X-Intercept

Starting with equation 5.6, show that the equations for the slope, y-intercept, and x-intercept in Figure 5.7(a) are correct. 

Since both Vq and Cs are known, the x-intercept can be used to calculate the analyte s concentration. 

In a standard addition the analyte s concentration is determined by extrapolating the calibration curve to find the x-intercept. In this case the value of X is... 

For volumes of titrant before the equivalence point, a plot of Vb X [H3O+] versus Vb is a straight line with an x-intercept equal to the volume of titrant at the end point and a slope equal to Results for the data in Table 9.5 are shown in Table 9.6 and plotted in Figure 9.13d. Plots such as this, which convert a portion of a titration curve into a straight line, are called Gran plots. 

The concentration of sodium in the standard addition samples is determined from the absolute value of the x-intercept (see Figure 5.7b) thus, substituting zero for the emission intensity gives the concentration of sodium as 1.44 ppm. The concentration of sodium in the salt substitute, therefore, is... 

A plot of 1/v versus 1/[S], which is called a double reciprocal, or Lineweaver-Burk plot, is a straight line with a slope of Km/Vmax) a y-intercept of 1/Vmax> and an x-intercept of-l/Km (Figure 13.11). 

Equation 1-111 is known as the Lineweaver-Burk or reciprocal plot. If the data fit this model, a plot of l/V versus 1/Cg will he linear with a slope K /V x intercept l/V x-... 

Pure noncompetitive inhibition occurs if Ki = Ki. This situation is relatively uncommon the Lineweaver-Burk plot for such an instance is given in Eigure 14.15. Note that K is unchanged by I (the x-intercept remains the same, with or without I). Note also that Tmax decreases. A similar pattern is seen if the amount of enzyme in the experiment is decreased. Thus, it is as if I lowered [E],... 

Intercepts are points where the curve of a function crosses the axes. The x intercepts are found by setting y = 0 and the y intercepts by setting x = 0. 

According to this expression, a plot of 1/v, versus l/[SJo will yield a straight line if the data follow the Michaelis-Menten mechanism. This line has a slope given by Km/Vmax, a y intercept of 1/Vmax, and an x intercept of -1 fKm. This is also illustrated in Fig. 4-7. Again, this treatment is valid when Eq. (4-107) applies whether or not the catalyst is an enzyme. The Lineweaver-Burk plot, Fig. 4-lb, is convenient for visualization but statistically unreliable for data fitting the form in Eq. (4-107) should be used for numerical analysis. 

FIG. 10 Nomalized unbalanced surface charge in a cylindrical pore with R = 5din the presence of an external potential The results, from left to right, are for original surface charge densities of —0.001, —0.005, —0.01, —0.02, —0.04, —0.05, —0.07123 C/m respectively. The x-intercepts are values of the corresponding equilibrium Donnan potentials. 

Equation (35) is the equation for a straight line, 7 = ax + b, wherey= l/v[andx= 1/[S]. Aplot of 1/v as j/as a function of 1/[S] as x therefore gives a straight line whose jy intercept is 1/l iax and whose slope is KJV. Such a plot is called a double reciprocal or Lineweaver-Burk plot (Figure 8-5). Setting thej/ term of equation (36) equal to zero and solving for x reveals that the x intercept is — IK. ... 

When the relation y = f x) is such that there is only one value of y for each acceptable value of x, f(x) is said to be a single-valued function of x. Thus, if the function is defined for, say, x = jc , the vertical line x = x intercepts the curve at one and only one point, as shown in Fig. S. However, in many cases a given value of x determines two or more distinct values of y. 

If the line is extrapolated to the x axis, the x intercept is the temperature at which zero volume is theoretically reached. That temperature is -273.1°C, also known as absolute zero, 0.0 K. 

Plotting 1/V versus 1/[S], one obtains a straight line having a slope of Km/Vmax with a y-axis intercept of l/VmAX and an x-intercept of - 1/Km as shown in Figure 2.13. Lineweaver-Burk plots of enzyme activity in the presence of an inhibitor can distinguish the type of inhibitor. Competitive inhibitors have a molecular structure similar to that of the substrate and will alter Km but not VnrAX because they compete with the substrate for binding at the enzyme s active site but do not change the enzyme s affinity for substrate. Noncompetitive inhibitors bear no structural similarity to the substrate but bind the free enzyme or enzyme-substrate... 

The task is to resolve (3.69) for y=0, i.e. to find the x-values that result in y=0, or in other words to find the intercepts of the curve with the x-axis. These x-intercepts are commonly call roots. There is a solution around x=20, another one near x=5 and possibly even more outside the window displayed in Figure 3-21. 

Figure 3-23 illustrates one potential disaster that strikes if the initial guess xo is in an unfortunate position, however, note, it is not very far from the solution. f(xi) is zero or very small and thus the x-intercept is at infinity or very far away. Most likely the iterative process will collapse. 

Figure 2.5 The amount of a chemical absorbed by a sampler through time, where the lag time (L) is the time represented by the x-intercept of the extension (dashed line) of the steady state I ine AB.

Don t bother memorizing these equations—they re all straight lines with an x and y intercept and a slope. The useful information (Tmax or Km) will either be on the x and y intercepts or encoded in the slope (y intercept/x intercept). Remember that v has the same units as Tmax, and Km has the same units as [S]. Look at the label on the axes and it will tell you what the intercept on this axis should give in terms of units. Then match these units with the units of Vmax and [S]. 

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