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Homework Help: Thermodynamics, Callen's problem, it doesn't make sense to me

  1. Mar 17, 2012 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    Two moles of a particular single-component system are found to have a dependence of internal energy U on pressure and volume given by [itex]U=APV^2[/itex] (for N=2) where [itex]A=10cm ^{-3}[/itex]. Note that doubling the system doubles the volume, energy, and the mole number but leaves the pressure unaltered, and write the complete dependence of U on P, V and N for arbitrary mole number. Answer: [itex]U=BPV^2/N[/itex] where [itex]B=20 cm ^3[/itex].


    2. Relevant equations
    Not sure there are any. To be found.


    3. The attempt at a solution
    First I tried to understand the problem and makes sense of the given expression [itex]U=APV^2[/itex]. I checked out the units in SI units and instead of joule I reach [itex]Jm^2[/itex]. So that this expression doesn't make sense to me, I cannot even proceed further.
    The book is considered of one of the most cited apparently for thermodynamics so I'm sure the book is right and I'm missing something. I just don't know what I'm missing, any help is appreciated.
     
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  3. Mar 17, 2012 #2

    vela

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    How'd you get that?
     
  4. Mar 17, 2012 #3

    fluidistic

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    By forgetting to divide the force by the area in the pressure expression. :smile:
     
  5. Mar 17, 2012 #4

    fluidistic

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    I don't know how to reach the answer. Attempt:
    Let [itex]V_0[/itex] be the volume when [itex]n=2[/itex]. So that [itex]V(n)=\frac{V_0 n}{2}[/itex] (linear relation). P is constant. I do the same relation for [itex]U(n)[/itex]: [itex]U(n)=\frac{U_0 n}{2}[/itex].
    The relation they provide becomes [itex]U_0=APV_0^2[/itex]. Written with my terms I get [itex]U(n)=\frac{2APV^2(n)}{n}=\frac{APnV_0 ^2}{2}[/itex].
    Something is definitely wrong, I shouldn't be able to get a dependence of U only in terms of n.
    I've tried other things but reached nothing like the result.
     
  6. Mar 18, 2012 #5

    rude man

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    Last edited: Mar 18, 2012
  7. Mar 18, 2012 #6

    fluidistic

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    Thank you for your help. But I think that's what I've done in my last post. I don't see anything wrong with my answer.
    When n=2 I get the given relation. When I duplicate n, I get twice the same energy and the volume duplicates too; the pressure remaining the same just as it should. Yet my answer differs from the one given (worth apparently [itex]\frac{200PV^2}{An}[/itex] if I write it in terms of A rather than B.)
     
  8. Mar 18, 2012 #7

    rude man

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    Last edited: Mar 18, 2012
  9. Mar 18, 2012 #8

    fluidistic

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    Ok rude man, thanks for helping me. I am the OP in case you didn't notice :rofl:
    If my answer [itex]U(n)=\frac{2APV^2(n)}{n}[/itex] is right and the answer provided by the textbook is (written in terms of A instead of B) [itex]\frac{200PV^2}{An}[/itex] which is right by definition, how would one shows they are equivalent?
    To me, this would mean that 2A=200/A. Wait, that means A=10 (I didn't check out the units though) which is... RIGHT!
    Wow.
    Problem solved!
     
  10. Mar 19, 2012 #9

    rude man

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    Yep, and my apologies for thinking you were not the OP! :blushing:
     
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