Derive the following general equation,which is similar to the Gibbs-Helmholtz equation: ( ∂ / ∂ T ) ( F / T ) = − U / T 2

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Derive the following general equation,which is similar to the Gibbs-Helmholtz equation:   (      ∂        /        ∂    T  )  (  F      /    T  )  =  −  U      /        T    2

percolarse2rzd · Accepted Answer

The Helmholtz free energy is a thermodynamic potential which is a measure of the useful work done by a system at a constant temperature. Mathematically, Helmholtz free energy is given by the relation where U, T and S are the internal energy, temperature, and entropy of a system respectively.The Helmholtz free energy is given by the relation   F  =  U  −  T  STaking differential of the above expression we get  d  F  =  d  U  −  T  d  S  −  S  d  TBy the combined first and second law of thermodynamics  T  d  S  =  d  U  +  P  d  VSubstituting this in the above equation we get  d  F  =  d  U  −  d  U  −  P  d  V  −  S  d  T    ⇒  d  F  =  −  P  d  V  −  S  d  TNow when we consider Helmholtz free energy as a function of temperature and volume which are considered as independent variables, we have  F  =  F  (  T  ,  V  )Taking differential of the above equation we get  d  F  =  (      ∂    F      ∂    T      )    V    d  T  +  (      ∂    F      ∂    V      )    T    d  VComparing the above expression with   d  F  =  −  P  d  V  −  S  d  T we get  P  =  −  (                    ∂            F                      ∂            V            )    T      S  =  −  (                    ∂            F                      ∂            T            )    V  Now from the definition of Helmholtz free energy   F  =  U  −  T  SSubstituting the expression for entropy S, we have  F  =  U  +  T  (                    ∂            F                      ∂            T            )    V      ⇒  U  =  F  −  T  (                    ∂            F                      ∂            T        )    ⇒  U  =  −      T    2                                ∂                                      ∂                    T                    (              F        T            )        V      ⇒                              ∂                                      ∂                    T                    (              F        T            )        V    =  −  U      T    2  This is the required result            ∂                      ∂            T        (      F    T        )    V    =  −  U      T    2

Derive the following general equation,which is similar to the Gibbs-Helmholtz equation: ( <mrow

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