Rigor changes following

Errors of omission result from an insutHcient rigor in following strict laboratory protocols. For example, to determine the rate kinetics of a gas-phase reaction, the necessary measurements include composition at the entrance and exit, reactor volume, flow rate, pressure, and temperature. When the reactor is operated at atmospheric pressure—that is, the effluent exits to the atmosphere—the actual operating pressure is invariably different from 1.01325 bar. Not only does it depend on the elevation above sea level but also the meteorological conditions. Within hours, the barometric pressure may change by as much as 5%. 

There now exist several methods for predicting the free energy associated with a compositional or conformational change.7 These can be crudely classified into two types "exact" and "approximate" free energy calculations. The former type, which we shall discuss in the following sections, is based directly on rigorous equations from classical statistical mechanics. The latter type, to be discussed later in this chapter, starts with statistical mechanics, but then combines these equations with assumptions and approximations to allow simulations to be carried out more rapidly. 

The mean reaction time or reaction timescale (also called relaxation timescale relaxation denotes the return of a system to equilibrium) is another characteristic time for a reaction. Roughly, the mean reaction time is the time it takes for the concentration to change from the initial value to 1/e toward the final (equilibrium) value. The mean reaction time is often denoted as x (or Xr where subscript "r" stands for reaction). The rigorous definition of x is through the following equation (Scherer, 1986 Zhang, 1994) ... 

In the usual kinetic experiments all these parameters must be rigorously constant so that the experimental C-t data, that is, the kinetic profile obtained following the change in concentration with time, can be easily fitted to the differential form of the first-order equation (2) or to its integrated, exponential [Equation (3)] or logarithmic [Equation (4)], forms to obtain the optimized values... 

Even though (6.105) is only an approximation, its symmetry properties (e.g., its parity) are the same as those of the true molecular wave function, since the symmetry properties of the wave function follow rigorously from the symmetry of the true Hamiltonian H the perturbation H cannot change the overall symmetry of p. 

A rigorous theoretical treatment of the non-alternant and heterocyclic indolizine is extremely difficult and, even for the related isoconjugate hydrocarbon, far from conclusive. Many questions, however, in which experimentalists are interested may be answered in a satisfactory way on the basis of a perturbational treatment. This approach has been used for a discussion of the electronic spectra of indolizine and some azaindolizines (63JCS3999). Following first-order PMO theory the 7r-stabilization which follows from aza substitution at the different positions of the model molecule depends on the ir-electron density qt as well as the change in electronegativity Sat (B-75MI30801). The perturbations caused by aza substitution of the indenyl anion are depicted in Scheme 1. 

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