Exponential Dependence

Since j is independent of x, the second term is a constant and may be replaced by its average value, giving 

The exponential dependence, however, allows of a valuable simplification by absorbing /3 into the definition of the dimensionless concentrations. Let 

These equations allow us to plot contours of constant W and constant j in the U, F-plane. It should be noted that, on account of the term log )8, U and V can be negative though u and v are always positive. 

It is convenient to consider also the concentration difference across the membrane 

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This expression corresponds to the Arrhenius equation with an exponential dependence on the tlireshold energy and the temperature T. The factor in front of the exponential function contains the collision cross section and implicitly also the mean velocity of the electrons. 

One can see tliat tlie Dexter exchange mechanism is exponentially dependent on tlie distance between tlie donor and acceptor, and as such it begins to play a visible role only at very short distances when tlie electron clouds begin to... 

The exponential dependencies in Eq. (14-195) represent averages of values reported by a number of studies with particular weight given to Lefebvre [Atomization and Sprays, Hemisphere, New York, (1989)]. Since viscosity can vary over a much broader range than surface tension, it has much more leverage on drop size. For example, it is common to find an oil with 1000 times the viscosity of water, while most liquids fall within a factor of 3 of its surface tension. Liquid density is generally even closer to that of water, and since the data are not clear mat a liquid density correction is needed, none is shown in Eq. 

TABLE 14-12 Exponential Dependence of Drop Size on Different Parameters in Two-Fluid Atomization... 

This method of writing D emphasises its exponential dependence on temperature, and gives a conveniently sized activation energy (expressed per mole of diffusing atoms rather than per atom). Thinking again of creep, the thing about eqn. (18.12) is that the exponential dependence of D on temperature has exactly the same form as the dependence of on temperature that we seek to explain. 

The exponent in this formula is readily obtained by calculating the difference of quasiclassical actions between the turning and crossing points for each term. The most remarkable difference between (2.65) and (2.66) is that the electron-transfer rate constant grows with increasing AE, while the RLT rate constant decreases. This exponential dependence k AE) [Siebrand 1967] known as the energy gap law, is exemplified in fig. 14 for ST conversion. 

In its most common mode of operation, STM employs a piezoelectric transducer to scan the tip across the sample (Figure 2a). A feedback loop operates on the scanner to maintain a constant separation between the tip and the sample. Monitoring the position of the scanner provides a precise measurement of the tip s position in three dimensions. The precision of the piezoelectric scanning elements, together with the exponential dependence of A upon c/means that STM is able to provide images of individual atoms. 

Integration shows the exponential dependence of probability, assuming P = 1 when t = 0. 

The breakthrough for molecular applications came with Boys s classic paper (1950) on the use of Gaussian-type orbitals (GTOs). These basis functions have an exponential dependence of exp (— (ar /al)) rather than exp(—( r/ao))-The quantity a is called the Gaussian exponent. Normalized Is and 2p GTOs are... 

A third functional form, which has an exponential dependence and the correct general shape, is the Morse potential, eq. (2.5). It does not have the dependence at long range, but as mentioned above, in reality there are also R, /f etc. terms. The D and a parameters of a Morse function describing vdw will of course be much smaller than for str, and Ro will be longer. 

A is a normalization constant and T/.m are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. 

Composilion has a marked effect on p. Dilution can cause (i to drop by orders of magnitude. The functional dependence is often expressed in terms of an exponential dependence on intersite distance R=at m as suggested by the homogeneous lattice gas model, p c)exponential distance dependence of intersite coupling precludes observing a percolation threshold for transport. 

L. L. Blyler and T. K. Kwei [39] proposed the direct opposite (to 4). In their reasoning, they proceeded from the known and generally acceptable Doolittle equation, which puts liquid viscosity in exponential dependence on the inverse value of the free volume of the latter. According to [39], gas has a volume of its own, the value of which it contributes to the free volume of the polymer when it dissolves therein as a result, viscosity falls. The theoretical formula obtained by the authors was experimentally confirmed in the same work. The authors measured pressure values at the entrance of cylindrical capillaries, through which melts of both pure polyethylene, and polyethylene with gas dissolved in it, extruded at a constant rate. 

