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Thermo - double Vrms, find heat required

  1. Feb 23, 2016 #1
    1. The problem statement, all variables and given/known data
    n moles of a diatomic gas with CV =5/2 R has initial pressure pi and volume Vi. The gas undergoes a process in which the pressure is directly proportional to the volume until the rms speed of the molecules has doubled.
    a. Show this process on a pV diagram.
    b. How much heat does this process require? Give your answer in terms of n, pi , and Vi

    2. Relevant equations
    PV = 2/3 N εavg
    εavg = 1/2 mv2
    PV = nRT
    E = 5/2 nRT
    Q = ΔE

    3. The attempt at a solution
    Part a was done simply by linearly increasing Pi and Vi to 4 Pi and 4 Vi respectively.

    Part b:
    [itex]ΔE = \frac{5}{2} nRΔT[/itex]
    [itex] = \frac{5}{2} nR(\frac{4P_iV_i}{nR} - \frac{P_iV_i}{nR})[/itex]
    [itex] = \frac{15}{2} P_iV_i[/itex]

    However, the answer in the back is:

    [itex]\frac{15n+3}{2} P_iV_i[/itex]

    Not sure where the n and +3 came from, maybe PV = 2/3 N εavg? Plugging that in doesn't seem to work though.
     
  2. jcsd
  3. Feb 23, 2016 #2

    TSny

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    Homework Helper
    Gold Member

    Welcome to PF!

    Does this equation account for any work done by the gas?

    This would cause PV to increase by a factor of 16.

    OK, now you have PV increasing by a factor of 4. I believe this is correct for ΔE. However, to get Q you will need to take into account any work done by the gas.

    I think there must be a misprint in this expression. Q should be proportional to the number of moles.
     
  4. Feb 23, 2016 #3
    You're right, it should be 2 Pi and 2 Vi.
    Okey dokey.

    [itex]\Delta E = \frac{15}{2} P_iV_i[/itex]
    [itex] W = \int pV [/itex]
    [itex]\Delta E = Q -W_s[/itex]
    [itex] \frac{15}{2} P_iV_i = Q - ( \frac{1}{2} (2V_i - V_i)(2P_i - P_i) + (P_i V_i) ) [/itex]
    [itex] Q = \frac{15}{2} P_iV_i + \frac{P_i V_i}{2} + P_i V_i [/itex]
    [itex] Q = \frac{18}{2} P_iV_i[/itex]

    That looks a little better. I would not be surprised if the answer is a misprint; that has happened a lot for this textbook. (Knight Physics 3rd ed.)

    Thanks for your help!
     
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