Titration curve slope

It has been shown that for most acid-base titrations the inflection point, which corresponds to the greatest slope in the titration curve, very nearly coincides with the equivalence point. The inflection point actually precedes the equivalence point, with the error approaching 0.1% for weak acids or weak bases with dissociation constants smaller than 10 , or for very dilute solutions. Equivalence points determined in this fashion are indicated on the titration curves in figure 9.8. 

The most obvious sensor for an acid-base titration is a pH electrode.For example, Table 9.5 lists values for the pH and volume of titrant obtained during the titration of a weak acid with NaOH. The resulting titration curve, which is called a potentiometric titration curve, is shown in Figure 9.13a. The simplest method for finding the end point is to visually locate the inflection point of the titration curve. This is also the least accurate method, particularly if the titration curve s slope at the equivalence point is small. 

Another method for finding the end point is to plot the first or second derivative of the titration curve. The slope of a titration curve reaches its maximum value at the inflection point. The first derivative of a titration curve, therefore, shows a separate peak for each end point. The first derivative is approximated as ApH/AV, where ApH is the change in pH between successive additions of titrant. For example, the initial point in the first derivative titration curve for the data in Table 9.5 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. 

In Fig. 15.7 are presented (a) the part of the experimental titration curve in the vicinity of the equivalence point (b) the first derivative curve, i.e. the slope of the titration curve as a function of V (the equivalence point is indicated by the maximum, which corresponds to the inflexion in the titration curve) and (c) the second derivative curve, i.e. the slope of curve (b) as a function of V (the second derivative becomes zero at the inflexion point and provides a more exact measurement of the equivalence point). 

The optimum volume increment AV depends upon the magnitude of the slope of the titration curve at the equivalence point and this can easily be estimated from a preliminary titration. In general, the greater the slope at the e.p., the smaller should A V be, but it should also be large enough so that the successive values of AE exhibit a significant difference. 

The slope of the tangent to the curve at the inflection point where oc = is thus inversely proportional to the number of electrons n. The E-oc curves are similar to the titration curves of weak acids or bases (pH-or). For neutralization curves, the slope dpH/doc characterizes the buffering capacity of the solution for redox potential curves, the differential dE/da characterizes the redox capacity of the system. If oc — for a buffer, then changes in pH produced by changes in a are the smallest possible. If a = in a redox system, then the potential changes produced by changes in oc are also minimal (the system is well poised ). 

The plot of light transmission versus volume of titrant added would be expected to be a step change, where the equivalence point might reasonably be taken as the position of greatest slope in the titration curve. 

The end point in a titration is a little different from the end of a reaction. What is desired is to know when all the add is titrated. This happens when the titration curve, shown in Figure 10.1, reaches its maximum slope. This change occurs at a pH of 7 for a typical strong acid-base titration), but occurs at pH... 

In most cases, a curve is not drawn and the end point is taken as the milliliters (mL) used just when the color change takes place. There should be a half-drop of titrant difference between the change from one color to the next. In many cases, the color change is very light but distinctive. If the titration curve is plotted, then the end point can be determined by inspection or by taking the first or second derivative of the data to find the point of maximum slope. 

Inflection point—A point in a titration curve at which the slope of the curve is a maximum. 

This corresponds to titrating citric acid with NaOH. The titration curve is very nearly linear from pH 2.2 to about pH 5.5 with a slope of 3.60. The effects of the three functional buffer groups of citric acid are smeared so that no S-shape or inflection points are apparent. 

The phase diagram of an electrode material may be determined from the slope of the coulometric titration curve. An electrode of N components shows activities which are independent of the composition as long as the maximum number of N phases are in equilibrium with each other. Relative changes in the amounts of the different phases do not change the activities of the components and therefore keep the cell voltage constant. This causes voltage plateaux for any region of the equilibrium of the maximum number of phases. 

The slope of the coulometric titration curve is accordingly proportional to the Wagner factor in the case of predominant electronic conductivity. 

