Acid-base equilibria titration curves

For the determination of the dissociation constant in the excited state, several methods have been used the Forster cycle,(109 m) the fluorescence titration curve/113 the triplet-triplet absorbance titration curve,014 but all involve the assumption that the acid-base equilibrium may be established during the lifetime of the excited state, which is by no means a common occurrence. A dynamic analysis using nanosecond or picosecond time-resolved spectroscopy is therefore often needed to obtain the correct pK a values.1(n5)... 

If the profile of the observed or the intrinsic rate constant plotted against pH resembles the profile for an acid-base titration curve, this strongly suggests that one of the reactants is involved in an acid-base equilibrium in that pH range. Such behavior is ftiirly common and is illustrated by the second-order reaction between the Co(II)-trien complex and O2 (Fig. 1.12). The limiting rate constants at the higher and low acidities correspond to the acidic and basic forms of the Co(II) reactant, probably. 

Acid solutions are often analyzed by titration with a solution of a strong base of known concentration similarly, solutions of bases are analyzed by titration with a strong acid. In either case, the measured pH is plotted as a function of the titrant volume. Calculation of a pH titration curve is a particularly good introduction to acid-base equilibrium calculations since a variety of calculations are involved. 

From what we have said above, it follows that the acid-base equilibrium in the solutions containing metal cations and oxide ions in different sections of the titration curve is described either by the dissociation constant (in unsaturated solutions) or by the values of solubility product (in saturated solutions). In Refs. [175, 330] we proposed a method based on the analysis of the scatter in the calculated equilibrium parameters corresponding to the titration process. Indeed, in the unsaturated solution section there is no oxide precipitation and the calculated value of the solubility product increases monotonously (the directed shift) whereas the calculated value of the dissociation constant fluctuates about a certain value, which is the concentration-based dissociation constant of the studied oxide. 

It is well-known that the K values can be estimated by means of the Fbrster cycle [10,17,19-21,47], the fluorescence titration curve [21,47], and the triplet-triplet absorption curve [27]. These methods involve the assumptions that proton transfer in the excited state is very fast and acid-base equilibrium may be established during the lifetime in the excited state. 

A laser study of the prototropic equilibrium of triplet benzophenone (BP) has been reported [111]. Acid-base properties in the triplet state of aromatic ketones in HjO-CHjCN (4 1) mixtures have been studied by means of nanosecond laser flash photolysis. The acidity constants p /T) in the triplet state are determined by means of the T T, absorbance titration curve, the Ware plot, and the Rayner-Wyatt plot, whose values agree well among them, showing that the acid-base equilibrium in the... 

Titrating Strong Acids and Strong Bases For our first titration curve let s consider the titration of 50.0 mb of 0.100 M HCl with 0.200 M NaOH. For the reaction of a strong base with a strong acid the only equilibrium reaction of importance is... 

Although not commonly used, thermometric titrations have one distinct advantage over methods based on the direct or indirect monitoring of plT. As discussed earlier, visual indicators and potentiometric titration curves are limited by the magnitude of the relevant equilibrium constants. For example, the titration of boric acid, ITaBOa, for which is 5.8 X 10 °, yields a poorly defined equivalence point (Figure 9.15a). The enthalpy of neutralization for boric acid with NaOlT, however, is only 23% less than that for a strong acid (-42.7 kj/mol... 

In the discussion of the relative acidity of carboxylic acids in Chapter 1, the thermodynamic acidity, expressed as the acid dissociation constant, was taken as the measure of acidity. It is straightforward to determine dissociation constants of such adds in aqueous solution by measurement of the titration curve with a pH-sensitive electrode (pH meter). Determination of the acidity of carbon acids is more difficult. Because most are very weak acids, very strong bases are required to cause deprotonation. Water and alcohols are far more acidic than most hydrocarbons and are unsuitable solvents for generation of hydrocarbon anions. Any strong base will deprotonate the solvent rather than the hydrocarbon. For synthetic purposes, aprotic solvents such as ether, tetrahydrofuran (THF), and dimethoxyethane (DME) are used, but for equilibrium measurements solvents that promote dissociation of ion pairs and ion clusters are preferred. Weakly acidic solvents such as DMSO and cyclohexylamine are used in the preparation of strongly basic carbanions. The high polarity and cation-solvating ability of DMSO facilitate dissociation... 

As the titration begins, mostly HAc is present, plus some H and Ac in amounts that can be calculated (see the Example on page 45). Addition of a solution of NaOH allows hydroxide ions to neutralize any H present. Note that reaction (2) as written is strongly favored its apparent equilibrium constant is greater than lO As H is neutralized, more HAc dissociates to H and Ac. As further NaOH is added, the pH gradually increases as Ac accumulates at the expense of diminishing HAc and the neutralization of H. At the point where half of the HAc has been neutralized, that is, where 0.5 equivalent of OH has been added, the concentrations of HAc and Ac are equal and pH = pV, for HAc. Thus, we have an experimental method for determining the pV, values of weak electrolytes. These p V, values lie at the midpoint of their respective titration curves. After all of the acid has been neutralized (that is, when one equivalent of base has been added), the pH rises exponentially. 

