Coefficient of curvature

The uniformity of the soil material, desirable in most cases, is quantified by reference to the uniformity coefficient and the coefficient of curvature. The uniformity coefficient (Cu) and the coefficient of curvature (Cq) are determined by the following equations ... 

Awareness of attributes of particle size distribution (i.e. percentage passing through a 0.075 mm sieve, d- o, d Q and djo), either coefficient of uniformity, Cu, or coefficient of curvature, Cq, as well as plasticity index (PI), is required for the classification of soils in some subgroups. 

The coefficient of curvature (Cq) ensures that the particle size distribution will have a concave curvature within relatively narrow limits for a given d Q and combination. As for well-graded gravels, Cq values should range from 1 to 3. 

Number of testing groups Dry density of test (g/cm ) Porosity (%) Testing gradation (%) Coefficient of curvature (Cc) Non-uniformity coefficient (Cu)... 

The grain size distribution has a significant influence on the density of a soil. The better the grading, the higher the density. The grading of a soil is generally described by the uniformity coefficient C = DJD q, and the coefficient of curvature Df). represent the sieve size for which x% of the... 

The theory and appHcation of SF BDV and COV have been studied in both uniform and nonuniform electric fields (37). The ionization potentials of SFg and electron attachment coefficients are the basis for one set of correlation equations. A critical field exists at 89 kV/ (cmkPa) above which coronas can appear. Relative field uniformity is characterized in terms of electrode radii of curvature. Peak voltages up to 100 kV can be sustained. A second BDV analysis (38) also uses electrode radii of curvature in rod-plane data at 60 Hz, and can be used to correlate results up to 150 kV. With d-c voltages (39), a similarity rule can be used to treat BDV in fields up to 500 kV/cm at pressures of 101—709 kPa (1—7 atm). It relates field strength, SF pressure, and electrode radii to coaxial electrodes having 2.5-cm gaps. At elevated pressures and large electrode areas, a faH-off from this rule appears. The BDV properties ofHquid SF are described in thehterature (40—41). 

The quantitative relationship between the degree of adsorption at a solution iaterface (7), G—L or L—L, and the lowering of the free-surface energy can be deduced by usiag an approximate form of the Gibbs adsorption isotherm (eq. 9), which is appHcable to dilute biaary solutions where the activity coefficient is unity and the radius of curvature of the surface is not too great ... 

This discussion of sources of curvature in Br insted-type plots should suggest caution in the interpretation of observed curvature. There is a related matter, concerning particularly item 5 in this list, namely, the effect of a change in transition state structure. Br nsted-type plots are sometimes linear over quite remarkable ranges, of the order 10 pK units, and this linearity has evoked interest because it seems to be incompatible with Marcus theory, which we reviewed in Section 5.3. The Marcus equation (Eq. 5-69) for the plot of log k against log K of the same reaction series requires curvature, the slope of the plot being the coefficient a. given by Eq. (5-67). A Brjinsted plot, however, is not a Marcus plot, because it correlates rates and equilibria of different reactions. The slope p of a Br nsted plot is defined p = d log kobs/d pK, which we can expand as... 

Another problem with the long distances comes about when trying to measure the radius of curvature. We previously showed that the radius of curvature of the segments would have to be correct to about 1 10 . Since the coefficient of expansion of most common materials is larger than this, holding the distance from the interferometer to the mirror over these distances is difficult. 

The simple harmonic oscillator picture of a vibrating molecule has important implications. First, knowing the frequency, one can immediately calculate the force constant of the bond. Note from Eq. (11) that k, as coefficient of r, corresponds to the curvature of the interatomic potential and not primarily to its depth, the bond energy. However, as the depth and the curvature of a potential usually change hand in hand, the infrared frequency is often taken as an indicator of the strength of the bond. Second, isotopic substitution can be useful in the assignment of frequencies to bonds in adsorbed species, because frequency shifts due to isotopic substitution (of for example D for H in adsorbed ethylene, or OD for OH in methanol) can be predicted directly. 

The rate of reduction of Tl(III) by Fe(II) was studied titrimetrically by John-son between 25 °C and 45 °C in aqueous perchloric acid (0.5 M to 2.0 M) at i = 3.00 M. At constant acidity the rate data in the initial stages of reaction conform to a second-order equation, the rate coefficient of which is not dependent on whether Tl(III) or Fe(II) is in excess. The second-order character of the reaction confirms early work on this system . A non-linearity in the second-order plots in the last 30 % of reaction was noted, and proved to be particularly significant. Ashurst and Higginson observed that Fe(III) retards the oxidation, thereby accounting for the curvature of the rate plots in the last stages of reaction. On the other hand, the addition of Tl(l) has no significant effect. On this basis, they proposed the scheme... 

So far, Santos has been able to express the relation between a set of coefficients af, aj J 6 / describing a vector field and the overall curvature of the stream lines of this vector field. Based on the curvature field, they constructed the measure E of the curvature distribution in the simulation box. Provided that the homogeneous curvature field of curvature c0 is the one that minimizes E, the problem of packing has been recast as a minimization problem. However, the lack of information about the gradient of the error function to be minimized does not facilitate the search. Fortunately, appropriate computer simulation schemes for similar minimization problems have been proposed in the literature [105-109]. 

Every minimization departs from an initial estimation for the vector field. The minimizations were carried out with a starting configuration obtained by randomizing the coefficients aj and a the resulting vector field has no preferential orientation and the distribution of curvature in the simulation box exhibits a long tail mainly due to abrupt changes in the direction of the stream lines (see Fig. 3.2A). 

The improvement in the fit from the quadratic polynomial applied to the nonlinear data indicated that the square term was indeed an important factor in fitting that data. In fact, including the quadratic term gives well-nigh a perfect fit to that data set, limited only by the computer truncation precision. The coefficient obtained for the quadratic term is comparable in magnitude to the one for linear term, as we might expect from the amount of curvature of the line we see in Anscombe s plot [7], The coefficient of the quadratic term for the normal data, on the other hand, is much smaller than for the linear term. 

A closer scrutiny of Figure 6 reveals the persistence of small, but consistent curvature in all of the plots. In order to verify the curvature, the transfer coefficient 8 was also determined independently from the width of the CV wave, as described by Nicholson and Shain (10). The potential dependence of 8 obtained in this manner is shown in Figure 7. The slopes 88/3E represent the unmistakable presence of curvature in Figure 6. 

ApA < 1. In Fig. 2 the region of curvature is much broader and extends beyond — 4 < ApA < + 4. One explanation for the poor agreement between the predictions in Fig. 3 and the behaviour observed for ionisation of acetic acid is that in the region around ApA = 0, the proton-transfer step in mechanism (8) is kinetically significant. In order to test this hypothesis and attempt to fit (9) and (10) to experimental data, it is necessary to assume values for the rate coefficients for the formation and breakdown of the hydrogen-bonded complexes in mechanism (8) and to propose a suitable relationship between the rate coefficients of the proton-transfer step and the equilibrium constant for the reaction. There are various ways in which the latter can be achieved. Experimental data for proton-transfer reactions are usually fitted quite well by the Bronsted relation (17). In (17), GB is a... 

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