Lecture 20

Equivalence Points Particularly simple relationships occur when the activities of two species are equal. These are determined by equilibrium constant expressions: CO 2 E.P.: [H + ] = [HCO 3 – ] Bicarbonate E.P. [CO 3 2– ] = [H 2 CO 3 ] carbonate speciation for ΣCO 2 =

pH of the Bicarbonate Equivalence Point Our constraint: [CO 3 2– ] = [H 2 CO 3 ] Reactions: H 2 CO 3 ⇄ H + + HCO 3 – HCO 3 – ⇄ H + + CO 3 2– Equilibrium Constants Solve the first for for a H+ : Solve the second for HCO 3 – : Substitute:

Equivalence Points Bicarbonate E.P. [CO 3 2– ] = [H 2 CO 3 ] Carbonate E.P. [OH – ] = [HCO 3 – ] Two others o [H 2 CO 3 ] = [HCO 3 – ], o [HCO 3 – ] = [CO 3 2- ] Note: some are independent of other constraints, some not. carbonate speciation for ΣCO 2 =

Conservative and Non- Conservative Ions Conservative ions are those whose concentrations are not affected by changes in pH, temperature, and pressure, assuming no precipitation or dissolution. o In natural waters, the principal conservative ions are Na +, K +, Ca 2+, Mg 2+, Cl –, SO 4 2– and NO 3 –. o These ions are conservative because they are fully dissociated (or nearly so) from their conjugate acids and bases over the normal range of pH of natural waters. NO 3 – is heavily biologically utilized –but that does not count here. Non-conservative ions are those that will undergo association and dissociation reactions in this pH range. These include the proton, hydroxyl, and carbonate species as well as B(OH) 4 –, H 3 SiO 4 –, HS –, NH 4 OH, phosphate species, and many organic anions. Virtually all the non-conservative species are anions, the principle exception being H +.

Alkalinity Alkalinity is an important and fairly readily measured property of natural waters. Alkalinity is a measure of acid-neutralizing capacity of a solution and is defined as the sum of the concentration (in equivalents) of bases that are titratable with strong acid. Mathematically, we define alkalinity as the negative of TOTH when the components are the principal species of the solution at the CO 2 equivalence point which are? H + = HCO 3 – principal species is H 2 CO 3, but since we have implicitly chosen H 2 O as a component, we chose what as a component? The acidity can be defined as the negative of alkalinity, and hence equal to TOTH. In this context, the equivalent of a substance is the amount of the substance multiplied by its valence. There are two equivalent moles of SO 4 2- for each mole of SO 4 2–, but only 1 equivalent for each mole, for example, of NH 4 +. In this context, the equivalent of a substance is the amount of the substance multiplied by its valence. There are two equivalent moles of SO 4 2- for each mole of SO 4 2–, but only 1 equivalent for each mole, for example, of NH 4 +.

Carbonate Alkalinity Consider a solution containing a fixed total dissolved concentration of CaCO 3. At the CO 2 equivalence point, H 2 CO 3 is the principal carbonate species, so we choose our components as H +, H 2 O, CO 2, and Ca 2+. Species are: H 2 CO 3 = H 2 O + CO 2 HCO 3 - = H 2 O + CO 2 - H + CO 3 2- = H 2 O + CO 2 - 2H + The proton mole balance equation is then: TOTH = [H + ] - [HCO 3 - ] - 2[CO 3 2- ] - [OH – ] The alkalinity is then: Alk = -TOTH = -[H + ] + [HCO 3 - ] + 2[CO 3 2- ] + [OH – ] The above is also the carbonate alkalinity: the alkalinity due to the presence of carbonate ions. o In many solutions, carbonate alkalinity is very nearly equal total alkalinity. To be clear, we sometimes call the alkalinity as defined previously Total Alkalinity. Every species containing H + contributes positively to TOTH while every species formed by subtracting H + contributes negatively to TOTH.

Alkalinity & Non- Conservative Ions Charge balance requires: Σ cations - Σ anions = 0 Σ cons. cations -Σ cons. anions +Σ non-cons. cations -Σ non-cons. anions = 0 Σ cons. cations -Σ cons. anions = -Σ non-cons. cations +Σ non-cons. anions The right hand side is equal to the alkalinity: Alk = Σ cons. cations -Σ cons. anions = -Σ non-cons. cations +Σ non-cons. anions This equation emphasizes an important point. The difference of the sum of conservative anions and cations is a conservative property (they cannot be changed except by the addition or removal of components). Since alkalinity is equal to this difference, alkalinity is also a conservative quantity (i.e., independent of pH, pressure and temperature). Thus total alkalinity is conservative, even though concentrations of individual species are not.

