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Adiabatic stretching of a rubber band

  1. Aug 10, 2017 #1
    1. The problem statement, all variables and given/known data
    For a stretched rubber band, it is observed experimentally that the tension ##f## is proportional to the temperature ##T## if the length ##L## is held constant. Prove that:

    (b) adiabatic stretching of the band results in an increase in temperature;
    (c) the band will contract if warmed while kept under constant tension.

    2. Relevant equations
    the first law:
    $$
    dU=Tds+fdL=C_L dT
    $$
    $$
    f=\left (\frac{\partial f}{\partial T}\right )_L T
    $$
    $$
    \left (\frac{\partial L}{\partial f}\right )_T>0
    $$

    3. The attempt at a solution

    (b)


    For an adiabatic process, entropy doesn't increase and so:
    $$
    dU=fdL=C_LdT
    $$

    The force is always positive and so temperature is positively increased by length.

    This feels too simple so i doubt I am correct. I have no idea for part (c).
     
  2. jcsd
  3. Aug 10, 2017 #2
    How do you know that dU=CdT?
     
  4. Aug 10, 2017 #3
    There was a part A that was to show this is the case
    $$dU=C_vdT+\left[f-T\left(\frac{\partial f}{\partial T}\right)_L\right]dL$$

    The second term drops out
     
  5. Aug 10, 2017 #4
    In part (a) it was shown that f = Tg(L), where g is an increasing function of L. This is the equation of state of the rubber. So if T increases at constant f, what happens to L?
     
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