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Maximum temperature reached by gas in expansion

  1. Feb 22, 2017 #1
    1. The problem statement, all variables and given/known data
    1 mole of ideal gas with internal energy U= 3/2 RT , expands from initial volume Vi = 1/10 Vo following the equation p=(− po / Vo ) V +po
    .
    Find
    (a) the highest temperature reached by the gas during the expansion and
    (b) the maximum amount of heat taken in by the gas.

    2. Relevant equations
    pv = nrt


    3. The attempt at a solution
    p=(− po / Vo ) V +po
    pV=(− po / Vo ) V2 +poV
    (pV/nr)=( (− po / Vo ) V2 +poV)/nr = T
    taking derivative of T with respect to v gets
    dT/dx= ((− po / Vo ) 2V +po)/nr
    setting derivative to 0 yields
    V = Vo/2
    plugging back into equation for temperature gets
    maximum temperature = poV0/4nr

    this is what i got is the right method or is there a way to get a numerical anser
     
  2. jcsd
  3. Feb 22, 2017 #2

    kuruman

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    The best you can do is express the maximum temperature as a number times the initial temperature.
     
  4. Feb 22, 2017 #3
    ok then how do i do part b
     
  5. Feb 22, 2017 #4

    kuruman

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    Use the first law.
     
  6. Feb 22, 2017 #5
    you mean llike Δu + w =q
     
  7. Feb 22, 2017 #6
    as Δu is easy but w = ∫(− po / Vo ) V +po dv
    so solving the integral do i plug in v as the v obtained for maximum temperature or what?
    is most heat added in to reach the highest temperature in this case
     
    Last edited: Feb 22, 2017
  8. Feb 22, 2017 #7
    You calculated that the highest temperature is 1/4 the initial temperature. How can that be if the initial temperature is one of the states that it passes through? Regarding application of the first law, why do't you just run the calculation and see what you get?
     
  9. Feb 23, 2017 #8
    no in this question po and vo are not the initial states they are just some constants
    initial volume = 1/10 vo
    initial pressure can be calculated using the above equation
    thus maximum temperature reached is 100/36 Ti

    now my my question is
    is maximum heat added to bring the system to state with highest temperature?
    if that is so then the problem is very easy hope you understand my question?
     
  10. Feb 23, 2017 #9
    Oops. Sorry. My mistake.
    The question is kind of ambiguous. I would just solve the problem as a function of V and see how it plays out. Let the math do the work for you.
     
  11. Feb 25, 2017 #10
    (− po / Vo ) V +po dv = w.
    [ (− p0 / V0 ) V2/2 + p0V ]vivf

    while vi is known what do i plug in for vf is it the volume that gives maximum temperature or what or should i differentiate the above integral to find when the derivative of work is zero and plug in that v. thanks!!
     
  12. Feb 25, 2017 #11
    I would plug in the volume that gives the maximum temperature.
     
  13. Feb 25, 2017 #12
    but how can we be sure that yields maximum works and is the other method just as equally correct
    and btw thanks for the fast replies
     
  14. Feb 25, 2017 #13
    Who said anything about maximum work?
     
  15. Feb 25, 2017 #14
    no because q = Δu + w
    so isnt maximum q added when both these quantities are maximum am i wrong please correct me if i am
     
  16. Feb 25, 2017 #15
    Isn't there also a change in internal energy?
     
  17. Feb 25, 2017 #16
    yeah change in internal energy is automatically maximum when the temperature is maximum but how are we supposed to know that is the case for work done by the gas too
     
  18. Feb 25, 2017 #17
    The problem is worded poorly. They want you to calculate the heat when the temperature reaches its maximum.
     
  19. Feb 25, 2017 #18
    ooh okay how did you arrive at that thanks anyways
     
  20. Feb 25, 2017 #19
    It's the only thing that makes sense to me.
     
  21. Feb 25, 2017 #20

    kuruman

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    What if you used the First Law to find an expression for the heat Q in terms of independent variable V and constants p0 and V0? Then you can find the value of V at which the heat that has entered the gas up to that point reaches a maximum before it starts going down, i.e. before heat starts being removed from the gas. The answer is a simple fraction of V0.
     
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