Suppose the countless atoms that make up an object are all moving with tiny, random motions in all directions at once. It's matter, and it's moving. So it's also energy. When it's in little pieces comprising a large object, we call it Heat.
Scientists have technical words they need to use such as "internal thermal energy," but we know that it's really just lots of little motions of lots of little objects in many directions at once. The atoms may be vibrating, spinning, or actually wandering about (if it's a liquid or a gas). It's not really different to the energy of a car whizzing down the street, just a lot smaller and a little trickier to keep track of. Remember, physics is mostly about being a good energy accountant.
Rather than a speedometer, we use a thermometer to keep track of heat energy. The temperature tells us the average speed of the many moving bits. The higher the temperature, the higher the average speed and therefore the more energy is in there. Trust me, it's easier that trying to put a little speedometer on each individual atom.
The temperature is higher when there's more energy. What if all the atoms were to stop at the same time? What would the temperature be then? Answer: -273 C. That's Zero on the absolute temperature scale, or 0 Kelvin.
I have a 1 kg block of ice in front of me that is at a temperature of negative 10 C, or ten below zero. If I add some heat to it, the temperature will go up. If I add 2 kiloJoules of energy (or heat) to it, the temperature will go up by 1 degree. If I add 20 kJ, the temperature goes up by 10 degrees. It's easy to do, but hard to keep track of. How do I add heat? By doing nothing.
Heat always spreads out. It does whatever it can to get away from high temperatures and move to lower temperatures. Just by leaving the block of ice sitting out, heat from the surrounding 20 C room moves towards the -10 C ice in whatever way it can. In this case, mostly through air currents.
Air next to the ice block gets cold. Cold air is heavier, and starts to sink down. Warm air then takes its place next to the ice, and the whole process repeats automatically. As the ice absorbs heat from the air in the room, the ice warms up. One result of that is that the rate of warming slows down. The other result is that the ice eventually starts to melt.
Ice melts at a temperature of 0 C. As we add more heat, more ice melts. But the temperature does not increase until all the ice is melted! Why not?
Water molecules in ice are stuck together and cannot move around. Water molecules automatically stick together when they are not moving very fast, in other words, when the temperature of the water is low (below 0 C, to be exact). If we want them to be unstuck and form a flowing liquid, then we have to give all the water molecules enough motion (meaning energy or heat) to move clear of each other. As we add heat, the temperature stays at 0 C until all the ice is melted. When we have added 335 kJ of heat, then the entire 1 kg block of ice will have melted. Scientists call that "The Latent Heat of Fusion," but we know it's just giving sufficient motion to the molecules to remain free. It's the same energy: motion.
I now no longer have a block of ice on my desk. I now have a litre of water in a pan which I had wisely placed under the ice. I knew what was going to happen, see. The water's temperature is still 0 C, but heat continues to flow towards it from the warmer surroundings. Every time the temperature increases by 1 degree, I know that another 4.2 kJ of heat has gone into the liquid water. Well, that's interesting! It takes more than twice as much energy to warm water as it does to warm ice! Why?
There are more ways for free water molecules to move. Up, down, left, right, forwards, backwards, and spinning in all directions. Previously, they could only wiggle a bit this way and that within the ice crystal structure. Each kind of motion takes a bit of energy to produce. Water molecules have to be doing all of them in order to make the temperature what it is. Therefore, raising the temperature of water requires more heat than it does for ice. Is that why the ice cubes in your drink never cause the entire drink to freeze? It's always the ice that turns to water, and not the other way 'round. How much ice would you have to put in a glass of water to make the entire glass freeze? Come on, accountants, get out your pencil and a calculator. It's not hard.
Eventually, when roughly 84 kJ of heat has entered the water, its temperature will be the same as the surroundings, and the flow of heat will trickle away to a stop. This happens increasingly gradually. Where the trip from 0 to 5 C may take only a few minutes, moving from 15 to 20 C may take hours.
One thing to remember about heat: our hands are not very good thermometers. They tell us when heat is moving into our hand (when we pick up something hot) or out of our hands (when we plunge them into ice water), but try picking up a handful of snow with a gloved hand. The insulation slows the heat flow out of our hand and we do not perceive the temperature as it actually is.
You already know that walking on carpet with bare feet does not feel as "cold" as walking on cement or tiles. But the carpet is exactly the same temperature as the tiles! Just something to think about.
Heat may be endlessly interesting, but never a mystery.
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