Relations between pressure and temperature I have several questions concerning thermodynamics and I order them in 4 points that may be related:


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*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?

*Consider a chamber containing a gas. Does an increase of pressure also increases the temperature of the gas? Is that a result of the mechanical action that puts the piston to move delivering energy to the gas through work?


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*But if one is able to displace the piston a little bit so that during this process no collision happens there will be no energy delivered, and the temperature doesn't rise. Is that possible?


*Excluding water and other special materials, why does a increase of pressure over a solid rises is melting point? 


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*My teacher said that pressure would decrease the molecules motion, so in order to melt the solid we should give more energy through heat. Is this correct? If yes, then see the link referred on the $4^{th}$ point (the document says that pressure rises temperature)  


*The inner core of the Earth is solid but is also at higher temperatures than the liquid outer core. I did some research and is often stated that pressure is the reason why the inner core is solid. But returning to the $3^{rd}$ point, 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)?


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*In this link the article states that pressure over the material inside earth increases the temperature. However if the pressure doesn't reduce the particles motion, then how can particles that jiggle so much be in the solid state?
 A: *

*Heat is energy in transit. It only comes into play when one body is transferring heat to other.
Feynman best explained in a video on youtube, it is just the vibrations of atoms at an atomic scale. So, if a more vibrating solid object comes in contact with solid with relatively low "jiggling", we say heat is transferred.

*[Ref]: Feynman lectures on physics vol.1 ch 39,40
Pressure is the total force by a gas on walls of chamber/Area. Simple calculation gives it's physical meaning as:
(number of atoms)*(average KE per molecule)/ Total inner surface area of chamber
Temperature is just (another way to write) average kinetic energy per molecule.
I don't think it's physically possible to move piston without giving energy.
But suppose we have to increase pressure. How can we do it?
By decreasing the surface area, or increasing number of atoms or increasing K.E.
If we somehow decrease surface area(volume of chamber, by pressing piston), very slowly so that not much force acts on particles increasing K.E, we can say Temperature didn't increase.
I'd like to have comments on my statement. 
3.
 If we increase pressure of gas-liquid equilibrium, we can easily imagine, gas will collide more and more with liquid and this tends to increase boiling point of a liquid compared to at 1 atmosphere. Similar reasoning is imaginable for your question.
4.
 Pressure doesn't reduce motion of particles. It is just an effect on chamber that we observe. 
A: 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.
