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Heat energy is transferred through solids by the process we call thermal conduction.

Hot matter and cold matter differ by the kinetic energy the atoms/molecules have.

Temperature (as measured on the absolute temperature scale) is proportional to the average kinetic energy of the particles in a material.

Normally we classify movement of heat according to three processes, thermal conduction, radiative transfer and convective heat transfer.

Convective heat transfer occurs in fluids because when fluids move the heat energy contained in the fluid moves with them. This is very important in home insulation and the main reason we insulate walls with fibers to inhibit motion of the air. Radiative transfer of heat occurs when the hot object emits electromagnetic radiation (light) and that electromagnetic radiation carries its energy elsewhere. We can see light energy that came from the sun or even a light bulb, but there is much more energy at longer wavelengths in the infrared part of the spectrum that we can't see. Radiative energy travels through materials (like air) that are transparent, or nearly so, for the particular part of the spectrum carrying the heat. (The distribution of energy emitted as light changes with the temperature of the object doing the emitting which is why we say something is "white hot" meaning it is at a high temperature.)

It is heat transfer through conduction that we observe when we say heat travels through a solid body. If one part of an object it hotter than another, the molecules in that part have more kinetic energy. If left undisturbed, heat energy in a hot spot gradually spreads out into the whole object through thermal conduction of heat. Thermal conduction is nothing more that the energetic particles having collisions with their less energetic neighbors and thereby passing on some energy

to them. The process happens many millions of times a second between neighboring molecules so only small amounts of energy need transferred from a "hot" molecule to a cooler neighbor for the over all process of neighbor to neighbor transfer to effectively carry that energy from a hot spot to cool spots. Eventually, the temperature of an isolated object will become uniform through this process of intermolecular energy transfer which we recognize as thermal conduction.

Aside: It should be mentioned that in metals some of the thermal conductivity is a result of kinetic energy transfer mediated by the conduction electrons.

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