Thermodynamic Properties and Statistical Mechanics

The equations of statistical mechanics are derived by applicationof Eq. (5.41) to a large number of macroscopicallyidenticalsystems(aneiisemble)iiiallof theirmany allowed quantum states. The entropy S as given by Eq. (5.42) is tlien a statistical average value for the ensemble. Ultimately, tlie result is an equationfor tlie entropy  

According to Stirling s formnla IniV = A hiA — Na also kNA = R- Making these substitutions gives  

Equation (16.20) for the molar entropy of an ideal gas allows calculation of absolute entropies for tile ideal-gas state. The data required for evaluation of the last two terms on tlie right are tlie bond distances and bond angles in the molecules, and the vibration frequencies associated witli tlie various bonds, as determined from spectroscopic data. The procedure lias been very successful in the evaluation of ideal-gas entropies for molecules whose atomic stractures are known. 

Botli tile classical and statistical equations [Eqs. (5.40) and (16.20)] yield absolute values of entropy. As is evidentfrom Table 16.3, good agreementbetween the statistical calculations and those based on calorimetric data is obtained. Results such as these provide impressive evidence for tile validity of statistical mechanics and quantum theory. In some instances results based on Eq. (16.20) are considered more reliable because of uncertainties in heat-capacity data or about tlie crystallinity of the substance near absolute zero. Absolute entropies provide much of tile data base for calculation of the equilibrium conversions of chemical reactions, as discussed in Chap. 13. 

The fundamental property relation most intimately connected with statistical mechanics is Eq. (6.9), which expresses the differential of the Helmholtz energy as a function of its canonical variables T and V  

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