Equivalence points

We have already discussed two methods of locating the equivalence points in titrations (1) as the center of the vertical portion (i.e., the most rapid change of pH with titrant volume) of the curve and (2) the steepest point in the plot of titration curve slope (ApH/AV vs. pH). Both of these methods require that very precise and closely spaced data be taken in the vicinity of the equivalence point. 

It is possible to significantly improve the reliability as well as simplicity of locating the equivalence point by converting the logarithmic expression of critical variable concentration, i.e., the pH, to a linear one, [H ]. This suggestion was first advanced in 1952 by Gunnar Gran, a Swedish analytical chemist but could not be exploited for several decades because commercial pH meters were not capable of sufficiently reliable pH measurements. 

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The problem in any quantitative volumetric analysis for ions in solution is to determine accurately the equivalence point. This is often found by using an indicator, but in redox reactions it can often... 

Experimentally, the aqueous iron(II) is titrated with cerium(IV) in aqueous solution in a burette. The arrangement is shown in Figure 4.6, the platinum indicator electrode changes its potential (with reference to a calomel half-cell as standard) as the solution is titrated. Figure 4.7 shows the graph of the cell e.m.f. against added cerium(IV). At the equivalence point the amount of the added Ce (aq) is equal to the original amount of Fe (aq) hence the amounts of Ce (aq) and Fe (aq) are also equal. Under these conditions the potential of the electrode in the mixture is ( - - f)/2 this, the equivalence point, occurs at the point indicated. 

H0CH3)3CNHH3 121.137 Tris(hydroxymethyl)aminomethane is available commercially as a primary standard. Dry at 100-103°C (<110°C). In titrations with a strong acid the equivalence point is at about pH 4.5-5. Equivalent weight is the formula weight. [J. H. Eossum, P. C. Markunas, and J. A. Riddick, Anal. Chem., 23 491 (1951).]... 

A selected list of redox indicators will be found in Table 8.26. A redox indicator should be selected so that its if" is approximately equal to the electrode potential at the equivalent point, or so that the color change will occur at an appropriate part of the titration curve. If n is the number of electrons involved in the transition from the reduced to the oxidized form of the indicator, the range in which the color change occurs is approximately given by if" 0.06/n volt (V) for a two-color indicator whose forms are equally intensely colored. Since hydrogen ions are involved in the redox equilibria of many indicators, it must be recognized that the color change interval of such an indicator will vary with pH. 

The determinate error in a titration due to the difference between the end point and the equivalence point. 

For a titration to be accurate we must add a stoichiometrically equivalent amount of titrant to a solution containing the analyte. We call this stoichiometric mixture the equivalence point. Unlike precipitation gravimetry, where the precipitant is added in excess, determining the exact volume of titrant needed to reach the equivalence point is essential. The product of the equivalence point volume, Veq> and the titrant s concentration, Cq, gives the moles of titrant reacting with the analyte. 

Almost any chemical reaction can serve as a titrimetric method provided that three conditions are met. The first condition is that all reactions involving the titrant and analyte must be of known stoichiometry. If this is not the case, then the moles of titrant used in reaching the end point cannot tell us how much analyte is in our sample. Second, the titration reaction must occur rapidly. If we add titrant at a rate that is faster than the reaction s rate, then the end point will exceed the equivalence point by a significant amount. Finally, a suitable method must be available for determining the end point with an acceptable level of accuracy. These are significant limitations and, for this reason, several titration strategies are commonly used. 

To find the end point we monitor some property of the titration reaction that has a well-defined value at the equivalence point. Eor example, the equivalence point for a titration of ITCl with NaOlT occurs at a plT of 7.0. We can find the end point. 

Suppose that the only available indicator changes color at a pH of 6.8. Is this end point close enough to the equivalence point that the titration error may be safely ignored To answer this question we need to know how the pH changes during the titration. 

Concentration is not the only property that may be used to construct a titration curve. Other parameters, such as temperature or the absorbance of light, may be used if they show a significant change in value at the equivalence point. Many titration reactions, for example, are exothermic. As the titrant and analyte react, the temperature of the system steadily increases. Once the titration is complete, further additions of titrant do not produce as exothermic a response, and the change in temperature levels off. A typical titration curve of temperature versus volume of titrant is shown in Figure 9.3. The titration curve contains two linear segments, the intersection of which marks the 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... 

