1. Why is there a definite sublimation temperature T?
First of all, as the saying goes, it's bad luck to prove things
that aren't true.
If you put a chunk of material into a box at constant /volume/
and let it sublimate, you do *not* get a constant temperature.
The idea only works at constant /pressure/, with the solid
nearly in equilibrium with its vapor.
With that added requirement, the explanation is pretty simple.
All physics is /local/. Look at what is going on locally at
the surface. The surface doesn't know how much solid is
behind it or how much vapor is in front of it; all it knows
is the local temperature, pressure, density, et cetera.
None of that changes as the reaction progresses.
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There are various higher-order effects that complicate the
picture. If the particles are very small and/or rough, there
will be surface-tension effects. In other words, energy is
not strictly an extensive property. For that matter, entropy
isn't strictly extensive either, for small systems. These
are however small corrections and don't change the basic
qualitative intuitive picture.
Also things go completely haywire if the system is far from
equilibrium, e.g. if you vigorously pump away the vapor.
There is still constant pressure, but it's not the
equilibrium pressure, so you get some screwy temperature
(assuming you can even define a temperature).