Exponential variation

Assume that an electric field varies exponentially with time as  

Further assuming that tq -C r and t tq, i.e. the rate of the field decay is considerably small and only times that widely exceed the time constant tq are considered, we have  

The volume density of charge S2 decays exponentially with time at exactly the same rate as the electric field, regardless of the conductivity of the medium. 

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In a first type of systems, redox centers are randomly distributed in a rigid matrix, glass or polymer [73, 74, 75], Donor or acceptor centers are initially created photochemically or by pulse radiolysis, and the study of the return to equilibrium of the system allows the determination of the law k(R) giving the rate variation as a function of the intercenter distance R. The experimental data are well described by an exponential law, which is considered as reflecting an exponential variation of the electronic factor ... 

Equation 7 shows the exponential variation of the homogeneous nucleation rate with an increasing supersaturation ratio. Equation 8, coming from the model of Strickland-Constable [59], establishes that a linear relation exists between growth rate and supersaturation ratio. The direct comparison of Nr and Gr is difficult because of their different units. However, Fig. 7, although only schematic, shows the variation of these two rates versus S, predicting that for a certain value of S the nucleation rate will become preponderant on the growth rate. Moreover, it also shows, that, in the case of preponderance... 

The interpretation of the anodic branch of LSV for p-Si is apparently more simple because the current increases following an exponential variation with a Tafel slope of 60-80 mV/decade. In this case, an accumulation layer is generated, and then the current is only controlled by the kinetics of the electrochemical reaction, which involves several successive steps. It is not necessary to account for the various reaction paths proposed by many authors. 

For the strong interactions dealt with in the last section, interatomic potentials are determined by overlap-dependent terms which have an inverse exponential variation with internuclear distance (R). At large separations the interactions are independent of overlap and may be represented by inverse powers ofR. There will be between these two an intermediate region, which encompasses the van der Waals minimum for non-bonded interactions, and we will take this to be contained in our definition of a weak-interaction. 

We have seen, that the crystallinity for these copolymers, wc, remains nearly constant as a function of temperature (see Sect. 7.3). In addition, the value of the crystal thickness, 1, is also constant below and above Tc. Since H and b are constants, one may assume that, for these copolymers, the quantity w H0/ (1 + b/1) = const for each composition. Hence, according to Eq. 4, for a given crystalline phase, one can essentially expect an exponential variation of H as a function of T. 

We now abandon the exponential variation on which we have focussed for a more general concentration dependence. In particular, if... 

Isothermal The temperature of the sample is maintained constant by adjusting the temperature of the surroundings in an appropriate way. The advantage is that the temperature effect, the exponential variation of the reaction rate, is eliminated during the measurement, which gives direct access to the conversion term of the reaction rate. The drawback is that there is no information on the temperature effect from one experiment alone. A series of experiments at different temperatures is required for this purpose (see Sections 11.4.2.1 and... 

P.7.C.I. Two semi-infinite media A symmetrically coated with a finite layers a of thickness D with exponential variation ea(z) perpendicular to the interface, retardation neglected... 

P.7.C.2. Exponential variation in finite layer of thickness D, symmetric structures, no discontinuities in s, retardation neglected... 

P.7.C.3. Exponential variation of dielectric response in an infinitely thick layer, no discontinuities in e, discontinuity in ds(z) at interface, retardation neglected... 

Because the temperature variation of 5 is to be compared with the strong exponential variation of e EIRT it is difficult to separate from experimental data the temperature dependence of 5 and E. For this reason it is a convenient working rule to assume that 5 is a constant. The activation energy E is then given by the slope of the line obtained by plotting 2.303 R log k against 1/T, as shown in Fig. 5. In most cases little error will be introduced by this simplification. 

