If you have two different materials at the same temperature but are at different states of matter, do they have the same amount of kinetic energy?
[Physics] If a liquid and a solid are at the same temperature, do they have the same kinetic energy
energystates-of-mattertemperaturethermodynamics
Related Solutions
Generally, bodies can have same internal energies, but have different temperatures, and vice versa, have the same temperature but different internal energies.
Consider for example ideal gasses, where the internal energy is given as a function of temperature and heat capacity: $U={C}_{V}T$. If we have a monoatomic gas consisting of N particles, its heat capacity is ${C}_{V}=\frac{3}{2}N{k}_{B}$, whereas for a gas consisting of diatomic molecules we have ${C}_{V}=\frac{5}{2}N{k}_{B}$ at room temperature (if you want to know where I got these explicit formulas, google the equipartition theorem). If you try to play a little with inserting different temperatures or plot those relations, you get something like this:
This graph is plotted for $N={N}_{A}$, namely for 1 mole of ideal gas.
Notice that you may have gasses with equal energy, but different temperature, and having the same temperature doesn't mean the internal energy of two systems is the same.
At temperatures below 100 K, the equipartition theorem doesn't hold and the heat capacities of both gasses slowly saturate to 0 at 0K (a consequence of the third law of thermodynamics). At higher temperatures (around 1000K), the heat capacity of the diatomic gas increases further because of vibrational degrees of freedom. Around room temperature the relation is linear, as shown.
As for your second question, assuming the two systems are in contact and able to transfer energy between them, they will continue to transfer energy between them until their temperatures are the same (that is, until they reach thermal equillibrium).
Not really, because the density varies already through thermal expansion.
But as there actually isn't any universal and exact definition for the difference of solid vs. liquid, you might consider that Carbon $C$ and Helium $He$ as such a substances. This thought can be reasoned from their abnormal triple points. For Carbon there practically isn't a liquid form, and for Helium there isn't solid form. (which is stable in low pressure)
The Mineral stability diagram (P-T diagram) of the system C (= Carbon) looks like this;
This study from 2015 summarizes this problematic quite good.
And the P-T diagram for HE-4 to comparison.
Another approach to the issue can be made through Bingham-plastic materials, which are either solid or liquid depending on the shear stress. This short of concludes, that the whole question is a problem in classical physics, because the modern physics hasn't just defined this question yet.
I have personally made some attempt to answer this question, and to define the phase transitions from the speed of light. This hypothesis found also support from observations.
Best Answer
Within a regime where the equipartition theorem provides a good description of the kinetic degrees of freedom in a material (generally this will mean at relatively high temperatures) each atom will have a mean kinetic energy of $\frac{3}{2}k_BT$, regardless of the state of matter. More generally, however, you would not expect there to be a general relationship between the kinetic energy and temperature. In particular in a solid at at low temperatures, where it is necessary to take account of the quatisation of atomic energy levels, the equipartition theorem will not hold.