Two-parameter approximation

Asymmetry was treated in the two-parameter approximation given by L.W. Finger, D.E. Cox, and A.P. Jephcoat, A correction for powder diffraction peak asymmetry due to axial divergence, J. Appl. Cryst. 27, 892 (1994). 

For individual components, a two-parameter approximation [7-9] of pressure vs. temperature dependence along the phase equilibrium curve is used ... 

In this two-parameter approximation three different models were tested, k = kff, k = k(filf and k = kjf/n. Hietanen found that the experimental data could be described equally well by these models with ko between 0.5 to 1.0 for the first two models and about 0.1 for the third. The value of logio for the three models were - 7.50, - 7.65 and -9.15, respectively. The core-and-link model is related to chain stractures found in solid state for so called basic salts , thorium hydroxide sulphate and thorium hydroxide chromate ([1949SIL/LUN], [1950LUN]). However, in solution systems, a model with a very large number of complexes does not make chemical sense and the model was also abandoned in later studies. 

Especially in protein crystals, but also in other structures that show relatively large voids or cavities, one can find solvent molecules that do not show any order at all. Such solvent regions can be interpreted as a liquid that is amorphously frozen during data collection (assuming that you collect the data at low temperature as you always should). Following the Babinet Principle, this extreme case of disorder is described as bulk solvent and can be refined using a two-parameter approximation (Moews and Kretsinger, 1975). See Example 5.3.5. 

As mentioned in Section 3.2 of Chapter 1, the Hamiltonian H of the Edwards continuous chain depends on L, b, /3c, and A. Hence, any equilibrium property of this chain in dilute solutions ought to be expressed as a function of these four parameters. Interestingly, eq 1.9 indicates that actually (i ) is governed by three combined parameters Lb, z, and X/L formed from the four (for continuous chains (R )o = Lb according to eq 1-3.15). Hence, if X/L < 1, essentially depends on Lb and (or or depends only on z). We can expect that the same holds for any equilibrium behavior of the Edwards continuous chain in dilute solutions. Any theory of dilute polymer solutions fitting to this expectation is called the two-parameter theory. We may say that it is a theory of the Edwards continuous chain subject to the condition A/L 1. The use of the Edwards Hamiltonicui combined with this condition is hereafter referred to as the two-parameter approximation. 

From later investigations by Krygowski and Fawcett [Kr 75] it has emerged that the correlation of the Tvalues relating to the different systems can be described well with a two-parameter approximation, since, in addition to the acidity of the solvent, there is another factor (donor property) playing a slight role in the determination of the reaction rate. 

All of these investigations show that solvent effects can be described correctly only by means of empirical parameters (with an accuracy corresponding to that of the experimental method), if both the donor and acceptor properties of the solvent can be taken into consideration, i.e., a two-parameter approximation is used. It is certain, however, that agreement with the experimental values of the data obtained by the combined consideration of these two effects shows merely that the donor-acceptor interactions predominate in the given systems, but other properties may also have effects. Attention is drawn to the letter by those systems in which the reactions cannot be described by the two-parameter models either. 

Results from simulations show that the next step is not easy. Compressed gases and low torque liquids may reasonably be modelled by Steele s torque approximation Powles information theory expression or by the J diffusion models, although the evidence suggests that the Fokker-Planck model is not very successful. But commonly occurring high torque liquids cannot be successfully modelled by these techniques and there seems to be no simple alternative. It is likely that at least two parameters will be needed even for spherical and linear molecules. Further work to compare two parameter approximations with simulation results will show the most satisfactory next approximation. 

As in the case of the chain expansion factors, terms of order Zj become insignificant in the perturbation expansion of the -second virial coefficient when P>P3- Hence the two-parameter approximation becomes acceptable and only the I and II terms are needed in equation (73). 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

The main difference between CNDO, INDO and NDDO is the treatment of the two-electron integrals. While CNDO and INDO reduce these to just two parameters (7AA 7ab), all the one- and two-center integrals are kept in the NDDO approximation. Within an sp-basis, however, there are only 27 different types of one- and two-center integrals, while the number rises to over 500 for a basis containing s-, p- and d-functions. 

