Say you have an object with a negative heat capacity. This means that the object increases in temperature as it radiates energy and decreases in temperature as it absorbs energy.
I don't quite understand why this object cannot be in thermal equilbrium with a reservoir.
Case 1: Let's say the object is hotter than the reservoir.
In this case, it radiates heat to the reservoir, and because of the negative heat capacity, the object's temperature increases. (which brings us back to Case 1 again, an infinite recurrence, so no equilibrium.)
Outcome: The object keeps getting hotter.
Case 2: Now, the object is cooler than the reservoir.
It takes up heat from the reservoir, which causes it to become even more cool. (back to Case 2 , an infinite recurrence again, so no equilibrium.)
Outcome: The object keeps getting cooler.
Case 3: Equal temperature.
The object stays in equilibrium. (because there is nothing causing a change in its temperature.)
Outcome: The object maintains its temperature.
The object can stay in equilibrium with the reservoir if it started out that way, but it can never reach equilibrium if it started out at a different temperature.
