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Minimal Helmholtz energy

  1. Mar 4, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that for a reaction occurring at constant T and V, F is minimized at equilibrium.

    2. Relevant equations
    ##F=U-TS##
    ##TdS=dU+pdV-\mu dN##
    3. The attempt at a solution
    ##dF=dU-d(TS)=dU-TdS-SdT=dU-dU -pdV+ \mu dN -S dT=-pdV - SdT + \mu dN##. At constant T and V this reduces to ##dF = \mu dN##. But I don't know what to do from here. Also in the next problem we have to use the fact that F is minimized at equilibrium to prove a relation between the chemical potentials of certain elements involved in a chemical reaction, so I assume i can't set ##dN=0##, as in a chemical reaction N changes and it seems that the fact that F is minimal at equilibrium holds in chemical reactions, too.
    How can I solve this? Thank you!
     
  2. jcsd
  3. Mar 6, 2017 #2

    Mapes

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    When multiple species exist, the last term must be ##\Sigma\mu_i \,dN_i##, no? That's quite different from ##\mu\,dN## in the context of your problem.

    Then, you have to prove that ##F## is minimized rather than maximized or at a saddle point. Use the second derivative to do this.
     
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