As discussed in Section 8.2 the relation between the chemical diffusion coefficient and diffusivity (sometimes also called the component diffusion coefficient) is given by the Wagner factor (which is also known in metallurgy in the special case of predominant electronic conductivity as the thermodynamic factor) W = d n ajd In where A represents the electroactive component. W may be readily derived from the slope of the coulometric titration curve since the activity of A is related to the cell voltage E (Nernst s law) and the concentration is proportional to the stoichiometry of the electrode material ... 

The potentiometric titration curves of gels, which relate the pH of the exterior solution to the degree of ionization of the gel, resemble the titration curves of monofunctional acids or bases. However, the dissociation constants differ, often by two orders of magnitude, from the expected value for the functional group, and the slope of the curves is not the usual one. Addition of neutral salt changes the picture markedly and brings the curves closer to expectation. In the case of weak or medium... 

Figure 4.32. Binding isotherms (a) and titration curves (b), and the corresponding slopes of and 6, for values of the binding constants calculated for the racemic form with k values from Table 4.8 (here, L is the trans form and H the cis-gauche form). Note the four peaks in the slope curves, corresponding to the four binding constants lo 2C be binding isotherms are plotted in...

The CMC of C14DAO is about 1 x 10 M at 25 C. Below the CMC a typical buffering action is observed (4 x ICT M), above the CMC the titration curves are slanted toward lower pH s with increasing HCl concentration 0.2 M having a steeper slope than 8 X 10 M. Addition of SDS to a solution of C DAO affects the HCl titration curve markedly and will be discussed later. 

When potentiometric titration is carried out, the volume of titrant added is plotted against the measured potential. Since the electrode takes time to equilibrate, the volume of titrant required to reach the end-point is first calculated and a volume of titrant is added to within ca 1 ml of the end-point. Then the titrant is added in 0.1 ml amounts until the steep inflection in the titration curve is passed. The end-point of the titration is the point where the slope of the titration curve is at its maximum. 

Buffers have their limits, however. The acid s proton reservoir, for excimple, can compensate for the addition of only a certain amount of base before it runs out of protons that can neutralize free hydroxide. At this point, a buffer has done all it can do, and the titration curve resumes its steep upward slope. 

Thus, the buffer capacity / at each point of the titration curve is inversely proportional to the slope of the tangential line to the curve. When homoconjugation occurs, / is expressed approximately by the following relation [11] ... 

Recently, Ta2Os- and Si3N4-type pH-ISFETs have been used in non-aqueous systems, by preparing them to be solvent-resistant [17]. In various polar non-aqueous solvents, they responded with Nernstian or near-Nernstian slopes and much faster than the glass electrode. The titration curves in Fig. 6.5 demonstrate the fast (almost instantaneous) response of the Si3N4-ISFET and the slow response of the glass electrode. Some applications of pH-ISFETs are discussed in Section 6.3.1. 

Titration curves in Figure 7-6 illustrate the effect of reactant concentration. The equivalence point is the steepest point of the curve. It is the point of maximum slope (a negative slope in this case) and is therefore an inflection point (at which the second derivative is 0) ... 

The complete titration curve in Figure 11-1 exhibits a rapid change in pH near the equivalence point. The equivalence point is where the slope (dpH/dVf) is greatest (and the second derivative is 0, which makes it an inflection point). To repeat an important statement, the pH at the equivalence point is 7.00 only in a strong-acid-strong-base titration. If one or both of the reactants are weak, the equivalence point pH is not 7.00. 

Figure 11 -5 shows an autotitrator, which performs the entire operation automatically.4 Titrant from the plastic bottle at the rear is dispensed in small increments by a syringe pump while pH is measured by electrodes in the beaker of analyte on the stirrer. (We will learn how these electrodes work in Chapter 15.) The instrument waits for pH to stabilize after each addition, before adding the next increment. The end point is computed automatically by finding the maximum slope in the titration curve. 

The measure of buffer capacity [dB/d(pH)] is the slope of the titration curve (pH plotted against increments of the base added) at any point. 

Calcium and magnesium influence the titration curves of milk because as the pH is raised they precipitate as colloidal phosphates, and as the pH is lowered, colloidal calcium and magnesium phosphates are solubilized. Since these changes in state are sluggish and the composition of the precipitates depends on the conditions (Boulet and Marier 1961), the slope of the titration curves and the position of the maximum buffering depend upon the speed of the titration. 

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