Slowing down equations

Coleman, Griffith and Heyland (1981) used a simple slowing-down equation to analyse the data, rather than the full diffusion equation (6.12). The positrons were assumed to start their slowing down at Ev and then lose energy at a rate... 

The general problem is formulated in 2 for homogeneous geometries and in 3 for heterogeneous ones. In the latter case a special assumption, the flat-flux hypothesis, has to be made, which is further discussed in 8. In 4 it is shown that the slowing-down equations can be solved by a simple numerical procedure. 

Integration of the slowing-down equations. While great efforts have been made to reduce equations of the type (2.10), (2.13), or (3.10) to differential equations, few attempts have been made to treat them directly. This is somewhat surprising since they are very suitable for direct machine computation [12]. 5... 

The RABBLE A/ertod —This is an Integral transport code with numerical integration Of the slowing down equation over the resonance range/.. ... 

At small driving forces a completely fiat interface cannot move at a constant speed. This is basically a result of the inherent scaling property of the diffusion equation, which scales lengths proportional to the square-root of time, so an advancing interface would slow down with time. 

Reality Check Notice that the concentration of 02, a product in the reaction, appears in the denominator. The rate is inversely proportional to the concentration of molecular oxygen, a feature that you would never have predicted from the balanced equation for the reaction. As the concentration of 02 builds up, the rate slows down. 

The following qualitative picture emerges from these considerations in weak flow where the molecular coils are essentially undeformed, the polymer solution should behave approximately as a Newtonian fluid. In strong flow of a highly dilute polymer solution where the macroscopic velocity field can still be approximated by the Navier-Stokes equation, it should be expected, nevertheless, that in the immediate proximity of a chain, the fluid will be slowed down because of the energy intake to stretch the molecular coil thus, the local velocity field may deviate from the macroscopic description. In the general case of polymer flow,... 

Equation 8b is plotted in Fig. 13.14. We see that the concentration of the reactant decreases rapidly at first but then changes more slowly than a first-order reaction with the same initial rate. This slowing down of second-order reactions has important environmental consequences because many pollutants degrade by second-order reactions, they remain at low concentration in the environment for long periods. Equation 8a can be written in the form of a linear equation ... 

As mentioned in Chapter 4, although this is a dynamic experiment where data are collected over time, we consider it as a simple algebraic equation model with two unknown parameters. The data were given for two different conditions (i) with 0.75 g and (ii) with 1.30 g of methanol as solvent. An initial guess of k =1.0 and k2=0.01 was used. The method converged in six and seven iterations respectively without the need for Marquardt s modification. Actually, if Mar-quardt s modification is used, the algorithm slows down somewhat. The estimated parameters are given in Table 16.1 In addition, the model-calculated values are... 

Figure 1 compares the conversion predicted for any reduced time I = k t with the use of Keller s theory and the above values of k /k and k2/k with experimental results obtained when polyacrylamide was exposed to 0.2N NaOH at 53 C. It may be seen that the reaction slows down at large x much more than predicted by Keller s model. In fact, this decrease of the reaction rate is even more pronounced than predicted by Keller s equations for the case where a single reacted nearest neighbor completely inhibits amide hydrolysis. We believe that this discrepancy is due to the repulsion of the catalyzing hydroxyl ions from amide residues by non-neighboring carboxylate groups. 

There had been some evidence that alkoxide ligands slow down reactions which involve elimination of a p-hydride from an alkyl ligand. a-01efins are dimerized to a mixture of head-to-tail and tail-to-tail dimers by olefin complexes of the type Tafr -CsMes)-(CH2=CHR)Cl2 (10). The p,p1- and a,p -disubstituted tantalacyclo-pentane complexes are intermediates in this reaction. Their decomposition involves the sequence shown in equation 5. When one... 

The addition of an hydroperoxide-breaking substance S to oxidized RH decreases the current concentration of ROOH. This cannot affect the rate of oxidation if the radicals are produced by radiant energy or other sources unreactive to S, but it slows down the oxidation if radicals are produced from hydroperoxide. The initial step of oxidation in the absence of a peroxy-trapping substance is described by the equation (see Chapter 4)... 

Raising a mixture of fuel and oxidizer to a given temperature might result in a combustion reaction according to the Arrhenius rate equation, Equation (4.1). This will depend on the ability to sustain a critical temperature and on the concentration of fuel and oxidizer. As the reaction proceeds, we use up both fuel and oxidizer, so the rate will slow down according to Arrhenius. Consequently, at some point, combustion will cease. Let us ignore the effect of concentration, i.e. we will take a zeroth-order reaction, and examine the concept of a critical temperature for combustion. We follow an approach due to Semenov [3],... 

We next consider the situation after a period of time has elapsed such that half of the magnesium has been consumed. By taking proportions, 0.5 mol of the acid has also reacted, so its new concentration is 0.5 mol dm-3. Accordingly, the value of rate from Equation (8.22) is smaller, meaning that the rate has slowed down. And so the rate at which hydrogen gas is produced will also decrease. 

The kinetics of the reaction have been studied [2], A first order has been found in the concentrations of the titanium catalyst, t-butylhydroperoxide, and the allylic alcohol. The reaction is slowed down by 2-propanol, or more accurately, it has an inverse second order in 2-propanol. Thus, the kinetic equation reads ... 

In these techniques, the concentrations at the electrode do not immediately attain their extreme values after the start of the experiment. Rather, they change with E ox t according to equation (1). While the steepness of the concentration profiles increases with E (forward scan), simultaneously 8 increases in the quiet solution. The latter effect slows down the increase of i with E, and finally (close to the limiting current region) leads to the formation of a peak with a characterishc asymmetric shape. On the reverse scan (after switching the scan direction ad. Ef), products formed in the forward scan can be detected (B, in the case discussed). 

Since the cross section for nonrelativistic Coulomb scattering is the same in classical and quantum mechanics, equation (2) must contain much of the essential physics in the slowing-down process. However, it also contains an undetermined minimum energy transfer rmin which is nominally zero and hence leads to an infinite stopping force. 

However, in bulk diffusion, ions cannot move independently of each other because electrical neutrality must be maintained. Consequently there is an electric potential between diffusing ions such that the faster ions tend to be slowed down by the slower ones and vice versa. The flux of a particular ion is therefore the sum of the diffusion due to its own concentration gradient and that due to the gradient of the diffusion potential arising from differences in the mobilities of the ions present. This is expressed by the Nemst-Planck equation along the x-axis ... 

The non-aqueous system of spherical micelles of poly(styrene)(PS)-poly-(isoprene)(PI) in decane has been investigated by Farago et al. and Kanaya et al. [298,299]. The data were interpreted in terms of corona brush fluctuations that are described by a differential equation formulated by de Gennes for the breathing mode of tethered polymer chains on a surface [300]. A fair description of S(Q,t) with a minimum number of parameters could be achieved. Kanaya et al. [299] extended the investigation to a concentrated (30%, PI volume fraction) PS-PI micelle system and found a significant slowing down of the relaxation. The latter is explained by a reduction of osmotic compressibihty in the corona due to chain overlap. 

For very high Cq the poison-free Monod equation just can t apply, for even if there is plenty enough food the cells will crowd out each other, and growth will slow down and will eventually stop. So, for very high cell concentration, we must go to product poison kinetics. 

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