glass expanded equally as they became warmer, the thermometer would simply change sizes slightly as its temperature increased. But the liquid expands more than the glass and can't simply remain in place. Some of it moves up the capillary. That's why the level of liquid in the thermometer rises as the thermometer's temperature rises.
A Galileo thermometer combines Archimedes' principle with the fact that liquids generally expand faster with increasing temperature than solids do. Each sphere in the thermometer has an average density (a mass divided by volume) that is very close to that of the fluid in the thermometer. As stated in Archimedes' principle, if the sphere's average density is less than that of the fluid, the sphere floats and if the sphere's average density is more than that of the fluid, it sinks. But the fluid's density changes relatively quickly with temperature, becoming less with each additional degree. Thus as the temperature of the thermometer rises, the spheres have more and more trouble floating. Each sphere's density is carefully adjusted so that it begins to sink as soon as the thermometer's temperature exceeds a certain value. At that value, the expanding fluid's density becomes less than the average density of the sphere and the sphere no longer floats. The spheres also expand with increasing temperature, but not as much as the fluid.
Here is a picture of a combined Galileo thermometer and simple barometer. In addition to measuring the temperature with floating spheres, this device measures the outside air pressure with a column of dark liquid. It has a trapped volume of air that pushes the liquid (visible at the bottom of the unit) up a vertical pipe when the outside air pressure drops. The owner of this unit would like to know its history and origin, so if you have any information about it, please let me know.
A typical thermostat turns on the furnace whenever the temperature falls below a certain temperature and turns the furnace off whenever the temperature rises above another temperature. Those two temperatures are slightly separated so that the furnace doesn't turn on and off too rapidly. In a typical home thermostat, a bimetallic coil tips a small mercury-filled glass bottle. The bimetallic coil is made from two different metal strips that have been sandwiched together and then rolled into a coil. As the temperature changes, the two metals expand differently and the coil winds or unwinds. As it does, it tips the glass bottle and the mercury rolls from one end of the bottle to the other. When the mercury falls to one end, it allows an electric current to flow between two wires and the furnace turns on. When the mercury falls to the other end of the bottle, the current stops flowing and the furnace turns off. So the winding and unwinding of the coil controls the furnace and the home temperature tends to hover at the point where the bottle of mercury is almost perfectly level. When you adjust the set point of the thermostat, you tilt the whole coil and bottle so that the average temperature in your home must shift in order for the bottle to be almost level.
Most modern liquid-in-glass thermometers do contain alcohol rather than mercury, but these aren't the digital thermometers you are referring to. The alcohol thermometers are the ones with the red line that moves upward in a glass tube as the temperature increases. I believe that the digital thermometers you're interested in are the ones with numbers that change colors as the temperature changes. For example, when its 72° F, the number "72" is brightly colored while the other numbers are essentially black. Those thermometers use liquid crystals to measure temperature. More specifically, they use chiral nematic liquid crystals—long asymmetric molecules that arrange themselves in orderly spirals in the liquid. When light strikes these spiral structures, some of it reflects. But the reflection is strongest when the light's wavelength is an integer or half integer multiple of the spiral's pitch—the distance between adjacent turns of the spiral. Since light's wavelength is related to its color, the light reflected by these liquid crystals is colored. Because the pitch of a chiral nematic liquid crystal changes with temperature, so does its color. Slightly different liquid crystals are inserted behind each number on the thermometer so that each number becomes colored at a different temperature.
The electronic fever thermometers that you can buy in a grocery store use a thermistor to measure temperature. A thermistor is a semiconductor device that acts as a temperature-sensitive electric resistor. At very low temperatures, a thermistor is essentially an insulator—it has no mobile electric charges and thus can't carry electricity. But as its temperature increases, thermal energy rearranges the charges in the thermistor and it has more and more mobile electric charges. Its ability to conduct electricity increases with temperature fairly dramatically—it gradually becomes an electric conductor. The thermistor used in a fever thermometer is designed to undergo this rapid change in electric resistance at temperatures near 98° F. A simple computer inside the thermometer measures the thermistor's electric resistance and determines the thermistor's temperature. It then uses a liquid crystal-based display to show you what that temperature is.
A simple thermostat turns the furnace on when the temperature it senses falls below a certain value and turns the furnace off when the temperature it senses rises above that value. Because it takes time for the furnace to respond to signals from the thermostat, for the heat from the furnace to travel to the thermostat, and for the thermostat to respond to changes in the temperature around it, the furnace tends to stay on for too long after the thermostat turns it on and then to stay off for too long after the thermostat turns it off. The result is an oscillation in temperature: the home or building alternately overheats and then overcools. To reduce this oscillation, a thermostat with a heat anticipator limits the amount of time that the furnace stays on. Since the furnace turns off earlier, the temperature doesn't overshoot as much on the high side and the furnace turns back on again more quickly once the home or building drifts below the set temperature of the thermostat. Overall, the temperature still oscillates above and below the set temperature, but those oscillations are smaller and faster.
The fancy ear thermometers used in doctor's offices are almost certainly measuring the thermal radiation emerging from inside the ear. They probably use a thermopile detector that responds very quickly to the thermal radiation that reaches them. Since the thermal radiation emitted by a black object (or from within a deep cavity such as the ear) is characteristic of the object's temperature, a quick study of that thermal radiation is enough to determine the person's temperature.
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