With the choice of a satisfying Eq. (4-122), it can be shown that the bound in Eq. (4-121), known as a Chernov bound, at least has the correct exponential dependence on N. 

In the search for a better approach, investigators realized that the ignition of a combustible material requires the initiation of exothermic chemical reactions such that the rate of heat generation exceeds the rate of energy loss from the ignition reaction zone. Once this condition is achieved, the reaction rates will continue to accelerate because of the exponential dependence of reaction rate on temperature. The basic problem is then one of critical reaction rates which are determined by local reactant concentrations and local temperatures. This approach is essentially an outgrowth of the bulk thermal-explosion theory reported by Fra nk-Kamenetskii (F2). 

The principal difficulty with these equations arises from the nonlinear term cb. Because of the exponential dependence of cb on temperature, these equations can be solved only by numerical methods. Nachbar has circumvented this difficulty by assuming very fast gas-phase reactions, and has thus obtained preliminary solutions to the mathematical model. He has also examined the implications of the two-temperature approach. Upon careful examination of the equations, he has shown that the model predicts that the slabs having the slowest regression rate will protrude above the material having the faster decomposition rate. The resulting surface then becomes one of alternate hills and valleys. The depth of each valley is then determined by the rate of the fast pyrolysis reaction relative to the slower reaction. 

The case of m = Q corresponds to classical Arrhenius theory m = 1/2 is derived from the collision theory of bimolecular gas-phase reactions and m = corresponds to activated complex or transition state theory. None of these theories is sufficiently well developed to predict reaction rates from first principles, and it is practically impossible to choose between them based on experimental measurements. The relatively small variation in rate constant due to the pre-exponential temperature dependence T is overwhelmed by the exponential dependence exp(—Tarf/T). For many reactions, a plot of In(fe) versus will be approximately linear, and the slope of this line can be used to calculate E. Plots of rt(k/T" ) versus 7 for the same reactions will also be approximately linear as well, which shows the futility of determining m by this approach. 

Particularly high stress occurs when bubbles burst on the surface of the liquid, whereby droplets are eruptive torn out of the surface [32-36]. According to theoretical calculations, maximum energy densities occur in the region of the boundary surface shortly before the droplets separate [36]. The results calculated by Boulton-Stone and Blake [34] show that these are exponentially dependent on bubble diameter dg. Whereas these authors found values of e = lO mVs with dg = 0.5 mm, these are only e 1 m /s with dg = 5 mm. The situation may be different regarding the droplet volume separated from the surface by the gas throughput and thus the number of particles which are exposed to high stress. The maximum for this value occurs with a bubble diameter of dg = 4 mm (see [34]), and it is therefore feasible that there could be an optimal bubble size. 

Fig. 32. Maximum energy dissipation rates produced throughout the bursting process, plotted against bubble radius. The logarithmic scale indicates an exponential dependence of maximum stress on bubble radius for large bubbles. The slight drop in the data point for the smallest bubble as compared to the next smallest may be because of the difficulty in locating the exact place and time of the peak, due to large spatial and temporal gradients beneath the forming jet [113]...

The exponential dependence is indicative of hydrophobic attractive forces between the glycerol-air interface and the glycerol-mica interface. This potential gives a negative disjoining pressnre of 1 atm at z close to 0. The strength of the force appears to be 100 times lower than that typical for water between hydrophobic surfaces [6]. 

In this case the usual exponential dependence analogous to Eq. (6.12) is obtained. 

The above equation then represents the balanced conditions for steady-state reactor operation. The rate of heat loss, Hl, and the rate of heat gain, Hq, terms may be calculated as functions of the reactor temperature. The rate of heat loss, Hl, plots as a linear function of temperature and the rate of heat gain, Hq, owing to the exponential dependence of the rate coefficient on temperature, plots as a sigmoidal curve, as shown in Fig. 3.14. The points of intersection of the rate of heat lost and the rate of heat gain curves thus represent potential steady-state operating conditions that satisfy the above steady-state heat balance criterion. 

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