Internal Energy and Entropy Boltzmanns Formula

Substituting the total energy of the system S of (D.14)2 into the Boltzmann distribution (D.19) yields 

Substituting the partition function (D.27) for the case of equi-difference energy, and differentiating this (recalling (D.26)2), we obtain the following expression for the total energy  

Here we should note that eo corresponds to the energy at the ground state. Substituting this into (D.29), we obtain 

Note that since we treat a closed system, drii = 0. Thus we obtain the following Boltzmann formula  

Substituting the above into (D.35), we obtain the Boltzmann formula of the molecular partition function as 

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