titration curve for reaction of a base analyte with an acid titrant; (d) corresponding dpH/dv plot
For the titration of a weak acid (analyte) by a strong base (titrant), for example:
at the end point the salt solution is weakly basic (pH >7), as the conjugate base of a weak acid is stronger than the conjugate acid of a strong base (Fig. 2a; see also Topic C3).
Furthermore, when strong base is added before the end point, the combined equilibrium between H2CO3 and in the weak acid solution:
shifts to the right towards the production of more hydronium ions, according to Le Chatelier’s principle (see Topic C1), offsetting the rise in OH−concentration and slowing the change in pH. Therefore the steepest rise in pH occurs at the end point, where there is no acid left to offset the rise in OH−concentration. This means that a graph of dpH/dV against V can again be used to determine the end point, which is where a maximum in dpH/dV occurs.
Fig. 2. pH titration curves for the reaction of (a) a weak acid solution (as analyte) with a strong base solution (as titrant), (b) a weak base solution (as analyte) with a strong acid solution (as titrant).
For the titration of a weak base (analyte) by a strong acid (titrant), for example:
at the end point the salt solution is weakly acidic (pH<7), as the conjugate acid of a weak base is stronger than the conjugate base of a strong acid (Fig. 2b; see also Topic C3).
Again, before the end point, the fall in pH due to the addition of the strong acid is partially offset by a shift in the weak base equilibrium:
to the right according to Le Chatelier’s principle (see Topic C1), so that the change in pH with added acid is greatest at the end point.
Again, a graph of dpH/dV against V could be used to determine the endpoint, as this is where a minimum in dpH/dV should occur. An alternative method for determining these end points does not involve the measurement of pH, but instead involves the use of an acid-base indicator.
Buffers
The ability of a weak acid to partially offset the rise in pH caused by the addition of a base and of a weak base to partially offset the fall in pH due to the addition of an acid is exploited in buffer solutions. These consist of a solution containing large and equal concentrations of both a weak acid, HA, and its conjugate weak base, A−, for which (see Topic C3) pH=pKa+log10(aA−/aHA), which in terms of concentrations is (see Topic C1):
where cHA and are the concentrations of acid and conjugate base, respectively, and γHA and are their respective activity coefficients. The concentrations of acid and base are typically of the order of 0.1 M such that the activity coefficients cannot be approximated to unity, but the effect of the activity term is generally small and hence:
This is known as the Henderson-Hasselbalch equation. For a buffer solution,
and so pH≈pKa. As large concentrations of HA and A−are present, addition (or production) of a relatively small amount of base compared with the amount of HA present results in the reaction HA(aq)+OH−(aq)→A−(aq)+H2O(1). This mops up the added hydroxide ion, whilst causing little change to the large values of cHA or , and hence little change to the solution pH. Similarly, addition (or pro duction) of a relatively small amount of acid produces the reaction A−(aq)+H3O+(aq)→HA(aq)+H2O(1) which mops up the added acid, whilst maintaining CHA, CA- and hence the solution pH constant. Judicious choice of the acid/conjugate base pair therefore allows the pH of a solution to be maintained at a desired value, determined by the pKa of the acid, even if relatively small amounts of hydroxide or hydronium ions are being added to or removed from the solution (Fig. 3). This is termed a buffered solution.
Acid-base titrations 85
Fig. 3. pH response of a buffered solution to the addition of acid or base.
Most biological systems are buffered solutions, with their pH maintained at or around a value of 7, the optimum value for physiological processes, despite the presence of variable amounts of acid-base species such as dissolved carbon dioxide (carbonic acid).
Acid-base indicators
An acid-base indicator is generally a large, soluble organic molecule which in its acid form (HIn) is colored and in its conjugate base form (In−) is differently colored. The Henderson-Hasselbalch equation for this species is:
and so if the solution pH changes from a value much less than pKa where to one much greater than pKa where , the indicator changes from its acid form (HIn) to its basic form (In−) and changes color. In fact this change is generally seen to take place between pH=pKa−1, where there is a ten-fold excess of HIn over In− and pH=pKa+1, where there is a ten-fold excess of In− over HIn. The abrupt change in pH at the end point of an acid-base titration is at least as large as two pH units and so the color change of a small amount of indicator added to the acid-base titration can be used to detect this end point. This will be possible as long as the pH at the end point is approximately equal to the pKa of the indicator.