The first task in constructing the titration curve is to calculate the volume of NaOIi needed to reach the equivalence point. At the equivalence point we know from reaction 9.1 that... 

Before the equivalence point, HCl is present in excess and the pH is determined by the concentration of excess HCl. Initially the solution is 0.100 M in HCl, which, since HCl is a strong acid, means that the pH is... 

At the equivalence point the moles of HCl and the moles of NaOH are equal. Since neither the acid nor the base is in excess, the pH is determined by the dissociation of water. 

Finally, for volumes of NaOH greater than the equivalence point volume, the pH is determined by the concentration of excess OH-. For example, after adding 30.0 mb of titrant the concentration of OH- is... 

Titrating a Weak Acid with a Strong Base For this example let s consider the titration of 50.0 mL of 0.100 M acetic acid, CH3COOH, with 0.100 M NaOH. Again, we start by calculating the volume of NaOH needed to reach the equivalence point thus... 

The equilibrium constant for reaction 9.2 is large K = KJK = 1.75 X 10 ), so we can treat the reaction as one that goes to completion. Before the equivalence point, the concentration of unreacted acetic acid is... 

At the equivalence point, the moles of acetic acid initially present and the moles of NaOH added are identical. Since their reaction effectively proceeds to completion, the predominate ion in solution is CH3COO-, which is a weak base. To calculate the pH we first determine the concentration of CH3COO-. 

After the equivalence point NaOH is present in excess, and the pH is determined in the same manner as in the titration of a strong acid with a strong base. For example, after adding 60.0 mb of NaOH, the concentration of OH is... 

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. 

We begin by drawing the axes for the titration curve (Figure 9.7a). We have already shown that the volume of NaOH needed to reach the equivalence point is 50 mb, so we draw a vertical line intersecting the x-axis at this volume (Figure 9.7b). 

Question 4 in the end-of-chapter problems asks you to consider why these pH limits correspond to approximately 10% and 90% of the equivalence point volume. 

After the equivalence point the pH is controlled by the concentration of excess NaOH. Again, we have already done this calculation. Using values from Table 9.3, we plot two additional points. 

Figure 9.8b shows a titration curve for a mixture consisting of two weak acids HA and HB. Again, there are two equivalence points. In this case, however, the equivalence points do not require the same volume of titrant because the concentration of HA is greater than that for HB. Since HA is the stronger of the two weak acids, it reacts first thus, the pH before the first equivalence point is controlled by the HA/A buffer. Between the two equivalence points the pH reflects the titration of HB and is determined by the HB/B buffer. Finally, after the second equivalence point, the excess strong base titrant is responsible for the pH. 

Earlier we made an important distinction between an end point and an equivalence point. The difference between these two terms is important and deserves repeating. The equivalence point occurs when stoichiometrically equal amounts of analyte and titrant react. For example, if the analyte is a triprotic weak acid, a titration with NaOH will have three equivalence points corresponding to the addition of one, two, and three moles of OH for each mole of the weak acid. An equivalence point, therefore, is a theoretical not an experimental value. 

An end point for a titration is determined experimentally and represents the analyst s best estimate of the corresponding equivalence point. Any difference between an equivalence point and its end point is a source of determinate error. As we shall see, it is even possible that an equivalence point will not have an associated end point. 

Where Is the Equivalence Point We have already learned how to calculate the equivalence point for the titration of a strong acid with a strong base, and for the titration of a weak acid with a strong base. We also have learned to sketch a titration curve with a minimum of calculations. Can we also locate the equivalence point without performing any calculations The answer, as you may have guessed, is often yes ... 

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 principal limitation to using a titration curve to locate the equivalence point is that an inflection point must be present. Sometimes, however, an inflection point may be missing or difficult to detect, figure 9.9, for example, demonstrates the influence of the acid dissociation constant, iQ, on the titration curve for a weak acid with a strong base titrant. The inflection point is visible, even if barely so, for acid dissociation constants larger than 10 , but is missing when is 10 k... 

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