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Raising temperature with no external work applied

  1. Jan 24, 2013 #1
    1. The problem statement, all variables and given/known data
    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.



    2. Relevant equations
    efficiency = 1- (Tc/Th)
    dU=dQ-dW

    3. 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.
     
  2. jcsd
  3. Jan 24, 2013 #2

    mfb

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    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.
     
  4. Jan 24, 2013 #3

    rude man

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    I would add that the OP can assume TCfinal = 488K and work back to show that the "conserved quantity" is indeed conserved, rather than solve for TCfinal = 488 K.
    .
     
    Last edited: Jan 24, 2013
  5. Jan 25, 2013 #4

    mfb

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    488 is not the exact answer, but it is possible to show that it has to be close to 488 with that approach.
     
  6. Jan 25, 2013 #5
    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.
     
  7. Jan 25, 2013 #6

    haruspex

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    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?
     
  8. Jan 25, 2013 #7

    mfb

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    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.
     
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