A Three-Step Gas Cycle

  1. 1. The problem statement, all variables and given/known data
    A monatomic ideal gas has pressure p_1 and temperature T_1. It is contained in a cylinder of volume V_1 with a movable piston, so that it can do work on the outside world.

    Consider the following three-step transformation of the gas:

    1. The gas is heated at constant volume until the pressure reaches Ap_1 (where A >1).
    2. The gas is then expanded at constant temperature until the pressure returns to p_1.
    3. The gas is then cooled at constant pressure until the volume has returned to V_1.

    It may be helpful to sketch this process on the pV plane.

    Part 1-
    How much heat DeltaQ_1 is added to the gas during step 1 of the process?
    Express the heat added in terms of p_1, V_1, and A.

    Part 2-
    How much work W_2 is done by the gas during step 2?
    Express the work done in terms of p_1, V_1, and A.

    Part 3-
    How much work W_3 is done by the gas during step 3?
    If you've drawn a graph of the process, you won't need to calculate an integral to answer this question.
    Express the work done in terms of p_1, V_1, and A.



    2. Relevant equations
    C_V = 12.47
    R = 8.31


    3. The attempt at a solution
    Part 1-
    I tried Q = p_1*V_1*(C_V/R) = 1.5*Ap_1*V_1 but I was told this is the final internal energy, not the change in internal energy. so I worked out that

    Q = [1.5*p_1*V_1*(AT_1-T_1)] / T_1 but the answer does not depend on AT_1 or T_1

    Part 2-
    all I've got so far is
    W = nRT*ln(V_f/V_i) = pV*ln(V_f/V_i)
    but thats about as far as I get.

    Part 3-
    I got Ap_1*V_1 but this is what the value would be if it were coming from V = 0. So I re-arranged pV=nRT to eventually get

    W = p_1[(p_1V_1)/(Ap_1) - V_1]
    but this is also wrong how do I take into account the initial state, wouldn't I just be able to write W = (Ap_1V_1) - V_1 ?
    Could someone please set me on the right path, I have been up late each night this week trying to work this out.
     
  2. jcsd
  3. Hello,
    (1)
    For monatomic ideal gas, the internal energy can be also writen as [tex]U=\frac{3}{2}PV[/tex]
    The first step is an isochor process, no work is done, the heat is just the change of internal energy.
    You can use the formula and try again.
    (2)
    For the second process, it is an isothermal one, which means [tex]PV=nRT=\text{constant}[/tex]
    Therefore [tex]AP_1\times V_1=P_1\times V_f\Rightarrow V_f=AV_1[/tex]
    Substitute this relation into your formula and get it.
    (3)
    Actually, i don't completely understand why you think from your words.
    But i know the work is the area under the curve on the P-V diagram.
    The work done by the gas in the third step is [tex]-P_1(AV_1-V_1)[/tex].
    (The work done by the gas in the first step is [tex]0[/tex] since no area.)
    The total work is the area inside the colse process on the P-V diagram.
    It is [tex]0+AP_1V_1\ln{A}-P_1(AV_1-V_1)[/tex].

    Hope these are helpful. Goodluck.
     
  4. So for part one, if delta_U = 3/2(pV)
    then delta_U should equal 3/2* the change in p * V = 3/2(Ap_1 - p_1)V_1. does this look right.

    for part 2, W = pV*ln(V_f/V_i)
    then W = p_1V_1*ln(AV_1/V_1), or should p and V be the changes in p and V, the feedback I get when I tried that answer was that N*k_B*T = A*p_1*V_1
     
  5. Hello,

    Part(1)
    You got it.

    Part(2)
    The formula [tex]W=pV\ln\left(\frac{V_f}{V_i}\right)[/tex] was derived in an isothermal process for ideal gas.
    In such an isothermal process, the product [tex]pV[/tex] is a constant, which means [tex]pV=p_iV_i=p_fV_f[/tex].
    The [tex]pV[/tex] substituted into the formula should be a product of some point on the process.
    [tex]p_1V_1[/tex] is not a point in the isothermal process, it's just a point in another process.

    Regards
     
  6. im working on the same problem now but im still a little confused. so what is the answer to part a,b,c.

    part a = 3/2(Ap1-p1)V1 ????

    part b = ??

    part c = ??


    im so lost! can anyone please help me? thanks a bunch!
     
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