Statistical Treatment of Entropy

To truly appreciate how thermodynamic principles apply to chemical systems, it is of great value to see how these principles arise from a statistical treatment of how microscopic behavior is reflected on the macroscopic scale. While this appendix by no means provides a complete introduction to the subject, it may provide a view of thermodynamics that is refreshing and exciting for readers not familiar with the deep roots of thermodynamics in statistical physics. The primary goal here is to provide rigorous derivations for the probability laws used in Chapter 1 to introduce thermodynamic quantities such as entropy and free energies. 

In our statistical treatment of an ideal elastomer, we have assumed that the elastic force is entirely attributable to the conformational entropy of deformation, energy effects being neglected. That the theory reproduces the essential features of the elasticity of real elastomers attests to the basic soundness of this assumption. On the other hand, we know that in real elastomers such energy effects cannot be entirely absent, and deviations from the ideal elastomer model may be expected to occur. Let us now examine in greater detail the extent to which the neglect of energy effects is justified. We can rewrite equation (6-28) ... 

There are two modes that are not applied to the above statistical treatment of thermodynamic properties that may be a major factor in some molecular species. The first is contribution to entropy, S, and heat capacity, Cp(T) from internal rotors, which for some species can be significant. Molecular species that do not have internal rotors are represented by the above statistical analysis representation. However, for molecular species that have hindered internal rotors, the contributions to S and Cp(T) need to be separately calculated and incorporated into the thermodynamic properties. One method to estimate the hinder rotor contributions is by using the vibration frequency for the torsion. 

The Gaussian theory considers the number of possible conformations of a chain having a specified end-to-end distance. A more accurate non-Gaussian statistical treatment of the random chain is based on the distribution of sin j, i.e. of the angle between the direction of a random link and of the end-to-end vector. From the probability of finding n links in the range AGj, ri2 in A 2 and so on, the entropy of a single chain is derived [2b] as... 

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

Treatment of Solutions by Statistical Mechanics. Since the vapor pressure is directly connected with the free energy, in the thermodynamic treatment the free energy is discussed first, and the entropy is derived from it. In the treatment by statistical mechanics, however, the entropy is discussed first, and the free energy is derived from it. Let us first consider an element that consists of a single isotope. When the particles share a certain total energy E, we are interested in the number of recog-... 

In equilibrium measurements, there is the possibility of determining the reaction enthalpy AH directly from calorimetry and of combining it with logK (i.e., AG°) to get the reaction entropy, AS . This case, advantageous and simple from the statistical point of view, was only mentioned in a previous paper (149). Since that time, this experimental approach has been widely used (59, 62-65, 74-78, 134, 137, 138, 210, 211) hence, a somewhat more detailed mathematical treatment seems appropriate. 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

Many years ago, Lacher (17) explained the P-C-T data shown in Figure 1 with a statistical mechanical model, a model that has formed the basis of subsequent treatments of nonstoichiometry in other systems as well. With the assumption that the hydrogen atoms are localized to each site, the resulting partial configurational entropy is given by... 

The foregoing discussion has neglected most of the details of thermochemical properties of the adsorbed species, for example, tacitly taking the entropy of the surface species Ss to be zero. Statistical thermodynamics allows a more rigorous treatment of surface processes such as these, discussed next. 

Summary. In conclusion, the advent of current generation computers has allowed the development of a new level of rigor in statistical thermodynamic and dynamic studies of organic and bio-molecular systems. We have discussed how we can now include the relaxation of molecular geometry, the treatment of conformational entropy, and the inclusion of solvent effects in theoretical treatments of biolmolecular systems, all of extreme importance in simulating the behavior of those systems. In addition we have indicated how the vibrational spectra can be calculated and its conformational dependence be used as a probe of conformation. It was pointed out that these developments have for the most part occurred within the last five years and in fact most publications in the area have been in the last year or two. Their full Impact on this exciting field is yet to be felt. 

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