Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Simple exponentials

Functional fonns used for the repulsion include the simple exponential multiplied by a linear combmation of powers (possibly non-mteger) of r, a generalized exponential function exp(-h(r)), where b r) is typically a polynomial in r, and a combination of these two ideas. [Pg.207]

The applications of this simple measure of surface adsorbate coverage have been quite widespread and diverse. It has been possible, for example, to measure adsorption isothemis in many systems. From these measurements, one may obtain important infomiation such as the adsorption free energy, A G° = -RTln(K ) [21]. One can also monitor tire kinetics of adsorption and desorption to obtain rates. In conjunction with temperature-dependent data, one may frirther infer activation energies and pre-exponential factors [73, 74]. Knowledge of such kinetic parameters is useful for teclmological applications, such as semiconductor growth and synthesis of chemical compounds [75]. Second-order nonlinear optics may also play a role in the investigation of physical kinetics, such as the rates and mechanisms of transport processes across interfaces [76]. [Pg.1289]

The quantitative analysis of the scattering profile in the high q range can be made by using the approach of Debye et aJ as in equation (B 1.9.52). As we assume tiiat the correlation fiinction y(r) has a simple exponential fomi y(r) = exp(-r/a ), where is the correlation length), the scattered intensity can be expressed as... [Pg.1415]

For this reaction alone, one would thus obtain a simple exponential relaxation witii relaxation time... [Pg.2119]

However, as can be seen from Figure 8 a simple exponential expected from first-order kinetics can be fitted to the data yielding a limiting concentration of 0.005, and a rate constant of 0.0003 a.u., which translates to 1.25 x 10 s at 300 K. [Pg.247]

This scheme requires the exponential only of matrices that are diagonal or transformed to diagonal form by fast Fourier transforms. Unfortunately, this matrix splitting leads to time step restrictions of the order of the inverse of the largest eigenvalue of T/fi. A simple, Verlet-like scheme that uses no matrix splitting, is the following ... [Pg.427]

VVe now need to consider how the form of the wavefunction varies with k. The first situation we consider corresponds to fc = 0, where the exponential terms are all equal to 1 and the overall wavefunction becomes a simple additive linear combination of the atomic orbitals ... [Pg.161]

Since the sinh function is less familiar than simple exponentials, an example illustrating Eq. (2.24) may be helpful. [Pg.94]

The conditions chosen make the reaction appear to be first-order overall, although the reaction is really not first-order overall, unlessjy and happen to be 2ero. If a simple exponential is actually observed over a reasonable extent (at least 90—95%) of decay the assumptions are considered vaUdated and is obtained with good precision. The pseudo-first-order rate constant is related to the k in the originally postulated rate law by... [Pg.508]

If the same measurement is repeated for different [BJ it should be possible to extractjy by plotting log vs log [BJ. This should be a straight line with slopejy. In a similar manner, can be obtained by varying [CJ. At the same time the assumption that x equals 1 is confirmed. Ideally, a variety of permutations should be tested. Even if xis not 1, and the integrated rate equation is not a simple exponential, a usefiil simplification stiU results from flooding all components except one. [Pg.508]

The solution of equation 16 is a decreasing, simple exponential where = k ([A ] + [P ]) + k. The perturbation approach generates small deviations in concentrations that permit use of the linearized differential equation and is another instance of pseudo-first-order behavior. Measurements over a range of [A ] + [T ] allow the kineticist to plot against that quantity and determine / ftom the slope and from the intercept. [Pg.510]

One simple equation form is the exponential equation, which gives a straight line on a log—log plot ... [Pg.441]

Representation of data by using a simple exponential relationship... [Pg.866]

Mixmre models have come up frequently in Bayesian statistical analysis in molecular and structural biology [16,28] as described below, so a description is useful here. Mixture models can be used when simple forms such as the exponential or Dirichlet function alone do not describe the data well. This is usually the case for a multimodal data distribution (as might be evident from a histogram of the data), when clearly a single Gaussian function will not suffice. A mixture is a sum of simple forms for the likelihood ... [Pg.327]

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. [Pg.20]

It is noteworthy that it is the lower cross-over temperature T 2 that is usually measured. The above simple analysis shows that this temperature is determined by the intermolecular vibration frequencies rather than by the properties of the gas-phase reaction complex or by the static barrier. It is not surprising then, that in most solid state reactions the observed value of T 2 is of order of the Debye temperature of the crystal. Although the result (2.77a) has been obtained in the approximation < ojo, the leading exponential term turns out to be exact for arbitrary cu [Benderskii et al. 1990, 1991a]. It is instructive to compare (2.77a) with (2.27) and see that friction slows tunneling down, while the q mode promotes it. [Pg.34]

It turns out that a rather simple description of this nonlinear relaxation in terms of a single relaxation time,, depending on the final average chain length Loo, is suggested by a scaling plot of L t) for different L o, as shown in Fig. 18 for an initial exponential MWD. It is evident from Fig. 18 that the response curves, L o — L t), for different L o may be collapsed onto a single master curve, 1 - L t)lLoo = /(V Loo) measured in units of a... [Pg.543]

At small N, correction terms come into play, which account for the ends of the cylinders. In particular, the aggregation number of cylindrical micelles in this simple picture must always be larger than M, the most probable aggregation number of a spherical micelle. Putting everything together, the expected size distribution has a peak at M which corresponds to spherical micelles, and an exponential tail at large N which is due to the contribution of cylindrical micelles. [Pg.653]

Evidently simple first-order behavior is predicted, the reactant concentration decaying exponentially with time toward its equilibrium value. In this case a complicated differential rate equation leads to a simple integrated form. The experi-... [Pg.60]

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. [Pg.170]

Alder and Wainwright gave MD treatments of particles whose pair potential was very simple, typically the square well potential and the hard sphere potential. Rahman (1964) simulated liquid argon in 1964, and the subject has shown exponential growth since then. The 1970s saw a transition from atomic systems... [Pg.65]

Behavior for a > aoo- What happens for a > Qoo The simple answer is that the logistic map exhibits a transition to chaos, with a variety of different attractors for Qoo < a < 4 exhibiting exponential divergence of nearby points. To leave it at that, however, would surely bo a great disservice to the extraordinarily beautiful manner in which this trairsition takes place. [Pg.182]

Except for simple cases, it is generally a nontrivial task to compute the Lyapunov exponents of a flow. In trying to estimate A(x(0)) in equation 4.59, for example, the exponentially increasing norm, V t), may lead to computer overflow problems. [Pg.202]


See other pages where Simple exponentials is mentioned: [Pg.7]    [Pg.7]    [Pg.27]    [Pg.1210]    [Pg.1501]    [Pg.2644]    [Pg.2830]    [Pg.2966]    [Pg.410]    [Pg.402]    [Pg.107]    [Pg.670]    [Pg.164]    [Pg.162]    [Pg.87]    [Pg.211]    [Pg.191]    [Pg.351]    [Pg.1905]    [Pg.5]    [Pg.387]    [Pg.116]    [Pg.1133]    [Pg.354]    [Pg.21]    [Pg.41]    [Pg.210]    [Pg.681]    [Pg.116]   
See also in sourсe #XX -- [ Pg.21 ]




SEARCH



Methods using a simple approximation for the exponential integral

Simple exponential smoothing method

© 2024 chempedia.info