Exponential phase, description

To model the experimental data we used a global-fit procedure to simulate EPS, integrated TG, heterodyne-detected TG, and the linear absorption spectrum simultaneously. The pulse shape and phase were explicitly taken into account, which is of paramount importance for the adequate description of the experimental data. We applied a stochastic modulation model with a bi-exponential frequency fluctuation correlation function of the following form ... 

In Chapter 2, the first chapter of the gas-phase part of the book, we began the transition from microscopic to macroscopic descriptions of chemical kinetics. In this last chapter of the gas-phase part, we will assume that the Arrhenius equation forms a useful parameterization of the rate constant, and consider the microscopic interpretation of the Arrhenius parameters, i.e., the pre-exponential factor (A) and the activation energy (Ea) defined by the Arrhenius equation k(T) = Aexp(—Ea/kBT). 

There are two versions of the physical description of this system. According to the Gouy-Chapman (G-C) theory, counter ion thermal energy runs counter to the electrostatic attraction and a secondary diffuse layer in which the potential decays almost exponentially because of screening effects is generated. Both layers are under dynamic equilibrium. The electrical potential difference, between the stationary phase and the bulk eluent can be theoretically estimated. Figure 3.2 depicts the G-C model. 

Equation (10-1) is based on the assumption of simple additivity of all interactions and a competitive nature of analyte/eluent interactions with the stationary phase. The paradox is that these assumptions are usually acceptable only as a first approximation, and their application in HPLC sometimes allows the description and prediction of the analyte retention versus the variation in elution composition or temperature. For most demanding separations where discrimination of related components is necessary, the accuracy of such prediction is not acceptable. It is obvious from the exponential nature of equation (10-1) that any minor errors in the estimation of interaction energy, or simple underestimation of mutual influence of molecular fragments (neglected in this model), will generate significant deviation from predicted retention factors. 

Doubling the volume fraction of one phase doubles the probability of solute interaction and, consequently, doubles its contribution to retention. There is another interesting outcome from the results of Purnell and his co-workers. Where a linear relationship existed between the retention volume and the volume fraction of the stationary phase, the linear functions of the distribution coefficients could be summed directly, but their logarithms could not. In many classical thermodynamic descriptions of the effect of the stationary-phase composition on solute retention, the stationary-phase composition is often taken into account by including an extra term in the expression for the standard free energy of distribution. The results of Purnell indicate that this is not acceptable, as the solute retention or distribution coefficient is linearly not exponentially related to the stationary-phase composition. The stationary phases of intermediate polarities can easily be constructed from... 

We may have defined Equation (44) in a slightly different manner as is usual in the literature. Instead of writing R in the argument of the exponential function, one can write R/ = R/ + [similarly to f,- = x vectors in Equation (29) for atom displacements]. In such a case the Fourier coefficients, Tiy-, of the new expression are related to those of Equation (44) by a phase factor, Sk/ = Tiyexp(—27rikxy), that depends on the atom positions inside the unit cell. We shall see that the convention we have adopted is more convenient for a unified description of commensurate and incommensurate magnetic structures. 

The anodic current is expected to increase exponentially with the electrode potential provided that the condition AI/e = A(Ac sc) again fulfilled. In such an anodic reaction in which electrons are transferred into the valence band, holes must be available at the surface. Frequently, scientists then argue in terms of hole transfer. This is only a rather lax description and has no real physical basis for the process at the boundary of two different phases. 

The deactivation process can therefore be described as the sum of two simultaneous processes deactivation by sintering following second order kinetics, and deactivation due to catalyst reduction. A mathematical description of this deactivation mechanism was not available, but the examination of the first phase of the deactivation curves seemed to indicate an exponential dependency. Therefore the following equation was proposed to describe the... 

---