For a degree of freedom whose energy is quadratic in just momentum (but flat in position, or flat with hard walls), the average energy classically is $kT/2$. That is the basic equipartition theorem for an ideal gas. However a lesser known result is that a classical degree of freedom with energy quadratic in both momentum and position has an average energy of $kT$. The atoms in a solid are in some sense each in a 3-way harmonic oscillator (this is the Einstein model) and hence one has $3NkT$ energy, i.e. $3Nk$ heat capacity.(†)
To understand this intuitively you should of course derive the equipartition theorem for yourself. But, basically, by having energy also quadratic in position you make the lower energy states less common; not only does the low energy require a small momentum, but also a particular position. By increasing energy, more and more positions become available. In contrast with a flat potential the position can always take on any value and so a low energy state only needs the momentum to be small.
So if you were to imagine each atom in a solid as instead as being inside its own little box with hard walls, then such a model would only give $3Nk/2$ heat capacity.
(†) Okay, actually the atoms are all coupled together however when you look at it this way, you can't so simply talk about the separate contributions of individual atoms anymore. Looking at these whole vibrations gives you phonons and the Debye model. Basically though, all of the atomic harmonic oscillators mix together into various modes, but of course the number of modes remains the same as the original number of individual oscillators. But, each mode is itself a harmonic oscillator so you get the $3Nk$ heat capacity at high temperature.(‡)
(‡) Actually, only $3(N-\frac{1}{2})k$ since three of the collective modes do not oscillate but rather correspond to the linear motion of the whole block of material. So, those three modes each give only $kT/2$.
I'll give brief answers to your questions. If you need more detail, you should ask your questions separately.
What's the difference between heat and work at the atomic level? Isn't heat simply work between particles colliding with different momentum against each other?
Treating a substance semi-classically, one can say that at the atomic level, the atoms have a certain position and momentum. Quantum mechanically, even that's dubious because position and momentum are conjugate variables. With regard to heat and work, these don't exist at the atomic level.
Heat and work are processes, not states. Atoms don't contain heat or work. Neither do individual collections of atoms. Heat and work are measures of quantities transferred amongst objects. Objects don't contain heat or work.
Does an increase of pressure also increases the temperature of the gas?
For an ideal gas being compressed adiabatically, the answer is an emphatic yes. For anything else, the answer is sometimes yes, sometimes no. The answer depends on how much heat is being transferred into or out of the gas and on the nature of the gas. If the gas is right at the triple point (ideal gases don't have a triple point), all that compressing the gas adiabatically is going to do is cause some of the gas to turn into liquid or solid.
Excluding water and other special materials, why does a increase of pressure over a solid rises is melting point?
What your teacher told you is nonsense. Increased pressure does not decrease the molecule's motion. What increasing the pressure does do is to decrease the intermolecular distance.
The reason most substances contract when they freeze is because the bonding forces that make a substance become a crystalline solid hold the atoms/molecules closer together than the intermolecular distance at the same temperature in the liquid phase. Increasing the pressure in these substances decreases the intermolecular distance, thereby making it easier for those intermolecular bonding forces that make a substance a solid to take hold.
Water is different. It expands upon freezing. The structure of ice (ice Ih) is very open thanks to the hydrogen-hydrogen bonds in ice. Because ice expands upon freezing at normal pressures, increasing the pressure reduces the freezing point. Increase the pressure beyond about 100 atmospheres and water/ice starts behaving like most other substances. Increase the pressure beyond 3000 atmospheres and something even weirder happens. Now the freezing point drops markedly with increasing pressure. Increase the pressure beyond that and something even weirder happens: The freezing point increases again, this time very sharply increasing with rising pressure. The freezing point is over 600K at a pressure of 100,000 atmospheres.
If the pressure reduces the motion of the particles, how can the inner core have material with higher temperatures (i.e. particles with higher average kinetic energy)?
What your teacher told you was wrong.
Best Answer
This description is misguiding in two ways.
First, the statement that
rotational energy does not contribute to temperature
makes an impression that temperature is a quantity that is closely connected with the translational kinetic energy, but not rotational kinetic energy. But that is false; according to classical theory (applicable when temperatures are high) in thermodynamic equilibrium, all quadratic degrees of freedom, translational and rotational, correspond to kinetic energy $k_BT/2$ on average.
It is only true that rotational energy does not contribute to translational kinetic energy of molecules, since the two energies are exclusive contributions to total kinetic energy.
Second, heat capacity when molecules are allowed to rotate is not higher because rotational energy does not contribute to translational kinetic energy of molecules.
It is higher because for the same temperature, such system has higher energy than system without rotation. This is because there are additional degrees of freedom, to which corresponds additional average kinetic energy.
Equilibrium implies temperature implies average energies of molecules. Value of average kinetic energies of molecules neither implies temperature exists nor implies temperature is only connected to translational kinetic energy.