Conceptual questions
Describe how a heat engine works and what changes when you make it a refrigerator.
A heat engine pull in heat from a hot reservoir and converts that heat into mechanical work. The excess energy gets dumped into a cold reservoir as heat.
To turn this into a refrigerator, just reverse the process. Now, put in work (often electrical) to suck heat out of the cold reservoir and dump heat into the hot reservoir.
Describe what happens to achieve mechanical equilibrium including what quantities are exchanged and which are the same.
Mechanical interactions are when volume is exchanged between objects. Mechanical equilibrium occurs when pressure is the same between the systems.
COP and efficiency both tell you how much benefit you will get from the engine/refrigerator based on how much energy you put into the system. For COP, the benefit is the cold heat and the cost is the work done, whereas for efficiency the benefit is the work done and the cost is the hot heat.
Additionally, efficiency is always <1 because of conservation of energy (the amount of heat we put in can't be more than the amount of work we get out). On the other hand, COP can be >1 because it is only limited by the amount of work you put in (it is not bound by conservation of energy).
Physically describe what's happening with this graph
As you add energy to the system, the entropy is increasing while its temperature decreases. A system behaving like this (e.g. stars) could take in energy and turn it into potential energy, thus increasing the multiplicity but decreasing the overall tendency for the system to give up heat.
Describe what makes a two-state paramagnet different than other systems (i.e. ideal gas, Einstein solid) in terms of energy, entropy, and temperature
Unlike ES and ideal gases, the total energy of a paramagnet is limited. This means that as the maximum allowed energy is approached, the multiplicity decreases.
Thus, as you add energy, the entropy of the system actually decreases. This leads to infinite/negative temperatures which act as higher temperatures than positive ones (i.e. a system with a negative temperature will have a tendency to give up heat to a system with positive temp)