Linearity plot

The Arrhenius relation given above for Are temperature dependence of air elementary reaction rate is used to find Are activation energy, E, aird Are pre-exponential factor. A, from the slope aird intercept, respectively, of a (linear) plot of n(l((T)) against 7 The stairdard enAralpv aird entropy chairges of Are trairsition state (at constairt... 

This last expression can be rearranged in several additional ways which yieh linear plots ... 

Fig. 1. Correlation of Dk with percent water (log-linear plot). 1 barrer = 10 cm O2 (at STP)-cm/(cm -s-mm Hg).

The unknown intermediate concentration C, has been mathematically ehminated from the last term. In this case, r can be solved for explicitly, but that is not always possible with surface rate equations of greater complexity. The mass transfer coefficient /ci is usually obtainable from correlations. When the experimental data are of (C, r) the other constants can be found by linear plotting. 

A linear plot of the reciprocal of the reaction rate versus 1/(S) will allow the determination of Km and from experimental data. 

The laboratory studies utilized small-scale (1-5-L) reactors. These are satisfactoiy because the reaction rates observed are independent of reac tor size. Several reac tors are operated in parallel on the waste, each at a different BSRT When steady state is reached after several weeks, data on the biomass level (X) in the system and the untreated waste level in the effluent (usually in terms of BOD or COD) are collected. These data can be plotted for equation forms that will yield linear plots on rec tangular coordinates. From the intercepts and the slope or the hnes, it is possible to determine values of the four pseudo constants. Table 25-42 presents some available data from the literature on these pseudo constants. Figure 25-53 illustrates the procedure for their determination from the laboratory studies discussed previously. 

For low-cycle fatigue of un-cracked components where (imax or iCT inl am above o-y, Basquin s Law no longer holds, as Fig. 15.2 shows. But a linear plot is obtained if the plastic strain range defined in Fig. 15.3, is plotted, on logarithmic scales, against the cycles to failure, Nf (Fig. 15.4). TTiis result is known as the Coffin-Manson Law ... 

Unfortunately there are very little reliable data on the frequency of wall perforation caused by corrosion in most cases comprehensive data about wall thickness, pipe coating, type of soil, etc. are lacking. The incidence of wall perforation is usually plotted on a logarithmic scale against the service life of the pipeline (see Fig. 22-3). Cases are also known where a linear plot gives a straight line. Curve 1 in... 

The general function giving linear plots on semilogarithmic graphs is ... 

The governing equations (Equations 42, 67, 74 and 81) describing the filtration mechanisms are expressed as linear relationships with parameters conveniently grouped into constants that are functions of the specific operating conditions. The exact form of the linear functional relationships depends on the filtration mechanism. Table 1 lists the coordinate systems that will provide linear plots of filtration data depending on the controlling mechanism. 

Therefore, the slope of the linear plot Cg versus gives the ratio kj/kj. Knowing kj -i- kj and kj/kj, the values of kj and kj ean be determined as shown in Figure 3-10. Coneentration profiles of eom-ponents A, B, and C in a bateh system using the differential Equations 3-95, 3-96, 3-97 and the Runge-Kutta fourth order numerieal method for the ease when Cgg =Cco = 0 nd kj > kj are reviewed in Chapter 5. 

Using Equation 11-26 for the bateh reaetor, the dependent variable 1/t ln(Cso/Cs) is plotted as a funetion of the independent variable (Cgo - Cs)/t. A linear plot gives a slope = -l/K and an intereept =... 

A log-linear plot of the idealized eontinuous MSMPR population density versus erystal size is shown in Figure 3.7. 

Thus, a zero-order reaction yields a linear plot of c vs. t, the slope being equal to — k. It is evident that a zero-order rate constant has the units of a rate, for example, moles per liter-second (M s ). 

Dole has rearranged Eq. (2-15) into Eq. (2-19), which is yet another linear plotting form. 

The Michaelis-Menten equation is, like Eq. (3-146), a rectangular hyperbola, and it can be cast into three linear plotting forms. The double-reciprocal form, Eq. (3-152), is called the Lineweaver-Burk plot in enzyme kinetics. ... 

Equation (3-150) and other equations of the same functional form can be placed in three linear plotting forms, of which the double-reciprocal form is shown as Eq. (3-152). Find the other two linear plotting forms. 

