Acid-base titration curves equivalence point

In the overview to this chapter we noted that the experimentally determined end point should coincide with the titration s equivalence point. For an acid-base titration, the equivalence point is characterized by a pH level that is a function of the acid-base strengths and concentrations of the analyte and titrant. The pH at the end point, however, may or may not correspond to the pH at the equivalence point. To understand the relationship between end points and equivalence points we must know how the pH changes during a titration. In this section we will learn how to construct titration curves for several important types of acid-base titrations. Our... 

Sketching an Acid—Base Titration Curve To evaluate the relationship between an equivalence point and an end point, we only need to construct a reasonable approximation to the titration curve. In this section we demonstrate a simple method for sketching any acid-base titration curve. Our goal is to sketch the titration curve quickly, using as few calculations as possible. 

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. 

When an indicator is used in a titration, the range of pH values at which its endpoint occurs must include, or be close to, the equivalence point. Some representative acid-base titration curves, shown in Figures 8.11, 8.12, and 8.13, will illustrate this point. 

Why does an acid-base titration curve (pH versus volume of titrant) have an abrupt change at the equivalence point ... 

When the equivalence point is reached, the Fe2+ will have been totally consumed (the large equilibrium constant ensures that this will be so), and the potential will then be controlled by the concentration ratio of Ce3+/Ce4+. The idea is that both species of a redox couple must be present in reasonable concentrations for a concentration to control the potential of an electrode of this kind. If one works out the actual cell potentials for various concentrations of all these species, the resulting titration curve looks much like the familiar acid-base titration curve. The end point is found not by measuring a particular cell voltage, but by finding what volume of titrant gives the steepest part of the curve. 

Analytically useful acid-base titration curves are characterized by a rather fast pH change near the equivalence point. This suggests that the location of the equivalence point might be determined experimentally from that of the maximum in its first derivative, d(pH)/dVfo, or the zero-crossing of its second derivative, d2(pH)/dVj,2. The advantage of such an approach is that it does not rely on any particular theoretical model, but instead exploits the characteristic feature of the titration curve, i.e., its fast pH change in the region around the equivalence point. The method does not even require that the pH meter is carefully calibrated. 

See Problem 21 for a spreadsheet calculation of the Ca-EDTA titration curve in Figure 9.3 at pH 10. As with calculated acid-base titration curves, the calculations here break down very near the equivalence point due to simplifying assumptions we have made. 

Fig. 5 shows the results of both titration experiments. The experimental results are in good agreement with the predictions based upon the equilibrium expressions for Kb the Ka for each indicator, and the mass and charge balances[13]. The data from the acid titration show a sharp equivalence point at approximately 10 m HCl, which suggests that B(OH)4 is still a strong base at 350°C and 0.622 g/mL and capable of neutralizing HCl. This strong acid base titration curve, as was also observed for HCl and KOH, may be contrasted with the weak acid-base behavior observed for the sulfuric acid-ammonia system at 380 C[41]. 

Denaturation and Renaturation When DNA molecules are heated to certain temperature (e.g., lOO C), the two polynucleotide strands separate. The transition from the double strand (original form) to the single strand (denatured form) can be observed by the change in optical density at 260 nm. The plot of the optical density versus temperature gives a sinusoidal curve that is similar to an acid-base titration curve. In Figure 17.8 the point Tin, which corresponds to the equivalence point in an acid-base titration, is the hypochromic point and denotes where a mixture of the native and denatured strands occurs. 

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. 

Where Is the Equivalence Point In discussing acid-base titrations and com-plexometric titrations, we noted that the equivalence point is almost identical with the inflection point located in the sharply rising part of the titration curve. If you look back at Figures 9.8 and 9.28, you will see that for acid-base and com-plexometric titrations the inflection point is also in the middle of the titration curve s sharp rise (we call this a symmetrical equivalence point). This makes it relatively easy to find the equivalence point when you sketch these titration curves. When the stoichiometry of a redox titration is symmetrical (one mole analyte per mole of titrant), then the equivalence point also is symmetrical. If the stoichiometry is not symmetrical, then the equivalence point will lie closer to the top or bottom of the titration curve s sharp rise. In this case the equivalence point is said to be asymmetrical. Example 9.12 shows how to calculate the equivalence point potential in this situation. 