The key to understanding the titrations of weak acids and bases is to be familiar with the species in solution and the dominant equilibrium at each point along the titration curve. Example reinforces these qualitative features. 

In these equations, HA symbolizes a weak acid and A symbolizes the anion of the weak acid. The calculations are beyond our scope. However, we can correlate the value of the equilibrium constant for a weak acid ionization, Ka, with the position of the titration curve. The weaker the acid, the smaller the IQ and the higher the level of the initial steady increase. Figure 5.2 shows a family of curves representing several acids at a concentration of 0.10 M titrated with a strong base. The curves for HC1 and acetic acid (represented as HAc) are shown, as well as two curves for two acids even weaker than acetic acid. (The IQ s are indicated.)... 

Whatever the aim of a particular titration, the computation of the position of a chemical equilibrium for a set of initial conditions (e.g. total concentrations) and equilibrium constants, is the crucial part. The complexity ranges from simple 1 1 interactions to the analysis of solution equilibria between several components (usually Lewis acids and bases) to form any number of species (complexes). A titration is nothing but a preparation of a series of solutions with different total concentrations. This chapter covers all the requirements for the modelling of titrations of any complexity. Model-based analysis of titration curves is discussed in the next chapter. The equilibrium computations introduced here are the innermost functions required by the fitting algorithms. 

In textbooks of computational chemistry you will invariably find examples calculating the pH = - lg [H+]/(mol/l)> in weak acid - strong base or strong acid - weak base solutions. Indeed, these examples are important in the study of acids, bases and of complex formation, as well as for calculating titration curves. Following (ref. 24) we consider here the aquous solution that contains a weak tribasic acid H A and its sodium salts NaH, Na HA and Na A in known initial concentrations. The dissociation reactions and equilibrium relations are given as follows. 

To extract acid dissociation constants from an acid-base titration curve, we can construct a difference plot, or Bjerrum plot, which is a graph of the mean fraction of bound protons, H, versus pH. This mean fraction can be measured from the quantities of reagents that were mixed and the measured pH. The theoretical shape of the difference plot is an expression in terms of fractional compositions. Use Excel SOLVER to vary equilibrium constants to obtain the best fit of the theoretical curve to the measured points. This process minimizes the sum of squares [nH(measured) -nH( theoretical) 2. 

As an example of a weak acid-strong base titration, let s consider the titration of 40.0 mL of 0.100 M acetic acid with 0.100 M NaOH. Calculation of the pH at selected points along the titration curve is straightforward because we ve already met all the equilibrium problems that arise. 

Point A on the titration curve is half-way to the first equivalence point with 0.5 mol of base added per mole of phosphoric acid. If the pH at this point were between pH > 5 and pH < 9, die concentration of H3P04 and of H2P04 ion could be assumed to be equal. However, because the pH is approximately pH 3, the concentration of H3P04 at this point is less than that of the H2P04 ion by an amount equivalent to about twice the hydronium ion concentration. Thus, the correct expressions at the half-equivalence point for the equilibrium concentrations are... 

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. 

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. 

The enthalpy change of some reactions can be measured directly, but for those that do not go to completion (as is common in acid-base reactions), thermodynamic data from reactions that do go to completion can be combined using Hess s law to obtain the needed data. For example, the enthalpy and entropy of ionization of a weak acid, HA, can be found by measming (1) the enthalpy of reaction of HA with NaOH, (2) the enthalpy of reaction of a strong acid (such as HCl) with NaOH, and (3) the equilibrium constant for dissociation of the acid (usually determined from the titration curve). 

Figure 3.10. Equilibrium composition, buffer intensity, and titration curve of diprotic acid-base system, (a) Species distribution, (b) Buffer intensity, (c) Titration curve. The equivalence points, x, y, and z (a), are representative of the composition of pure solutions of H2L NaHL, and Na2L respectively, and correspond to minima in the buffer intensity. The smaller the buffer intensity, the steeper is the titration curve.

Firstly, ion exchange resins when hydrated generally dissociate to yield equivalent amounts of oppositely charged ions. Secondly, as with conventional aqueous acid or alkali solutions, resins in their acid or base forms may be neutralized to give the appropriate salt form. Finally, the degree of dissociation can be expressed in the form of an apparent equilibrium constant (or pK value) which defines the electrolyte strength of the exchanger and is usually derived from a theoretical treatment of pH titration curves. ... 

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