This is the general expression for film growth under an electric field. The same basic relationship can be derived if the forward and reverse rate constants, k, are regarded as different, and the forward and reverse activation energies, AG are correspondingly different these parameters are equilibrium parameters, and are both incorporated into the constant A. The parameters A and B are constants for a particular oxide A has units of current density (Am" ) and B has units of reciprocal electric field (mV ). Equation 1.114 has two limiting approximations. 

If a data set containing k T) pairs is fitted to this equation, the values of these two parameters are obtained. They are A, the pre-exponential factor (less desirably called the frequency factor), and Ea, the Arrhenius activation energy or sometimes simply the activation energy. Both A and Ea are usually assumed to be temperature-independent in most instances, this approximation proves to be a very good one, at least over a modest temperature range. The second equation used to express the temperature dependence of a rate constant results from transition state theory (TST). Its form is... 

Assuming an approximately constant cohesive energy per C-C bond, that trend is understandable. With clusters on the above general type, the number of carbon atoms is 6N, the number of dangling bonds is 6N, and the number of C-C bonds is 9N -3N. The energy per bond shows a smoother trend, the numbers being 71.0, 77.6 and 79.9 kcal/mol, respectively. Alternatively, the energies can be fitted to a two-parameter expression of the form... 

Hence by assigning two parameters, a Q and an c, to each of a set of monomers, it should be possible according to this scheme to compute reactivity ratios ri and V2 for any pair. In consideration of the number of monomer pairs which may be selected from n monomers—about n /2—the advantages of such a scheme over copolymerization experiments on each pair are obvious. Price has assigned approximate values to Q and e for 31 monomers, based on copolymerization of 64 pairs. The latitude of uncertainty is unfortunately large assignment of more accurate values is hampered by lack of better experimental data. Approximate agreement between observed and predicted reactivity ratios is indicated, however. 

The computation of the above surface in the parameter space is not trivial. For the two-parameter case (p=2), the joint confidence region on the krk2 plane can be determined by using any contouring method. The contour line is approximated from many function evaluations of S(k) over a dense grid of (k, k2) values. 

Stratified flow. A separated flow model for stratified flow was presented by Taitel and Dukler (1976a) in which the holdup and the dimensionless pressure drop, = (dpldz)TPl(dpldz)GS is calculated as a function of the Lockhart-Martinelli parameter only. (The results, however, differ from those of Martinelli and compare better with experimental data.) This model uses two basic approximations ... 

The Na I D-lines have wavelengths and oscillator strengths A,i = 5896 A, /i = 1 /3, and X2 = 5889 A, f2 — 2/3. In a certain interstellar cloud, their equivalent widths are measured to be 230 mA and 370 mA respectively, with a maximum error of 30 mA in each case. Assuming a single cloud with a Gaussian velocity dispersion, use the exponential curve of growth to find preferred values of Na I column density and b, and approximate error limits for each of these two parameters. (Doublet ratio method.)... 

The temperature dependent T data are shown in Fig. 9. 7j values decrease from 28 ms at 21°C with increasing temperature, and show a minimum of 6.4 ms at 80° C. These results indicate the presence of the motion with a Larmor frequency of 30 MHz at this temperature. This minimum was found to be attributed to the flipping motion of a phenyl ring from the result of our other experiments discussed in later section.13 The jump rates of the flipping motion were estimated with a two-site jump model that a C-2H bond jumps between two equivalent sites separated by 180°, and that the angle made by the C-2H bond and the rotational axis is 60°. The quadrupole coupling constant of 180 kHz and the asymmetry parameter approximated to zero were used in the calculation. The calculated values for fitting with the... 

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