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Gas work -- Heating the gas in a cylinder with a weighted piston on top

  1. Mar 28, 2017 #1
    1. The problem statement, all variables and given/known data
    In the open cylindrical chamber is the piston of the total mass m. The initial air pressure inside the container is pa, the initial temperature T0. The initial height of the piston above the bottom h0. Now we start the gas supply heat to the moment when the piston reaches the height h above the bottom of the container .Air is ideal diatomic gas.

    a / determine the final temperature T of the air inside the container

    b / specify the total heat Q which air is received in the container

    c / determine the efficiency of the piston stroke - ratio performed mechanical work and the total heat to the air inside the cylinder accepted

    upload_2017-3-28_9-13-53.png
    2. Relevant equations
    F = mgΔh
    W = F * ( h-h0) = p * S * ( h-h0) = p * ΔV
    pV = nRT
    paV0/T0 = p V/T
    3. The attempt at a solution

    work necessary for the piston stroke of the mass m ......W = F*Δh = m*g*Δh = m*g* ( h-h0)
     
    Last edited: Mar 28, 2017
  2. jcsd
  3. Mar 28, 2017 #2
    it is an adiabatic process? pVκ = konst.
     
    Last edited: Mar 28, 2017
  4. Mar 28, 2017 #3
    How can it be adiabatic if you are adding heat?
     
  5. Mar 28, 2017 #4
    Is there vacuum outside the container? If so, then you have calculated the work correctly.

    From a force balance on the piston, how is mg related to pa and S? Does the gas pressure change? What is the work in terms of pa, S, and Δh? What is the initial volume in terms of h0 and S? What is the final volume in terms of h and S? From the ideal gas law, what is the final temperature in terms of h, h0, T0?
     
  6. Mar 28, 2017 #5
    yes, there occurs heat exchange, I understand......


    no vacuum, out is atmospheric pressure......

    the initial volume V0 = S*h0
    the final volume V = S*h

    so answer a/ V/V0 = T/T0.....T = T0*h/h0
     
    Last edited: Mar 28, 2017
  7. Mar 28, 2017 #6
    b/ The first law of thermodynamics
    Q = W + ΔU system

    W = F*Δh = m*g*Δh = m*g* ( h-h0)

    ΔU = Ek = 3/2 nRT.....?
     
  8. Mar 28, 2017 #7
    This is on the right track. But, if there is air outside then the work you calculated is not correct. From a force balance on the piston, $$P_aS=P_0S+mg$$where ##P_a## is the gas pressure and ##P_0## is the outside pressure. So the work the gas does on the piston is $$W=P_aS(h-h_0)=(P_0S+mg)(h-h_0)$$

    Also, for a diatomic gas, the change in internal energy should be $$\Delta U=\frac{5}{2}nR(T-T_0)$$
     
  9. Mar 28, 2017 #8
    aha, thank you, now I see it.....



    c/ η = W/Q....?
     
  10. Mar 29, 2017 #9
    But I have a problem, I do not know molar amount n, I can not therefore calculate ΔU :-(
     
  11. Mar 29, 2017 #10
    Really? I think you have enough information to determine n. What do you think?
     
  12. Mar 29, 2017 #11
    Can I express it from the initial state?

    pa * S * h0 = nRTo...... n = (pa * V0)/R *T 0
     
  13. Mar 29, 2017 #12
    Yes.
     
  14. Mar 29, 2017 #13
    Q = W + ΔU = (p0S + mg ) ( h-h0 ) + 5/2 R ( T-T0) * ( (p0Sh0)/RT0 )
     
  15. Mar 29, 2017 #14
    The ##\Delta U## should have pa=p0, not p0. And, in that term, you should eliminate T and T0. Also, in the W term, you should use paS in place of (p0S + mg ).
     
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