What is temperature? This question comes up quite a bit—especially in introductory science courses. The most common answer is something like this:
Temperature is a measure of the average kinetic energy of the particles in an object. When temperature increases, the motion of these particles also increases.
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It's not a terrible definition, but it's not the best either. There are plenty of other crazy things about temperature that you should probably know.
If temperature is a measure of the average kinetic energy, shouldn't thermal energy and temperature be the same thing? No. Thermal energy is the total energy an object has due to the internal motions of its particles. The temperature is related to the average kinetic energy—not the total kinetic energy.
Here's a classic example that you can try at home. Put a piece of cold pizza on top of a sheet of aluminum foil and then stick it in the oven to heat up. After about 10 minutes, the pizza should be nice and hot—the aluminum foil is the approximately the same temperature. You can pull the aluminum foil out with your fingers, but not the pizza. Although the aluminum foil has a high temperature, its low mass means it doesn't have much thermal energy. Without a lot of thermal energy in the foil, your fingers won't get burned. Meaning? Thermal energy and temperature are different things.
More Definitions of Temperature?
You already have one definition from above, but I am going to give you two more definitions. The first one is the historical version. It goes like this:
Temperature is the quantity that two objects have in common after being in contact for a long time.
This definition is based on the idea of thermal equilibrium. If you put an aluminum ball into some water, eventually the water and the ball will have the same temperature. They won't have the same thermal energy, but they will have the same temperature. It's a very operational definition of temperature—and that's not a bad thing.
But really, this temperature is the basis of most thermometers. Take your basic mercury or alcohol thermometer (the mercury ones are not so common because—you know, they contain mercury). When you put this thermometer in a liquid or something else, the temperature of the liquid inside the thermometer changes until it is the same as the object. Since both mercury and alcohol expand with an increase in temperature, you can determine the temperature based on this thermal expansion (or contraction). Really, you could say that the thermometer even came before the idea of temperature.
Now for the second definition of temperature. This one is pretty tough, so hold on to something.
Temperature is the rate that internal energy changes with respect to entropy.
It's short, but there's a lot in there. First, what is entropy? I could try to explain entropy, but this would be a complete new blog post. Instead, you can just check out this very awesome post by Aatish Bhatia in which he explains entropy using sheep. Yes, it's really good.
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So, instead of a full explanation of entropy, I will just give some interesting aspects of it. Thermal equilibrium is not a purely energy phenomena. Energy is conserved when two objects reach thermal equilibrium, but it would also be satisfied if one object got hot and the other one became cold. Thermal equilibrium is a statistical process. It just so happens that the most probable outcome for two objects in contact is that they reach the same temperature. The other weird cases (one getting hot and one getting cold) can also technically happen, but their chances are way less than you winning the lottery (and your chance of winning the lottery is essentially zero).