In an ingenious ultramicroscopic study of the components of turbulent flow near a pipe wall, Page and Townend (19) showed that a film of liquid at the wall possesses a small degree of Jerky, laminar flow, while the v - component normal to the interface fails to very small values. To account for the exponential variation of k with D, King (lib) assumed that v might be proportional to a small power, say the cube, of the distance d (Fig. i) ... 

An exaggerated emphasis on heats of chemisorption has probably been harmful in the proper understanding of the role of chemisorption in surface phenomena. Thus the marked nonuniformity of all surfaces with respect to heats of chemisorption has led to rather elaborate treatments where models of surface heterogeneity (statistical distribution of energy sites) or, less successfully, specific forces of interaction between adsorbed species have been invoked to explain the non-Langmuirian adsorption isotherms. For instance the Frumkin isotherm can be obtained with a linear variation of heats of adsorption with coverage, and the Freundlich isotherm is attributed to an exponential variation of heats of adsorption. 

For an oxide layer, Young assumed that the nonstoichiometry of tiie oxide layer resulted in an exponential variation of the conductivity with the normal distance to the electrode as... 

It can be expected that the higher and the thicker the barrier in Fig. 2.7 the less is the probability that the particle be found outside the box. Equally expected is that the heavier the particle the smaller that probability. In fact, the exponential variation of inside the barrier (see Problem 2.13)... 

Physically, the reason for the dramatic difference between performances of cathode and anode active layers is the exchange current density ia at the anode the latter is 10 orders of magnitude higher than at the cathode [6]. Due to the large ia, the electrode potential r]a is small. The anode of PEFC, hence, operates in the linear regime, when both exponential terms in the Butler-Volmer equation can be expanded [178]. This leads to exponential variation of rja across the catalyst layer with the characteristic length (in the exponent)... 

Hydrazine yields for the various reactor geometries and the operating variables employed are given in Figures 2 to 6. From the data presented in Figures 2 to 4, it is evident that the yield varies inversely as an exponential function of the power density at pressures under 100 mm. of mercury. The effect of pressure also follows a negative exponential variation and, therefore, a general equation of the form... 

The discussion of luminescence has, up to the present, been based on the properties of dilute solutions in which the analyte molecules were presumed not to interact with one another. It has already been established that at high absorbance at the wavelength of excitation, deviations from linearity of the fluorescence in-tensity-versus-concentration relationship may occur because of the exponential variation of luminescence intensity with concentration. However, over a wide range of solute concentrations, solute-solute interactions may also account for loss of luminescence intensity with increasing solute concentration. 

Figure 2,a shows the variation of the FLI with time for 4 different kinds of orbits. The upper curve, with initial conditions (10-4, 0) in the chaotic zone just described, shows an exponential variation of the FLI with time. The upper value of 20 is a computational threshold that allows to avoid floating overflow. 

Heat Transfer on the Walls With Exponential Heat Flux. Heat transfer on walls with exponential wall heat flux is denoted as the boundary condition. According to the analysis by Siegel et al. [25], the local Nusselt number for a circular duct with exponential variation of the wall heat flux, as represented by qZ = q" exp(mjt ), can be determined using the following formula ... 

Between parallel linear molecules in well-defined arrays, exponential variation is again observed (11-13). The work of transfer of water from such an array is expressed in terms of a chemical potential of water or, equivalently, of an osmotic stress IIosm on the lattice as a function of a lattice parameter such as lattice interaxial distance d ... 

Barriol and Rivail 80) also derived an equation of the Elovich form using a model in which it was assumed that all sites in a particular region must be simultaneously unoccupied before adsorption could occur, and obtained appropriate expressions for the probable existence of such regions. Meller 81) also obtained an exponential variation of the number of adsorption sites by another different approach. 

For an isothermal process the Elovichian equation is still valid for a situation in which there is an exponential variation of the number of sites on the homogeneous patches and also a linear variation of activation energy with patch number, i.e., (i) Us = rio ex-p( bs) and (ii) Es = Eo- - as, where s is the reference number of the patch and a, b are temperature-independent constants. 

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