It is important that the concentration of indicator is very much smaller than the concentration of acid and of base in the titration. This ensures that very little extra titrant is required to effect the indicator acid-base color change, which ensures the accuracy of the end-point determination is unaffected. This can easily be achieved, as indicators are highly colored.
C5 SOLUBILITY
Key Notes
Partially soluble salts only partly dissolve in solution. An equilibrium between the ions and the solid salt is established and a saturated solution of the ions is produced. The equilibrium dissociation constant for this process is called the solubility product, Ksp. For partially soluble salts, the solubility of the salt, s, is simply determined by Ksp.
When a common ion (an ion which is part of the equilibrium reaction) is added to the solution, the solubility of the salt decreases. This is consistent with Le Chatelier’s principle, as the equilibrium position changes to remove the ion from solution.
When an inert ion (which takes no part in the equilibrium reaction) is added, the solubility of the salt increases. This is due to the energetically favorable electrostatic interactions between the inert ions and the salt ions, which stabilize the ions in solution, favoring the dissociation of more salt.
Fundamentals of equilibria
Partially (or sparingly) soluble salts are salts that only partly dissolve, forming a saturated solution of ions. For these systems an equilibrium exists between the solid salt and the dissolved ions:
The equilibrium constant for this reaction is often called the solubility product, Ksp, and is given by:
since MX is a pure solid (see Topic C1), where and are the activities of the M+ ion (the cation, see Topic E1) and the X−ion (the anion, see Topic E1). A good example would be solid silver chloride, which only partially dissolves into Ag+ and Cl− ions. By substituting for the activity of the ions (see Topics C1 and E2):
where and are the activity coefficients of M+and X− and is the standard concentration of 1 mol dm−3. For sparingly soluble salts, such as silver chloride, which have concentrations much less than 0.001 mol dm−3, there is negligible interaction between ions in solution and the activity coefficients can be approximated to unity. The equation then becomes:
The solubility, s, of the salt is the concentration of dissolved salt in the solution. For a salt MX, as one mole of M+ and X− ions is produced by the dissolution of one mole of salt. Therefore:
which allows the solubility of the salt in water to be determined from Ksp. For sparingly soluble salts containing ions with differing stoichiometries, a similar expression can be obtained. For example for silver sulfide, Ag2S, the solubility equilibrium is:
One mole of silver sulfide dissolves to form one mole of sulfide ions and two moles of silver ions.
The common ion effect
The common ion effect considers the effect on the solubility of the salt MX of adding either M+ or X−. An example is the addition of NaCl to a saturated AgCl solution. Le Chatelier’s principle predicts that the equilibrium:
will shift to the left to counteract the increase in chloride ion concentration and that the solubility will decrease. Quantitatively, if a concentration, c, of NaCl is added which is enough to swamp the original concentration of Cl− in solution, then . The
solubility of the salt would then be given by , as only the silver ions in solution must have come from the silver chloride salt. Hence:
and
This confirms the shift to the left of the equilibrium with the solubility, s, decreasing as c increases. This is the common ion effect.
It must be remembered that this equation only rigorously applies if c is sufficiently small (of the order of 0.001 mol dm−3or less) to ensure that there is no interaction between the ions in solution. If c becomes larger than this, the energetically favorable interactions between ions seen in the inert ion effect become increasingly important (see Topic E1) and the effects of activity cannot be neglected. In this case, the solubility equation becomes
which as γ<1 (see Topic E2) allows for the small increase in solubility due to the electrostatic stabilization of the ions. However, this effect is relatively small and is dominated by the common ion effect when adding a common ion and so even at higher concentrations, an overall decrease in solubility is seen in this case.
The inert ion effect
When inert ions, which take no part in the solubility equilibrium, are added to the solution, these tend to cause an increase in the solubility of the salt, called the inert ion effect. An example is adding an NaNO3 solution to the saturated AgCl solution. In this case
and since
Solubility 89
As the concentration of inert ions is increased to 0.001 M and above, the effects of electrostatic ion interaction become increasingly important, stabilizing the ions in solution (see Topics E1 and E2) and leading to greater ion dissociation. Thus the activity coefficients, γ, become significantly less than unity and the solubility, s, increases. Values of γ at any ion concentration can be calculated by using Debye-Hückel theory (see Topic E2), which allows the calculation of s.