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Challenging Thermodynamics Problem

  1. Feb 23, 2007 #1
    «Challenging» Thermodynamics Problem

    1. The problem statement, all variables and given/known data

    Consider a cubical vessel of edge a, having a small hole in one of its walls. The total thermal resistance of the wall is [itex] \varphi [/itex] [itex] \mbox{At time} \ t=0 [/itex], it contains air at atmospheric pressure [itex] p_a [/itex] and temperature [itex] \theta_0 [/itex]The temperature of the surrounding is [itex] \theta_a ( > \theta_0 ) [/itex] Find the amount of gas in moles in the vessel at time t. Take [itex] C_v = \frac{5R}{2} [/itex]


    2. Relevant equations

    [tex] dQ=dW+dU [/tex]
    [tex]dQ= \frac{i_{th}}{\varphi} [/tex]
    [tex]
    PV=nR \theta \mbox{ideal gas eqn} [/tex]

    3. The attempt at a solution

    I assumed pressure to be constant throughout the problem.
    [tex] P=P_a [/tex]
    Initially,
    [tex] i_{th} = \frac{\theta_a - \theta_0}{\varphi} [/tex]
    Now since volume and pressure both are constant,
    PV=const.
    or,
    [itex]nRd\theta + R\theta dN = 0[/itex]

    [itex] \frac{d\theta}{\theta} = -\frac{dn}{n} [/itex]

    Now i try to apply first law, which gives,
    [tex] \frac{\theta_a - \theta}{\varphi} dt = nC_vd\theta + \theta C_v n [/tex] where [itex] \theta\ is\ temperature \ at\ time \ t[/itex]
    But since these rates are also varying, i have no idea how to continue. Specially if someone could throw light on the integration part.
    Thanks for any assistance.
     
    Last edited: Feb 24, 2007
  2. jcsd
  3. Feb 23, 2007 #2
    what level is this question?
     
  4. Feb 24, 2007 #3
    Senior secondary
     
  5. Feb 25, 2007 #4
    Any1 got an idea?
     
  6. Mar 31, 2007 #5
    Long time....if any1 knows a tactic for this do tell :cry:
     
  7. Mar 31, 2007 #6
    Mabye you should try posting this in the Advanced Physics forum instead?
     
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