# Third law of thermodynamics conundrum with cyclic machines

Consider 3 objects, each with different heat capacities ($C_1$, $C_2$, $C_3$) and initial temperatures $T_1$, $T_2$, and $T_3$. Heat engines (HE1 and HE2) are placed between the objects to generate work, which powers a heat pump (HP) to transfer heat from object 2 to object 4. Are the final temperatures of objects 1, 2 and 3 all the same? Or is object 2 colder than objects 1 and 3?

I think the final temperatures of objects 1-3 should be the same, since the heat engines will keep running until there are no temperature differences. HE1 will run until $T_1 = T_2$, and HE2 will run until $T_2=T_3$. By the third law of thermodynamics, we therefore have $T_1=T_3$ at this final stage. I don't think the heat pump should influence this result.

Intuitively, however, it seems like object 2 should be colder in the final stage, since it is the only one rejecting heat to a heat pump.

• Have you tried analyzing this problem quantatively (I.e., modeling the problem)? – Chet Miller Sep 18 '18 at 0:28
• @Chester_Miller yes, using $d U = \delta Q + \delta W$ on each system is only useful if I know the final temperature of each system. So I take objects 1 and 2 and realize the heat engine runs until $T_1=T_2$. Well, the same happens with the heat engine between objects 2 and 3. Third law therefore says that $T_1=T_3$. That's really the only analysis I know how to do for this problem. I'm not trying to solve for the final temperature, I just want to determine if the 3rd law applies between objects 1, 2 and 3. – Drew Sep 18 '18 at 0:39
• Suppose you assume that the engines and heat pump operate in instantaneous reversible Carnot cycles, based on the current temperatures. I think you can do something quantitative with that. – Chet Miller Sep 18 '18 at 0:46
• Heat reservoirs are normally considered to be "infinite". This means that T3>T2>T1 for the "normal" thermodynamic case. It also means that the three temperatures are constant. – David White Sep 18 '18 at 1:38
• @David_White, these heat "reservoirs" are actually of finite heat capacity (i.e., normal objects that can change temperature). I made this problem up, so the problem itself may not even make sense or be solvable with the given parameters. – Drew Sep 18 '18 at 1:45