Jones plot

In their Figure 2 Slough and Jones plotted z-r versus A //° of reactions,... 

The Hardin Jones plot for some presumably homogeneous cohorts of cancer patients shows some curvature, such as to suggest a moderate amount of heterogeneity. A reasonable assumption is that there is an error function distribution of the activation energy of the rate constant, a, about a mean value of the activation energy corresponding to an intermediate value a<, of the rate constant. This assumption leads on expansion and integration to the introduction of a quadratic term ... 

Still another method is to calculate the slope of the line connecting the points on the Hardin Jones plot with the origin the reciprocal of this slope for each point is a value of... 

Another method makes use of three points, 52(12). and 53(13), on a smoothed Hardin Jones plot of In 5 vs. t. For one set of 130 breast cancer patients (4), this treatment with 5 = 0.5, 0.25, and 0.0625 gave/i = 0.45,/2 = 0.55, t, = 29 days, and T2 = 83 days. This method is unreliable for small cohorts. An alternative is to make a least-squares fit to all the points on a Hardin Jones plot with a two-term function or, if the cohort is very large, to carry out a Laplace transformation. 

Figure 7-12. Plot of the van der Waals interaction energy according to the Lennard-Jones potential given in Eq. (27) (Sj, = 2.0 kcal mol , / (, = 1.5 A). The calculated collision diameter tr is 1.34 A.

Fig. 4.22 Apparent specific surface A(app) of carbons obtained from the decomposition of polymers, plotted against the carbonization temperature, (a) Polyfurfuryl carbons (b) dibenzanthrone carbons (c) polyvinylchloride carbons. O, A(app) estimated from CO2 isotherm at 195 K (a fCOj) = 17-0 A ) A. /f(app) estimated from N2 isotherm at 77 K = 16-2 A ). (Courtesy Marsh and Wynne Jones.)...

Thus, in the purely agglomerative proeess with a typieal value of the parameter a > 1, the number density distribution n L) aeeording to equation 6.11 is linear when plotted in logarithmie eo-ordinates, i.e. In n L) versus logL, thus exhibits an initial slope equal to —5/2, as shown in Figure 6.12. Note If the population density is defined on a unit solid volume, rather than size, basis, then the eorresponding slope is —3/2, see also Jones etal., 1996.)... 

Fig. 11(a) displays plots of the in-plane pair correlation function for s = 2. and 3.0 well outside the regime where K exhibits its first maximum (see Fig. 12). The plots indicate that the transverse structures of one- and two-layer fluids (see Fig. 10) are essentially identical and typical of dense Lennard-Jones fluids. However, the transverse structure of a two-layer fluid is significantly affected as the peak of K is approached, as can be seen in Fig. 11(b) where g (zi,pi2) is plotted for s = 2.55 and 2.75, which points...

Figure 8-1. A plot of Eq. (8-16), the Lennard-Jones 6-12 potential energy function.

Jones et al. [73] have provided an alternative approach to the linearization of data using the tabulated reduced time values given by Sharp et al. [70]. The experimental data are expressed in the form ae as a function of (t/t0.s)ei where the subscript e refers to the experimental data. Three broadly equivalent methods of plotting can be used. 

Fig. 1. The Lennard Jones 12 6 pair potential plotted for a pair of CH2 united atoms using the OPTS united force field. Enonbond = 4e((o /r) (o /r) ), where s is the well depth for the potential and cr is the distance at which the repulsive energy exactly cancels the attractive energy...

A typical Lennard-Jones (6-12) function is plotted in Fig. 9. Often, the second team on the right-hand side of Eq. (100) is added to represent an effective potential function, viz. 

Mass defect plots utilized by Jones and coworkers44 highlight impossible mass defects for the compounds considered and suggest chemical com-... 

Figure 3.12 The distribution of a particle with zero charge constrained to move in a ring relative to two others of the same spin. Plots a-c show the probability i/i2 of finding the third particle at a given location on the ring which is a maximum when all three particles are located at 120° intervals around the ring. In d-f Q indicates the most probable location of the third particle with respect to the other two. (Reproduced with permission from J. E. Lennard-Jones, Adv. Sci. 11, No. 54, 1956.)...

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

Figure 3 shows an ordering map for this Lennard-Jones system, with the translational order t (of Eq. [2]) plotted against the bond-orientational order Qs (of Eq. [5]). It can be observed that the data, collected over a wide range of temperatures and densities, collapse onto two distinct equilibrium branches... 

Fig. 19 Bond length-reactivity plots for phosphate triesters and for monoester dianions. Note the increased slope for the monoesters, which undergo SNl(P)-type P-O cleavage in solution. Reprinted with permission from Jones and Kirby (1984). Copyright 1984 American Chemical Society.

Fig. 20 Plot of P-OX bond length against pKHox for five esters [101], and for the symmetrical pyrophosphate. Reprinted with permission from Jones et al. (1985).

The low reactivity of the triesters [101] allowed the plot shown in Fig. 19 to be extended to better leaving groups, and in this case, uniquely, there is some evidence that the bond-lengthening effect may reach a limit. As shown by Fig. 20, the P-OX bond lengths of the three most reactive compounds, with pXHox < 6, are identical within experimental error. The significance of this interesting but isolated observation is not clear (Jones et al., 1985). 

Fig. 25 Plots of the sensitivity parameters and intercepts obtained from bond length-reactivity correlations for axial tetrahydropyranyl- [96], methoxymethyl- and a-glucosyl-OX versus their relative reactivities towards hydrolysis. Reprinted with permission from Kirby and Jones (1986).

The relationship, between magnitudes of these sensitivity parameters and the reactivity of the system concerned, was explored quantitatively for three sets of acetal structures, and a further linear relationship was found (Kirby and Jones, 1986). Figure 25 shows a plot of the sensitivity parameter against relative reactivity for series of compounds, tetrahydropyranyl-, methoxymethyl- and a-glucosyl-OX, known to react with C-OX cleavage at very different rates (relative rates of hydrolysis approximately 1, 103 5 and 106 5, respectively). The correlation is good for both the intercepts (i.e. the... 

Figure 2.2 Illustrative plot of the Lennard-Jones-Devonshire interatomic potential showing the force and the modulus curve for the pair interaction. Positive values indicate repulsion and negative values indicate attraction...

A plot of the Lennard-Jones 9-3 form of Equations 7 and 8 for ST2 water interacting with smectite and mica surfaces is shown in Figure 1. Values for the parameters used in Figure 1 are given in Tables II and III, and in reference (23). The water molecule is oriented so that its protons face the surface and its lone pair electrons face away from the surface, and the protons are equidistant from the surface. Note that the depth of the potential well in Figure 1 for interactions with the smectite surface and mica surface are... 

Figure 4.4 shows a section of a perfect single crystal surface [such as Pt(lll)] which is approached by a diatomic molecule (say 02) undergoing dissociative chemisorption. The progress of this process is illustrated by a contour plot of the energetics as a function of the distance x of the molecule from the surface and of the separation y between the two atoms, together with the well-known one-dimensional Lennard-Jones potential diagram. (The molecular axis is assumed to be parallel to the surface... 

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