Been looking at heat equation and it's derivation, according to Wikipedia it uses 2 mathematical assumptions. My problem is that although it all seems OK, what is the physics of heat transfer in solids? so far haven't seen answer to the following questions regarding heat equation:
Given 2 adjacent atoms, how does one measure the hot and the colds one? ( theoretically at least, no need for a physical thermometer)
How does heat transfers from hot to the cold one? ( most likely by photons, but do electrons from one atom get knocked out and knockout the electrons of the other? is that considered a temperature increase or just change of charges?)
If this transfer occurs with discreet amount of energy being moved from the hot one to the cold one, what are the intervals (time wise) between each transfer? (In other words what's the time interval between photons being emitted within a solid?)
Answer
Temperature is not a concept that has a lot of utility at the level of single atoms because it represents the mean kinetic energy of a group of particles (to within a coefficient).
You can define it, it just doesn't help much.
At the level of two atoms you revert to a more fundamental model such as the forces between them.
One atom transfers energy to another through electromagnetic forces between them. When that energy manifests as randomized kinetic energy at the microscopic scale we refer to it as "heat" at the macroscopic scale.
In a solid it is usually reasonable to treat the forces between individual pairs of atoms as being spring-like (i.e. they obey $F_{i,j} = -k(r_{i,j} - r_0)$). Starting from there you can build various models of solid behavior. For instance the Einstein model of a crystal.
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