The scale of operations, accuracy, precision, sensitivity, time, and cost of methods involving redox titrations are similar to those described earlier in the chapter for acid-base and complexometric titrimetric methods. As with acid-base titrations, redox titrations can be extended to the analysis of mixtures if there is a significant difference in the ease with which the analytes can be oxidized or reduced. Figure 9.40 shows an example of the titration curve for a mixture of Fe + and Sn +, using Ce + as the titrant. The titration of a mixture of analytes whose standard-state potentials or formal potentials differ by at least 200 mV will result in a separate equivalence point for each analyte. 

An analogous effect on the sharpness at the equivalence point as for an acid-base titration (cf., eqn. 2.30) may not be overlooked in the case of curve II because... 

Figure 6.3 A schematic pH curve for the titration of a strong acid with a strong base. At the equivalence point, the amount of alkali added is the same as the amount of acid in solution initially, allowing for an accurate calculation of the acid s concentration. Note how the end point is determined by extrapolating the linear regions, and drawing a third parallel line between them...

Before continuing with other examples, it is important to consider how the equivalence point in an acid-base titration is found and what relationship this has with titration curves. As we have said, the inflection point at the center of these curves occurs at the equivalence point, the point at which all of the substance titrated has been exactly consumed by the titrant. The exact position for this in the case... 

O O Sketch the pH curve for the titration of a weak acid with a strong base. Show the equivalence point on your sketch. Suggest an indicator that might be used, and explain your selection. 

FIGURE 16.6 A strong acid-strong base titration curve, (a) In this pH titration, 0.100 M NaOH is added slowly from a buret to an HC1 solution of unknown concentration. The pH of the solution is measured with a pH meter and is recorded as a function of the volume of NaOH added, (b) The pH titration curve for titration of 40.0 mL of 0.100 M HC1 with 0.100 M NaOH. The pH increases gradually in the regions before and after the equivalence point, but increases rapidly in the region near the equivalence point. The equivalence point comes after addition of 40.0 mL of 0.100 M NaOH. The pH at the equivalence point is 7.00. 

A pH titration curve is a plot of the pH of a solution as a function of the volume of base (or acid) added in the course of an acid-base titration. For a strong acid-strong base titration, the titration curve exhibits a sharp change in pH in the region of the equivalence point, the point at which stoichiometri-... 

Obviously one could measure the pH of a known concentration of a weak acid and obtain a value of its hydronium ion activity, which would permit a direct evaluation of its dissociation constant. However, this would be a one-point evaluation and subject to greater errors than by titrating the acid halfway to the equivalence point. The latter approach uses a well-buffered region where the pH measurement represents the average of a large number of data points. Similar arguments can be made for the evaluation of solubility products and stability constants of complex ions. The appropriate expression for the evaluation of solubility products again is based on the half-equivalence point of the titration curve for the particular precipitation reaction [AgI(OH2)2h represents the titrant] ... 

In the use of potentiometry for the evaluation of stability constants for complex ions, the expressions can become extremely complicated if multiequilibria are present. For a simple one-to-one complex a direct potentiometric titration curve again provides die most satisfactory route to an accurate evaluation of the constant. The curve looks similar to that for an acid-base titration, and the appropriate point to pick is the half-equivalence point. If the complex is extremely stable, then die amount of free metal ion at this point on die dtration curve (ligand titrated with metal ion) is sufficiently low that it can be disregarded. If not, it must be handled in a way similar to the first point on the titration curve for phosphoric acid. Assuming that it is a stable complex, at the first half-equivalence point the concentration of complexed metal ion will be equivalent to that of the free ligand. The potential will give a direct measure of the free metal ion and allow the stability constant for the complex to be evaluated at the half-equivalence point ... 

