Slope, equations

The operational forcing function for variable slope (Equation 3.14.4) yields... 

Thus, In k will vary linearly with 1 fT. This plot yields A from the intercept and Ea from the slope. Equation (7-1) or (7-3) provides a definition of the Arrhenius activation energy, which is expressed as... 

Equation (1.3) follows readily from the point-slope equation of a line, applied to the tangent at (ay, /(x )), namely to... 

The potency of compounds derived from concentration response assays is expressed most commonly as IC50 or EC50 defined as the compound concentration that produces half maximum response. A common model is the four-parameter Hill-slope equation (Table 14.1). A three-parameter model can be used if a maximum or minimum asymptote is not available because compound potency falls outside the concentration range. One recommendation is to fit the logarithm (loglO) of IC50 or EC50 instead of the untransformed concentration because the concentration response errors are normally... 

One may plot Equations 2.69, 2.71, and 2.73 as pAlhydroxy species versus pH. This will produce three linear plots with different slopes. Equation 2.69 will produce a plot with slope 3, whereas Equations 2.71 and 2.73 will produce plots with slopes 2 and -1, respectively. The sum of all three aluminum species as a function of pH would give total dissolved aluminum. This is demonstrated in Figure 2.12, which describes the pH behavior of eight Al-hydroxy species. The following three points can be made based on Figure 2.12 (1) aluminum-hydroxide solubility exhibits a U-shaped behavior, (2) aluminum in solution never becomes zero, and (3) different aluminum species Predominate at different pH values. 

Figure 3 indicates that for the trans-olefins CH3CH=CHR, C2H5CH=CHR, i-C3H7CH=CHR, and terf-C4H9CH=CHR the slopes (Equation 8) are similar, within experimental error. The same observation also applies to the cis-olefins represented in Figure 4. This suggests...

The c/a versus c function is plotted here, the slope is l/z, the intercept is l/(bz). This linear form is fairly suitable for the determination of the maximum adsorption capacity since it is calculated from the slope. Equation 1.64 is used more frequently than Equation 1.63. 

The plot of overpotential versus current density in log scale gives the parameters a, b, and io (h is called the Tafel slope). Equation 3.28, which is only valid for i > io, suggests that the exchange current density io can be also regarded as the current density value at which the overpotential begins to exert its function to make possible the electrochemical reaction, becoming different from zero. 

The most striking feature of the family of curves in Fig. 2 is their constant slope. Equation (5) shows that this is in accord with the principle of a linearly increasing activation energy, since RT/a is independent of pressure. The slopes of Fig. 2 give RT/a = 11.7 pg, so that a = 50 cal./pg. 

When the mass of a material is plotted against its volume, the slope of the line represents the material s density. An example of this is shown in Figure 2.16a. To calculate the slope of the line, substitute the x and y values for Points A and B in the slope equation and solve. 

If the reaction follows a simple nth order rate law and the temperature dependence of the rate constant is described by Arrhenius equation the thermogravimetric loss rate is given by the slope equation ... 

Harcourt, an English chemist, was Reader in Chemistry at Oxford University. He was a very competent experimentalist but knew little mathematics. Because rate equations are differential equations—slope equations—and a topic for treatment with calculus, he enlisted the aid of the mathematician Esson. Together they came up with methods for interpreting reaction rates that are essentially the same as those used today. 

Hence, the critical temperature 7, . is deteriiiiued using the equality betwc en the tangent line (Equation 78) at x = 0 and the slope (Equation 77), i.e. 

Another common theory was proposed by Manson (27) and is referred to as the universal slopes equation. In this model, the plastic strain or permanent deformation, is considered as a measure of the damage imposed in the material. On this basis, the true plastic strain amplitude can be used as a measure of the fatigue behavior. Moreover, the fatigue curve can be predicted in terms of the monotonic stress-strain curve. This empirical approach was initially statistically correlated with many metals and takes the following form ... 

Opp and co-workers (14) attempted to extend the universal slopes equation to predict the behavior of polymer fatigue. However, in their attempts they showed that a form of equation 6 was not useful for predicting polymer behavior and instead developed a model to predict the low cycle fatigue behavior based on hysteretic heating. And in this sense is consistent with those presented in equation 3 and References 15-17. 

Different choices of b yield different second-order Runge-Kutta methods. If 6 = the method is called the improved Euler s or Heun s method. If 6 = 1, the method is called the improved polygon or modified Euler s method. As demonstrated by this development, Runge-Kutta methods are not unique since they involve the choice of an arbitrary constant. All second-order methods involve the evaluation of two slopes ki and A)2 (equation (3.1.38) and equation (3.1.39)) and the value of is a weighted average of these two slopes (equations (3.1.35) and (3.1.36)). 

Alternatively, for wave basins with mildly sloping bottoms, the mild slope equation may be applied according to ... 

Similar approaches were developed by Lejeune et al and Yu. The regular mild slope equation was modified through a parameter to account for the bottom friction. Shorling and breaking effects were included in many works such as by Balas and Inan and Massel. The nonlinear effects of higher order were investigated by Mei. ... 

Various extended approaches (so-called extended mild slope equation) were generated to improve the performance of the regular mild slope equation for abrupt and undulating topography. Terms of bottom curvature and slope were included into the regular mild slope equation by groups of researchers. Although different approaches were used to derive the equations, they all obtained equivalent formulae as... 

Although the mild slope equation was obtained with the assumption of mild slope, the work by Booji showed that the regular mild slope equation is applicable for bottom slope as large as 1/3. Since the mild slope equation can be conveniently implemented in a finite element model, we will apply it to real harbors in this presentation. The basics of the model will be shown in the following section. 

A Computer Model Using the Mild Slope Equation... 

The governing equation is the mild slope equation first derived by BerkhoflP ... 

The wave potential in the inner region is then obtained by solving the mild slope equation and by matching the solutions with outer region at dA. 

P. G. Chamberlain and D. Porter, The modified mild slope equation, J. Fluid Mech. 291, 393-407 (1995). 

C. Lee and S. B. Yoon, Effect of higher-order bottom variation terms on the refraction of water waves in the extended mild slope equation. Ocean Eng. 31, 865 882 (2004). 

Curved Brpnsted plots. Although most slow reactions of carbon acids and bases give linear Brpnsted plots, there are some instances with an exceptionally long range of Ap T where curvature can be detected. A striking example is shown in Figure 8.5(b). The simplest way to extend Equation (8.11) to cover such plots is to add a term linear in AG to the expression for the slope (Equation (8.13)). We then write ... 

That is, from the ideal gas law, we are able to determine how one state variable varies with respect to another in an analytic fashion (that is, with a specific mathematical expression). A plot of pressure versus temperature is shown in Figure 1.6. Consider what equation 1.14 is telling you. A derivative is a slope. Equation 1.14 gives you the plot of pressure (y-axis) versus temperature (x-axis). If you took a sample of an ideal gas, measured its pressure at different temperatures but at constant volume, and plotted the data, you would get a straight line. The slope of that straight line should be equal to nR/V. The numerical value of this slope would depend on the volume and number of moles of the ideal gas. 

If the cell current is high, the polarization curves of CLs are similar, when either feed or ion transport is insufficient. In both the cases, the curve exhibits doubling of the Tafel slope (Equations 4.88 and 4.134). This makes it difficult to distinguish the physical origin of this doubling, by measuring the polarization curve only. 

Mlcoulaut M., The slope equations A universal relationship between local structure and glass transition temperature, Eur. Phys.J. B, 1,277-294 (1998). 

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