Entropy and Chemical Change. I. Characterization of Product (and Reactant) Energy Distributions in Reactive Molecular Collisions: Information and Entropy Deficiency

Abstract

The present paper considers optimal means of characterizing the distribution of product energy states resulting from reactive collisions of molecules with restricted distributions of initial states, and vice versa, i.e., characterizing the particular reactant state distribution which yields a given set of product states, at a specified total energy E. The S-matrix, or reaction probability matrix P(E), ``global'' in nature, contains much more detail than necessary to reproduce the results of any single specific experiment or computer simulation thereof (via classical mechanical trajectory calculations). Moreover, since reactant and/or product state resolution is always experimentally limited (to a greater or lesser degree), data are necessarily coarse-grained accordingly. Many quantal features are thereby lost and the results are often at a level appropriate for comparison with classical calculations (e.g., in the form of low-resolution contour maps of energy disposal). Such contour plots of the yield function Y or the averaged transition probability ω (the ``poor-man's'' P-matrix) nevertheless contain the essence of the dynamical results. It is suggested to represent the energy dependence of global-type results in the form of square-faced prism plots (contour maps vs E), and of data for specific-type experiments (or computer simulations) as triangular-faced prismatic plots (contour maps vs E). The essential parameters defining the internal state distribution are isolated, and the information content I(E) of such a distribution (for a microcanonical ensemble) is put on a quantitative basis. The relationship between the information content, the surprisal, and the entropy of the continuous distribution is established, thereby making the link between microscopic collision theory and nonequilibrium statistical mechanics. The concept of an ``entropy deficiency'' ΔS′, which characterizes the specificity of product state formation, is suggested as a useful measure of the deviance from statistical (``phase-space dominated'') behavior. The degradation of information by experimental averaging is considered, leading to bounds on the entropy deficiency. The present analysis is useful for the characterization of both experimental results and theoretical models.