Slopes of the lines method

Fig. 13.2. Results of the reaction path shown in Figure 13.1, plotted to allow the overall reaction to be extracted using the slopes-of-the-lines method.

We can write the overall reaction by which hematite forms, using the slopes-of-the-lines method discussed in Chapter 13. Initially, the reaction proceeds as,... 

We can readily derive the overall reactions in the buffer from Figure 15.3 using the slopes-of-the-lines method, described in Chapter 13. In the first stage, the overall reaction is,... 

Fig. 15.3. Species masses in the Na2C()3 solution plotted so that the overall reaction can be determined using the slopes-of-the-lines method.

Fig. 13. The standard addition method where MB is the confidence interval for the slope of the line = k, and represents 95% confidence interval (14).

In each following step Ot is the slope of the line joining [xk,fixi)] to the most recently determined point where/(x,) has the opposite sign from that of/(xi). This method is of first order. If one uses the most recently determined point (regardless of sign), the method is a secant method. 

A linear plot indicates that the luminescence decay is exponential. The slope of the line gives kt, and rt can be calculated as above. The lifetime obtained by measuring the decay of P-type delayed fluorescence is equal to one-half the lifetime of the triplet state (see Section 5.2). Since in fluid solution at room temperature phosphorescence is generally much weaker than delayed fluorescence, the measurement of delayed fluorescence decay offers a convenient method for determining the lifetime of triplets at room temperature. 

The method assumes that the data are independent and normally distributed, and it is sensitive to outliers. The Y-axis (or horizontal) component plays an extremely important part in developing the least square fit. All points have equal weight in determining the height of a regression line, but extreme x-axis values unduly influence the slope of the line. 

Figure 13.7 shows the application of this method for a typical calibration and experiment, respectively. A value of 4>obs = 1 060 is obtained by the ratio between the slopes of the lines (b) and (a), which refer to equations 13.20 and 13.19, respectively. 

Many automated instruments measure enzyme activity using fixed time colorimetric methods. Some, however, can be classed as reaction rate analysers, e.g. the centrifugal analysers, and these instruments determine the reaction rate from either the initial slope of the reaction curve or from repeat measurements at fixed intervals. In both methods the slope of the line is taken to represent the activity of the enzyme. 

Also referred to as the Hanes-Hultin plot and the Hanes-Woolf (or, Woolf-Hanes) plot, the method is based on a transformation of the Michaelis-Menten equation i.e., the expression for the Uni Uni mechanism) [A]/v = (i a/ max) + ([A]/Umax) whcrc U ax IS the maximum forward velocity and is the Michaelis constant for A. In the Hanes plot, the slope of the line is numerically equal to Umax, the vertical intercept is equivalent to, ... 

Retaining the latest estimates for x and X2, the slope of the line follows more closely the form of the function than in the false position method. The order of convergence can be shown to be 1.61B, the "golden ratio", which we will encounter in Section 2.2.1. The root, however, is not necessarily bracketed, and the next estimate x3 may be far away if the function value ffx ) is close to f(x2>. Therefore we may run into trouble when starting the search in a region where the function is not monotonic. 

Figure 3.26 The log 0 plasma concentration against time profile for a compound after oral administration. The absorption rate constant kab can be determined from the slope of the line (dotted) plotted using the method 60 of residuals as described in the text. (Thus, A" = A -A, etc.)...

The method of testing this equation is to plot the logarithm of the velocity constant against the reciprocal of the absolute temperature. A straight line is obtained if the equation is satisfied. The slope of the line gives the value of A. The quantity A has the dimensions of energy, and if R is taken as 1 98 calories, then A is measured in calories per gram molecule. 

The choice of which variable isx and which lsy is arbitrary but, in the method of least squares, whichever is chosen asy is assumed to possess all of the error, andx is assumed to possess none If the variables were interchanged (that is, if.v and y were plotted the other way around), this assumption would be reversed, and the slope of the line would be given by... 

If AH 0 can be assumed constant over the temperature range of an experiment, a plot of In K vs 1/T provides a convenient estimate of AH 0 (or AH if In K is plotted). The slope of the line will be -AH0/R. Since AG0 can be calculated from K, the method also permits evaluation of AS0 using Eq. 6-15. However, unless great care is taken the method is of low accuracy23 and it is preferable to establish AH by direct calorimetry. Also, especially for proteins, the assumption that AH0 is constant over a significant temperature range may be erroneous. 

The procedure adopted to portray the scope and utility of a linear free-energy relationship for aromatic substitution involves first a determination of the p-values for the reactions. These parameters are evaluated by plotting the values of log (k/ka) for a series of substituted benzenes against the values based on the solvolysis studies (Section IV, B). The resultant slope of the line is p, the reaction constant. The procedure is then reversed to assess the reliability and validity of the Extended Selectivity Treatment. In this approach the log ( K/ H) observations for a single substituent are plotted against p for a variety of reactions. This method assays the linear or non-linear response of each substituent to variations in the selectivity of the reagents and conditions. Unfortunately, insufficient data are available to allow the assignment of p for many reactions. It is more practical in these cases to adopt the Selectivity Factor S as a substitute for p and revert to the more empirical Selectivity Treatment for an examination of the behavior of the substituents. 

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