Express in terms of the temperature function and the partition function the form of the Helmholtz equation.

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Abigailf91er · Accepted Answer

The entropy is to be evaluated in the first case, then the Helmholtz free energyThe partition function is Z and the temperature is T.  s  =  k  [  N  ln  ⁡  N  −      ∑          i      =      0            N    i    ln  ⁡      N    i    ]    =  k  [  N  ln  ⁡  N  −  N      ∑          i                  −              ϵ        i            l              k        B            T        Z    ln  ⁡  N  (            −              ϵ        l                    k        B            T        Z    )  ]    =  k  [  N  ln  ⁡  N  −  N  ln  ⁡  N  +  N  ln  ⁡  Z  +      N                  k        B            T                  ∑              ϵ        i                    e                  −                      ϵ            i                    l          k                      B            T                                Z    ]  S  =      k    B    [  N  ln  ⁡  Z  +      U                  K        B            T        ]    =  N      k    B    ln  ⁡      Z    n    +      U    T      =      k    B    ln  ⁡      Z    N    +      U    T      =      k    B    ln  ⁡  z  +      U    T      U  =            N              ∑                  i          =          0                            ϵ        i            l              k        B            T        Z  for indistinguishable particles,   z  =      Z    N  for Helmholtz free energy,   F  =  U  −  T  S    =  U  −  T  [      U    T    +      k    B    ln  ⁡  z  ]    =  −      k    B    T  ln  ⁡  zAnswer: Helmholtz energy F in terms of T and z is   F  =  −      k    B    T  ln  ⁡  z

Express in terms of the temperature function and the partition

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