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Heat Engine Question

  1. Dec 10, 2013 #1

    I am trying to figure out heat engines. I don't understand why heat from a hot reservoir MUST exhaust heat into a cold reservoir. How does that satisfy the second law of thermodynamics? Why can't all energy from the hot reservoir be used to do work?

  2. jcsd
  3. Dec 10, 2013 #2
    If all the heat from the hot reservoir is used to produce work and that work is used to run a heat pump that pumps heat from a cold reservoir back to the hot reservoir the hot reservoir gets even hotter and you created a perpetual motion machine of the second kind which is forbidden by the second law of thermodynamics.
  4. Dec 10, 2013 #3
    Thanks :) Can you explain why putting all the energy from the hot reservoir to work would cause entropy of the system to decrease?
  5. Dec 10, 2013 #4
    The formula for the change in entropy is [itex]dS=\frac{\delta Q}{T}[/itex], where [itex]\delta Q[/itex] is the heat lost and is negative while [itex]T[/itex] is the temperature and is positive. Clearly [itex]dS[/itex] is negative representing a decrease of entropy forbidden by the 2nd. If some of the heat goes to a second reservoir at lower temperature than there is a second term in the expression for the entropy. [itex]dS=\frac{\delta Q_1}{T}_1 + \frac{\delta Q_2}{T}_2[/itex], [itex]\delta Q_1[/itex] is the heat lost at the higher temperature reservoir [itex]T_1[/itex] and is still negative, but [itex]\delta Q_2[/itex] is the heat gained at the lower temperature reservoir [itex]T_2[/itex] and is positive. Since [itex]T_2[/itex] is smaller than [itex]T_1[/itex] the second term can counter the first leading to a positive increase of entropy and there will be some heat (but not all) left over to produce work.
  6. Dec 10, 2013 #5
    Thank you :)
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