Concave down curve

The plot of IhAtoh/I vs T for the base hydrolysis of trans-Co (i SSI [14]aneN4)Cl2 is curved slightly (concave down). Bearing in mind the conjugate mechanism for base hydrolysis give a plausible explanation for this behavior. 

Since d yjdx <0 everywhere, the curves in the two-stage Pb evolution model are concave down everywhere. According to Eq. (12.29), the extent of the curvature during the second stage is dependent on Mi ... 

An inflection point is where a curve changes from concave up to concave down. Mathematically, at an inflection point of a curve f(x), d2f /dx2 = 0. 

Graphically, this means that the Gibbs free energy is concave down with respect to, i 2 (e.g., see the solid line in Fig. 7.2 between x 2 and x. In this situation, any line that connects two points on the curve will lie below the curve, and thus the mixture will want to phase separate. 

Note that the curve is concave up at first, and then concave down this corresponds to the initial acceleration for x < Jt/2, followed by the deceleration toward x = 7t. 

It has the same shape as the graph of the logistic map. Both curves are smooth, concave down, and have a single maximum. Such maps are called unimodal. 

The use of this relation to evaluate experimental data is essentially curve htting since it has no mechanistic base. Values of K provide an index of the extent of sorption and often are listed without units. However, if sorption (x/m) is expressed as pmol/g and equilibrium concentration (Ce) in pmol/mL, K would have units (pmol) " mL" g The exponent, n, simply indicates whether the relation between xlm and Ce is linear (n = 1), concave down (n < 1) or concave up (n > 1), Figure 3.4. Values for the Freundlich K and exponent n derived by a simple regression from the experimental data summarized in Table 3.2 are given in Table 3.3. 

Suen, S.Y., An isotherm model describing concave-down Scatchard curve for heterogeneous affinity adsorption, J. Chem. Technol. Biotechnol., 70(3), 278-286 (1997). 

The data in Figure 6-2 form a relatively smooth curve that is concave down. This suggests that the order of the reaction with respect to A is less than 2. The downward curvature incficates that — rA is a weaker function of Ca than assumed. If the curvature of the data on the same plot had been concave up, a stronger dependency of — ta on Ca would be suggested. 

In the case that the % value of a binary system is high, a mixed solution could be stable, unstable, or metastable depending on its composition. As shown in Figure 15.2b, if the composition of the solution is located at the concave-down section of the AG curve (i.e., < 0), a small fluctuation in composition from 0oto 0j and... 

We now wish to examine when a single phase will spontaneously split into different liquid phases. The criterion for instability of a single liquid phase is given when the curve for the total solution Gibbs energy is concave down. Mathematically, this is expressed by ... 

A point on a curve at which the concavity of the curve changes (e.g., up to down or vice versa). The second derivative of the equation of the curve will be zero at this point. A number of curves found in enzyme kinetic... 

Because the second derivative d yfdx ) of a function y = f x)) determines whether the curve of the function is concave upward or down, to answer the question of this section, we need to derive the second derivatives of the Tera-Wasserburg Concordia plot. 

It is often necessary to make hardness measurements on curved surfaces, e.g. rollers or O rings. In the first example the product may be large enough for the hardness instrument to rest upon it, whilst in the second it would usually be possible to rest the product on the specimen table. In either case, some form of jig is required to locate the test piece and suitable examples and precautions to be taken are described in ISO 48. Any of the standard methods could be used for curved surfaces except that it is not possible to use a foot on concave surfaces. For large cylindrical surfaces, the hardness tester is either fitted with feet movable in universal joints which rest on the curved surface or the base of the instrument is fitted with two cylindrical rods which rest on the curved surface. The latter method can be used for surfaces with radius of curvature down to 50 mm. For surfaces having double curvature, only the method using movable feet is suitable. For small products and where the radius of curvature is too small to rest the instrument on the surface, the test piece is placed on the base of the instrument as with... 

It will be observed that some of the curves are concave upward, while others are concave downward. While the mathematical proof for this is not given, the physical explanation is not hard to find. In the case of the type 1 curves, the surface must approach a horizontal asymptote as the velocity is progressively slowed down owing to the increasing depth. Likewise, all curves that approach the normal or uniform depth line j... 

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