Can heat be viewed as a particle
Is it heat that causes particles to vibrate, or is it the vibrations that cause heat?
In addition to these nice answers, I want to point out that the words "heat" and "heating up" are examples where the term "heat" is used in very different ways, so one needs to be careful.
In general, "warmth" means that you have energy in transit, with the added connotation that the shape of the energy is highly random, or "thermal" as the other answers pointed out. But "heat content" is something of a misnomer because heat is not a state variable, which means that you cannot tell me the state of the system and thus know how much heat was put into the system to get to that state ( because if you start with some systems at very low temperatures but different volumes, you can get them all to reach the state you want by releasing different amounts of heat depending on the course.
When the concept of "heat" is used interchangeably with something like "random energy content", it usually means that it is with constant volume There arrives (i.e. no work to worry about), and then the second law says the internal energy U. This is a state variable and corresponds to the added heat Q. In this case, U also corresponds to the internal random kinetic energy.
But even then there is a fold in quantum systems - U will not necessarily be what we mean by "thermal energy content" as the gas may exhibit quantum mechanical effects such as becoming "degenerate". When it is degenerate, its thermal energy content is much less than its internal energy, and the heat added at constant volume relates to the latter, not the former.
The term "heating up" is even more ambiguous because what people usually mean by it is nothing more than "temperature rise". However, if you include work you can actually raise the temperature, by giving them Withdraw heat. A classic example is the formation of a star in which there is a gas cloud that is always warm loses, while it "warms up". in the sense of increasing temperature.
So these terms are generally difficult! But in the simplest case of classic free particles with constant volume or classic springs, your picture works perfectly. By adding heat, the internal random kinetic energy is increased, and by the presence of the random internal energy you have a concept of "heat content" "that is the heat you need to add to get into this state from a very low temperature.
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