Can we show that the helmholtz free energy at equilibrium is minimized from its second derivative?

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Damarion Pierce · Accepted Answer

The first derivative at an extremum point       x    0   is 0 only when evaluated at       x    0  . Same goes with the second derivative. So you first have to evaluate the second derivative, then you plug in the specific   x  =      x    0  For example, take   y  =      x    2                      d            y                      d            x        =  2  x  and                               d                2            y                                d          2                    x        =  2At   x  =  0, the minimum,                     d            y                      d            x                  |        0    =  0, and                               d                2            y                                d          2                    x                  |        0    =  2If we were to apply your logic however, we would have                               d                2            y                                d          2                    x        =            d                      d            x                                    (                                                    d                            y                                                      d                            x                                )                ⏟                    =      0        =  0.So, you have to expand your last line.            d        2    F  =  −      d    P      d    V  −  P            d        2    V  −      d    S      d    T  −  S            d        2    T  .Now you apply the condition that you are at equilibrium, so       d    T            |                      e        q              =      d    V            |                      e        q              =      d    P            |                      e        q              =      d    S            |                      e        q              =  0, so that:            d        2    F            |                      e        q              =  −  P            d        2    V            |                      e        q              −  S            d        2    T            |                      e        q              .Then I guess that if pressure is positive, then the volume is a maximum so             d        2    V  &amp;lt;  0 so that the first term is positive.

Can we show that the helmholtz free energy at equilibrium is minimized from its second derivative?

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