A titration curve is a plot of a solution s pH charted against the volume of an added acid or base. Titration curves are obtained if a pH meter is used to monitor the titration instead of an indicator. At the equivalence point, the titration curve is nearly vertical. This is the point where a rapid change in pH occurs. In addition to determining the equivalence point, the shape of titration curves may be interpreted to determine acid/base strength and the presence of a polyprotic acid. 

Any titration involves the progressive change of the activities (or concentrations) of the titrated and titrating species and, in principle, can be done potentiometrically. However, for an accurate determination it is necessary that there is a fairly rapid variation in equilibrium potential in the region of the equivalence point. Useful applications are redox, complexation, precipitation, acid-base titrations, etc. From the titration curve it is possible to calculate values of the formal potentials of the titrated and titrating species, as explained below. 

Fig. 13.2. Methods for determining the equivalence point of a potentiometric titration curve (including acid-base titrations), (a) First derivative (b) Second derivative (c) Gran plot for titration of a strong acid with a strong base Vx is the initial volume of acid and V the volume of base added.

The acid-base behavior of amino acids may also be illustrated via titration curves. If one started with aspartic acid hydrochloride, that is, aspartic acid crystallized from solution in hydrochloric acid, one would require 3 mol base to remove completely the protons from 1 mol aspartic acid. The titration curve obtained with structures at each step of the reaction series is shown in Figure 4.1. Note that the isoelectric point is attained after one proton equivalent has been removed from the molecule. At this point, aspartic acid contains one positive and one negative charge it is zwitterionic. 

When the progression of an acid-base titration is graphed as a function of pH vs the volume of acid or base added, the curve will appear as shown below. If we recall, from general chemistry coursework, that the steepest point on the curve represents the equivalence point of the titration (the point where the amount of acid and base are equal), we can locate the point on the curve that represents the midpoint of the titration. This point is found at half the concentration of base added to acid (or acid added to base) to reach the equivalence point. Once we have done this, we recall the Henderson-Hesselbach equation (Fig. 2.8)—specifically, the term dealing with the concentrations of the ionic and the neutral species. Realizing that at the midpoint of the titration, these concentrations are equal, the logarithmic term in the Henderson-Hesselbach equation reduces to log(l), which is equal to zero. Therefore, the equation reduces to pA a = pH at the midpoint of the titration. 

The methylol derivatives are stronger acids (weaker bases) than are the original unsubstituted amino groups. In other words, the pKo value for the substituted amino acid is lower than the pifo, value for the original amino acid. The titration curves are sketched in Figure 1-7. Note that formaldehyde has no effect on the amounts of KOH required to titrate the amino acid to pKa, pKo (or pKi,), and the equivalence points. Also note that only the pK value is shifted formaldehyde has no effect on the a-COOH group. 

The proton conditions of equations 48 and 49 correspond to the two equivalence points in acid-base titration systems. The half-titration point is usually (not always) given by pH = pAT. Thus the qualitative shape of the titration curve can be sketched readily along these three points (Figure 3,3a). 

Acid-base titrations of humic substances reflect the nature of the different p/Tfl values, hence the smeared out appearance of these titration curves. While no unique equivalence points are observed, different p regions of carboxylic and phenolic groups can be discerned. Similarly, in metal titrations, metal ions are bound differently by the different ligand groups. The extent of metal-ion binding depends on the ratio of metal ions to humic substances, [M7]/ [L7-]. In titrating humic or fiilvic acids with metal ions (at fixed pH), the metal is bound first to the highest affinity sites. 

Titration curves for strong bases are derived in an analogous way to those for strong acids. Short of the equivalence point, the solution is highly basic, the hydroxide ion concentration being numerically related to the analytical molarity of the base. The solution is neutral at the equivalence point and becomes acidic in the region beyond the equivalence point then the hydronium ion concentration is equal to the analytical concentration of the excess strong acid. 

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