Zero intercept

The ratio n/c2 is called the reduced osmotic pressure-and can be plotted with or without the RT-and the zero-intercept value (subscript 0) is the limiting value of the reduced osmotic pressure. Quite an assortment of different pressure units are used in the literature in reporting n values, and the units of R in Eq. (8.88) must be reconciled with these pressure (as well as concentration) units. 

There are any number of variations that can be applied to the CLS technique. Here we will only consider the most important one non-zero intercepts. If you are interested in some of the other variations, you may wish to consult the references in the CLS section of the bibliography. 

Referring to equation [40], we can see that we require the absorbance at each wavelength to equal zero whenever the concentrations of all the components in a sample are equal to zero. We can add some flexibility to the CLS calibration by eliminating this constraint. This will add one additional degree of freedom to the equations. To allow these non-zero intercepts, we simply rewrite equation [40] with a constant term for each wavelength ... 

We perform CLS on A6 to produce 2 calibrations. K6 and K6, are the matrices holding the pure component spectra and calibration coefficients, respectively, for CLS with zero intercepts. K6a and K6aMl are the corresponding matrices for CLS with nonzero intercepts. 

We can see that the ILS calibrations are noticeably better than the CLS calibrations done with zero intercept. And they are as good or somewhat better that the CLS calibrations with nonzero intercept. This is remarkable when we consider how badly we have degraded the spectra when we condensed them The main reason for the advantage of ILS over CLS can be seen in equation [47],... 

CLS calibration with non-zero intercept from Al/Cl predicts concentrations for AI... 

CLS with non-zero intercept estimated pure component spectra... 

Comparisons of relative rate constants obtained with Mv s of the total polymer and M s of the HMWF for the same samples show similar trends negligible transfer and termination control of molecular weights for the f-BuCl/Et2AlCl/MeCl system in the —40° to —60 °C range and also for the f-BuBr/Et2AlCl/MeCl at —50 °C (Table 7). For the samples prepared with the f-BuCl/Et AlCl system Mayo plots based on Mv s show zero intercept while that based on Mn s of the HMWF shows a small but finite intercept, z., ktr/kp = 1.91 x 10-5 and 2.14 x 10-s at —50° and -60 °C. Similarly, for the samples prepared with the t-BuBr/Et2AlCl system the Mayo plot based on Mn s of HMWF shows zero intercept while the Mayo plot based on Mv s show a very small intercept, ie., ktr/kp = 5.0 x 10-s at —50 °C. The reasons for this small discrepancy are not known. 

It is important to note that the fitting according to eq. (1) requires zero intercept behavior i.e., F =. 00 for H (for which Oj = Or =. 00). While we recognize that the data for the unsubstituted (H) member of a set may be as subject to experimental error as any other member, such error is generally relatively small for a set of reliable data. Any constant error from this source will be distributed among all of the substituents in such a manner as to achieve best fit. Any loss in precision of fitting of the set which may result by such a procedure we believe is a small price to pay compared to the violence done by introduction in eq. (I) of a completely variable constant parameter. The latter procedure has been utilized by other authors both in treatments by the simple Hammett equation and by the dual substituent parameter equation. 

Thus, a plot of 3 vs should yield a line with slope 2N. k jk j and a zero Intercept while a plot of i/ - vs. 1 will yield a line with ... 

Typically, at least two different values of m1 (besides ml = 0) are used because there are invariably phase shifts that arise from various factors that do not depend on the gradient moments, resulting in a non-zero intercept of c > versus mx. Thus, a velocity image is time consuming because each set of measurements with a value of m1 is an image in itself. 

The plot of k against the hydrogen ion concentration is linear, virtually zero intercept. Thus, first order with... 

A graphical method such as this is usually superior to a simplistic calculation of k2 = k -l [NaOH] (e.g. in the Worked Example 8.16), because scatter and/or chemical back or side reactions will not be detected by a single calculation. Also, the involvement of a back reaction (see next section) would be seen most straightforwardly as a non-zero intercept in a plot of k (as y ) against [reagent in excess] (as V). 

Equation 45 and 47 as well as equation 28 account for the quadratic dependence of kj with [B], with a zero intercept if the uncatalysed decomposition is assumed to be negligible. However, the peculiar kinetic behaviour observed in some systems (which has just been described) can only be explained by equation 28. Hirst148 and Banjoko170... 

This suggests that a plot of r0 vs pA (Fig. 4) or r0/pA vs pA should be linear. It is very difficult to reject this model on the basis of data curvature, even though it is evident that some curvature could exist in Fig. 4. However, Eq. (16) demands that Fig. 4 also exhibit a zero intercept. In fact, the 99.99% confidence interval on the intercept of a least-square line through the data does not contain zero. Hence the model could be rejected with 99.99% certainty. 

Figure 6.17 A Randles-SevCik plot of 7p against Data refer to the oxidation of aqueous ferrous ion at a stationary platinum wire electrode. The non-linearity at the higher scan rates represents the demand for flux at the working electrode being too great since i is too fast, while the non-zero intercept is caused by non-faradaic currents contributing...

Figure 6.17 broadly follows the Randles-Sevcik equation insofar as most of the plot of p against v is linear. The figure does, however, have a non-zero intercept, and is not linear at the highest scan rates. 

The simplest reason for a non-zero intercept is non-faradaic current, the causes of which are discussed in Section 6.8.2. 

In the oxidation of Fe(II)P + (5) by excess O2 there is a second-order loss of Fe(II). The second-order rate constant Ar is linearly dependent on [Oj] with a zero intercept for the [O2] plot. An overall third-order rate law therefore holds. 

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