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PV Diagram

  1. May 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider the following PV diagram of one mole of a monatomic gas:

    hMivkKO.png

    The units are in atm and litres.

    Find the change in internal energy, heat, and work for all 3 processes and for the cycle as well.

    2. Relevant equations

    PV = nRT
    ΔU = Q - W
    ΔU = (3/2)*R*ΔT
    Q = (5/2)*R*ΔT (for constant pressure)

    3. The attempt at a solution

    For the work, I just found the areas:

    A -> B: 0 J
    B -> C: +3.5 J
    C -> A: -1 J

    For the cycle, the work done is just the total work added:
    3.5 - 1 = 2.5 J

    For the cycle, the Q is the same as the Work:
    2.5 J

    For the cycle, the ΔU = 0.

    So far I have this (unless I did something wrong):

    3535rkr.png

    Since for a constant volume process the ΔU = Q, I decided to do that A -> B first.

    T_A = (0.5 atm)(1 L) = (1)(0.0821)T, T = 6.09 K
    T_B = (0.5 atm)(1 L) = (1)(0.0821)T, T = 36.5 K

    ΔT = 30.41 K

    ΔU = (3/2)*R*ΔT = (3/2)*0.0821*30.41 = 3.75 J = Q

    So:

    x45zO8h.png

    The constant pressure process looks easier so I'll do C -> A next:

    T_C = (0.5 atm)(3 L) = (1)(0.0821)T, T = 18.27 K
    T_A = (0.5 atm)(1 L) = (1)(0.0821)T, T = 6.09 K

    ΔT = -12.18

    Q = (5/2)*R*ΔT = (5/2)*0.0821*(-12.18) = -2.5 J

    ΔU = Q - W = (-2.5) - (-1) = -1.5

    So:

    FW9DwiX.png

    Now I can just add up the unknown columns:

    ΔU for B -> C: 3.75 + (-1.5) + x = 0, x = -2.25

    Q for B -> C: 3.75 + (-2.5) + x = 2.5, x = 1.25

    So I got:

    ZiCNBTq.png

    But I got this problem incorrect. What errors have I made?
     
    Last edited: May 19, 2013
  2. jcsd
  3. May 19, 2013 #2

    ehild

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    Homework Helper
    Gold Member

    Your work looks correct. Are you sure the work asked is the work done by the gas?


    ehild
     
  4. May 19, 2013 #3

    Andrew Mason

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    Science Advisor
    Homework Helper

    Your method is correct but you have to check your units. In the MKS system the units for pressure are Pascals and the units for volume are m3. Your PV diagram is in atmospheres and litres.

    So, for example, the work done by the gas from B to C is not 3.5 Joules but 3.5 atm. litres = 3.5(101325Pa) 10-3 m3 =350 J.

    AM
     
  5. May 20, 2013 #4

    ehild

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    Homework Helper
    Gold Member

    AM is right. You used litres and atm as units, then you got also the energy, heat, and and work in units (Latm) instead of joules.

    ehild
     
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