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What is the second law of thermodynamics

  1. Jul 24, 2014 #1
    Definition/Summary

    The second law has various forms. I shall give these here then show how they are logically connected.

    Entropic Statement of the Second Law:

    There exists an additive function of thermodynamic state called entropy which never decreases for a thermally isolated system.

    Clausius' Statement of the Second Law:

    No process exists in which heat is transferred from a cold body to a less cold body in such a way that the constraints on the bodies remain unaltered and the thermodynamic state of the rest of the universe does not change.

    Equations

    [tex]\Delta S_{tot} = \Delta S_{sys} + \Delta S_{surr} \geq 0 [/tex]

    Extended explanation

    The entropic statement of the second law requires that systems are thermally isolated. This is rarely the case and so the second law becomes:

    [tex]\Delta S_{tot} = \Delta S_{sys} + \Delta S_{surr} \geq 0 [/tex]

    Thermodynamic processes can be reversible and so [itex] \Delta S_{sys}[/itex] can be < 0.

    Lets consider the transfer of heat from body B to body A in such a way that the constraints on those two bodies never change.

    From the second law:

    [tex] dS_{tot} = dS_A+dS_B = \left(\frac{\partial S_A}{\partial E_A}\right)_{PV}dE_A + \left(\frac{\partial S_B}{\partial E_B}\right)_{PV} dE_B[/tex]

    therefore;

    [tex] dS_{tot} = \left[\left(\frac{\partial S_A}{\partial E_A}\right)_{PV} -\left(\frac{\partial S_B}{\partial E_B}\right)_{PV}\right]dq_A \geq 0 [/tex]

    where, [itex] dq_A = - dq_B[/itex], from the first law. Also the work is zero since [itex]P[/itex] and [itex]V[/itex] don't change.

    We now define the coldness of a body to be:

    [tex] \frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_{PV}[/tex]

    and thus:

    [tex] \left(\frac{1}{T_A} - \frac{1}{T_B} \right) dq_A \geq 0 [/tex]

    So for [itex] dq_A > 0 [/itex] we must have:

    [tex] \frac{1}{T_B} < \frac{1}{T_A}[/tex]

    Body B must be less cold than body A and we conclude Clausius' statement of the second law.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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