Not all free energy relationships (Hammett plots) are linear. Aside from occasional scatter or unexplained deviations from an otherwise linear plot, it sometimes happens that the plot is smooth but nonlinear. Two causes have been identified, and many examples are known. One cause is a change in a rate-determining... 

Series 7 gives a linear plot with p = —5.27, consistent with the S l mechanism. Series 8, however, shows a discontinuity, which Gassmann and Fentiman interpreted as a change in mechanism. Compounds in series 8 are capable of intramolecular assistance (neighboring group participation) by electron donation from the double bond to stabilize the cation, as in 9. 

Thus, a linear plot of log 2 against log ki for a reaction series implies an isokinetic relationship for the series. The reason that this plot is a reliable test for such a relationship is that the errors in and 2 are independent (unlike the errors in A// and AS ). From the slope b of the straight line the isokinetic temperature p can be found ... 

Another method for studying solvent effects is the extrathermodynamic approach that we described in Chapter 7 for the study of structure-reactivity relationships. For example, we might seek a correlation between og(,kA/l ) for a reaction A carried out in a series of solvents and log(/ R/A R) for a reference or model reaction carried out in the same series of solvents. A linear plot of og(k/iJk ) against log(/ R/ linear free energy relationship (LFER). Such plots have in fact been made. As with structure-reactivity relationships, these solvent-reactivity relationships can be useful to us, but they have limitations. 

The quantity on the left side of the equation is referred to as the logarithm of the rate constant corrected for protonation often the correction term is negligible. If the activity coefficient term on the right side is negligible, Eq. (8-104) predicts a linear relationship between the corrected log kobs and —//o, the slope being unity. A similar treatment of the A-Se2 mechanism also predicts a linear plot of log fcobs against —Ho-... 

A plot of the left side of Eq. (8-108) against X is linear for A1 and nonlinear for A2 reactions. The curved A2 plot can be transformed to a linear plot by... 

Linear Plots Can Be Derived from the Michaelis-Menten Equation... 

The FTS mechanism could be considered a simple polymerization reaction, the monomer being a Ci species derived from carbon monoxide. This polymerization follows an Anderson-Schulz-Flory distribution of molecular weights. This distribution gives a linear plot of the logarithm of yield of product (in moles) versus carbon number. Under the assumptions of this model, the entire product distribution is determined by one parameter, a, the probability of the addition of a carbon atom to a chain (Figure 4-7). ... 

Polarisation Resistance slope of the linear plot of overpotential versus current density measured at potentials close to the corrosion potential, or the tangent of such a curve at the corrosion potential if the plot is not linear. If a small change in potential, A , gives rise to a change in current density. A/, then the polarisation resistance is / p(Q m ) = AE/Ai. 

The values of Kd(H) were determined by the extrapolation to n = 0 of the virtually linear plots of log 1/Kd vs. n obtained for cyclodextrin complexes with methyl-,... 

Other complexes of 4c, 10, and 11 show linear plots with slopes close to unity in the a pH range 6.5-8.5. These linear plots also seem to represent the ionization of hydroxyl groups, but their pKa values must be higher than 8.5. Unfortunately, it is difficult to examine higher pH s due to precipitation of the metal hydroxide. 

Order Rate Expression Conc.-Time Relation Half-Life Linear Plot... 

If it is not obvious from this table that the only linear plot is that of 1/[HI] versus time, that point should be clear from Figure 11.5. This is a second-order reaction. 

This is referred to as a double reciprocal or lineweaver Burk plot. From this linear plot, Kd = slope/intercept and the 1 /intercept = Bmax. Finally, a linear plot can be achieved with... 

FIGURE 5.19 Method of Barlow for measurement of affinity of a partial agonist, (a) Guinea pig ileal smooth muscle contraction to histamine (filled circles) and partial histamine receptor agonist E-2-P (N,N-diethyl-2-(l-pyridyl)ethylamine (open circles). Dotted lines show equiactive concentrations of each agonist used for the double reciprocal plot shown in panel b. (b) Double reciprocal plot of equiactive concentrations of histamine (ordinates) and E-2-P (abscissae). Linear plot has a slope of 55.47 and an intercept of 1.79 x 10s. This yields a KB (1 — tp/ta) = 30.9 pM. (c) Variant of double reciprocal plot according to Equation 5.8. (d) Variant of double reciprocal plot according to Equation 5.10. Data redrawn from [10],... 

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