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Pressure-Volume Diagrams

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Practice

practice problem 1

One mole of an ideal, monatomic gas runs through a four step cycle. All processes are either isobaric or isochoric. The pressure and volume of the gas at the extreme points in the cycle are given in the table below.
  1. Sketch the PV graph of this cycle.
  2. Determine the temperature at state A, B, C, and D.
  3. Calculate W, Q, and ΔU on the path A→B, B→C, C→D, D→A and for one complete cycle. (Include the algebraic sign with each value.)
  4. Does this cycle behave more like an engine or a refrigerator?
A four step cycle (isobaric & isochoric)
state A B C D  
P (Pa) 100,000 100,000 200,000 200,000
V (m3) 0.020 0.060 0.060 0.020
T (K)
path A→B B→C C→D D→A ABCDA
descrip­tion iso­baric iso­choric iso­baric iso­choric closed cycle
∆U (J)
Q (J)
W (J)

solution

  1. Plot the four given points. Connect them with straight lines. Isobars are horizontal lines. Isochors are vertical lines. Add arrows to show the direction of each path. The overall cycle runs counterclockwise.

    Line graph

  2. Compute the temperature at each point using the ideal gas law.

    T =  PV  ⇐  PV = nRT
    nR

    Do it four times. Note that the temperature changes between points are all simple multiples (2×, 3×) or simple fractions (½, ⅓). That's because the pressures and volumes always changed by simple amounts. The numbers were rigged to make it easy to check your work.

    TA =  (100,000 Pa)(0.020 m3)  = 240 K
    (1 mol)(8.31 J/mol K)
    TB =  (100,000 Pa)(0.060 m3)  = 720 K
    (1 mol)(8.31 J/mol K)
    TC =  (200,000 Pa)(0.060 m3)  = 1440 K
    (1 mol)(8.31 J/mol K)
    TD =  (200,000 Pa)(0.020 m3)  = 480 K
    (1 mol)(8.31 J/mol K)

  3. Compute the change in internal energy on each segment using the function of state.

    ∆U = 32nR∆T

    Do this four times, and watch the signs. They matter for the computations that follow.

    ∆UA→B =  32(1 mol)(8.31 J/mol K)(720 K − 240 K)
    ∆UA→B =  + 6,000 J
       
    ∆UB→C =  32(1 mol)(8.31 J/mol K)(1440 K − 720 K)
    ∆UB→C =  + 9,000 J
       
    ∆UC→D =  32(1 mol)(8.31 J/mol K)(480 K − 1440 K)
    ∆UC→D =  − 12,000 J
       
    ∆UD→A =  32(1 mol)(8.31 J/mol K)(240 K − 480 K)
    ∆UD→A =  − 3,000 J

    Let's start filling in the table. Include a sign with all numbers associated with a path. The pressure volume, and temperature numbers are always positive, so adding a sign to them would be redundant. I like to use red for negative values in a table, but that's just a personal preference.

    A four step cycle (isobaric & isochoric)
    state A B C D  
    P (Pa) 100,000 100,000 200,000 200,000
    V (m3) 0.020 0.060 0.060 0.020
    T (K) 240 720 1440 480
    path A→B B→C C→D D→A ABCDA
    descrip­tion iso­baric iso­choric iso­baric iso­choric closed cycle
    ΔU (J) +6,000 +9,000 −12,000 −3,000
    Q (J)
    W (J)

    Compute the work done on each segment next. That's just an area with a sign switch. The areas under the isochors are both zero. The areas under the isobars are both rectangles. When you find an area under a segment running in the positive direction it's positive. When you go the other way, it's negative. If you prefer to do calculations systematically, compute the rectangular areas using base times height, where "base" is final minus initial volume (order of subtraction matters) and "height" is volume.

    W = −area = −bh = −(V2 − V1)P

    Do this twice.

    WA→B =  −(0.060 m3 − 0.020 m3)100,000 Pa
    WA→B =  −4,000 J
       
    WC→D =  −(0.020 m3 − 0.060 m3)200,000 Pa
    WC→D =  +8,000 J

    Add your results to the table.

    A four step cycle (isobaric & isochoric)
    state A B C D  
    P (Pa) 100,000 100,000 200,000 200,000
    V (m3) 0.020 0.060 0.060 0.020
    T (K) 240 720 1440 480
    path A→B B→C C→D D→A ABCDA
    descrip­tion iso­baric iso­choric iso­baric iso­choric closed cycle
    ΔU (J) +6,000 +9,000 −12,000 −3,000
    Q (J)
    W (J) −4,000 0 +8,000 0

    The remaining empty cells are filled in using the first law of thermodynamics…

    ∆U = Q + W

    or, in the case of the whole cycle ABCD, by just doing a sum across a row.

    A four step cycle (isobaric & isochoric)
    state A B C D  
    P (Pa) 100,000 100,000 200,000 200,000
    V (m3) 0.020 0.060 0.060 0.020
    T (K) 240 720 1440 480
    path A→B B→C C→D D→A ABCDA
    descrip­tion iso­baric iso­choric iso­baric iso­choric closed cycle
    ΔU (J) +6,000 +9,000 −12,000 −3,000 0
    Q (J) +10,000 +9,000 −20,000 −3,000 −4,000
    W (J) −4,000 0 +8,000 0 +4,000

Let's add some of what we've learned about this cycle to our graph.

Line graph

  1. This cycle is more like a refrigerator than an engine. The net work done was positive, which means work was being done on the gas. The net heat was positive, which means more heat was released to the environment than was generated by doing work on the gas. That heat came from something and was deposited into the environment. This is what a refrigerator does. It uses work to extract heat from something and then deposits that heat outside itself.

practice problem 2

Write something else.

solution

Answer it.

practice problem 3

Write something different.

solution

Answer it.

practice problem 4

Write something completely different.

solution

Answer it.