Raising temperature with no external work applied

[BOYLANATOR]

Homework Statement

A system consists of three bodies with the same heat capacity. These are
initially at 200 K,400 K and 400 K respectively. Show that, without the
supply of any mechanical energy or heat from outside the system, the highest
temperature to which any of the bodies can be raised by operating heat
engines between them is 488 K.

Homework Equations

efficiency = 1- (T_c/T_h)
dU=dQ-dW

The Attempt at a Solution

I believe the solution may have something to do with the work lost in one heat engine being used as useful work in another.

 

[mfb]

With A=200K, B=400K, C=400K, you could heat A and cool B - this gives work to heat C and cool B. And if you find a quantity which is conserved in such a process, you can show that no other process can be better.

 

[rude man]

I would add that the OP can assume T_Cfinal = 488K and work back to show that the "conserved quantity" is indeed conserved, rather than solve for T_Cfinal = 488 K.
.

 

[mfb]

488 is not the exact answer, but it is possible to show that it has to be close to 488 with that approach.

 

[BOYLANATOR]

Ok I'm guessing the conserved quantity is energy and the rounding error may be from 487.5 which is like 7/8 but I'm not sure how to prove this.

 

[haruspex]

Ok I'm guessing the conserved quantity is energy and the rounding error may be from 487.5 which is like 7/8 but I'm not sure how to prove this.

Since it's a closed system, energy will be conserved. E.g. you could just let two bodies equalise in temperature. What thermodynamic quantity is conserved in 100% efficient processes?

 

[mfb]

Energy conserved, but there is another conserved quantity, which is very important in thermodynamics.
Energy conservation alone would allow to cool two bodies to 0K and to heat the third one to 1000K.