The Tableau Method of Morel & Hering In determining alkalinity, we need to correctly write the TOTH equation. To do so, we need to decide on a proper system of components (as usual, we want the minimum number of components) and then decide which species are formed by adding H + and which are formed by subtracting H +. The Morel & Hering method is to produce a matrix-like table (a ‘tableau’) with components across the top and species listed vertically. Entries in the Tableau are the stoichiometric coefficient of each component needed to form each species. Consider a solution containing H 3 SiO 4 –, H 4 SiO 4, B(OH) 3, B(OH) 4 –, H 2 S, HS –, HPO 4 2-, H 2 CO 3, HCO 3 –, CO 3 2–, and, H +, and OH –. We also recall that we always chose H 2 O as a component.

H+H+ H2OH2OCO 2 H 2 PO 4 – H 4 SiO 4 B(OH) 3 H2SH2S H+H+ 1 OH – H 2 CO 3 11 HCO 3 – 11 CO 3 2– -211 HPO H 2 PO 4 – 1 H 3 SiO 4 – 1 H 4 SiO 4 1 B(OH) 4 – 11 B(OH) 3 1 H2SH2S1 HS – 1 Alk = -TOTH = -{[H + ] - [OH - ] - [HCO 3 – ] - 2[CO 3 2- ] - [HPO 4 2- ] - [H 3 SiO 4 - ] - [HS - ]}

Titration Determination of Alkalinity We have learned how to calculate alkalinity, but it can also be measured by titration. Titration is the process of progressively adding a strong acid or base to a solution until a specified pH, known as an end-point, is reached. In the case of the determination of alkalinity, this end-point is the CO 2 equivalence point, as the definition suggests. The analytical definition of alkalinity is its acid neutralizing capacity when the end-point of the titration is the CO 2 equivalence point.

Titration Example Consider a solution containing sodium carbonate (Na 2 CO 3 ). Because the carbonate ion can act as a proton acceptor, Na 2 CO 3 is a base. o We assume ideal behavior, complete dissociation, and a volume of solution large enough that the titration results in trivial dilution. The charge balance equation during the titration is: [Na + ] + [H + ] = [Cl – ] + [ HCO 3 – ] + 2[CO 3 2- ] + [OH – ] o Since the Cl – concentration is conservative, it will be equal to the total amount of HCl added. Using equilibrium constant expressions to eliminate some species, e.g., and noting mass balance requires 2[Na]=ΣCO 2, we derive the following:

Titrating Alkalinity This plot shows the how carbonate species in a 5 mM Na 2 CO 3 solution change with pH as HCl is added during titration. Notice how the HCl curve flattens at the equivalence points - pH changes rapidly with small additions of HCl. o This means we don’t have to hit the pH of the E.P. with great precision to obtain the alkalinity. o In practice, a solution indicator or pH meter would be used. The end-point is reached here at 10 mM HCl.

Calculated and Titrated Alkalinity In this example, titration yielded an alkalinity of 10 mM. Can we obtain the same result from calculation? TOTH = [H+] - [OH - ] - [HCO 3 – ] - 2[CO 3 2- ] Charge balance is (before addition of HCl): [Na+] + [H + ] = [HCO 3 – ] + 2[CO 3 2- ] + [OH – ] Solving the two equations, we find: Alk = -TOTH = [Na + ] = 10 mM

Buffer Intensity The carbonate system is a good example of a pH buffer - a system of reactions that tends to maintain constant pH. If we add acid (H + ), carbonate is converted to bicarbonate and bicarbonate is converted to carbonic acid, both reactions consuming H + and driving pH higher. We define the buffer intensity of a solution as the inverse of change in pH per amount of strong base (or acid) added: where C B and C A are the concentrations, in equivalents*, of strong base or acid respectively. *Equivalents are the number of moles of an acid (base) times the number of H + (OH - ) ions it will release upon complete dissociation. A 1M solution of HCl is a 1N solution, but a 1M solution of H 2 SO 4 is a 2N solution, as is a 1M solution of Ca(OH) 2.

Buffer Intensity The buffer capacity of the carbonate system depends strongly on pH and also on the concentration of the carbonate species and the concentration of other ions in solution. In pure water containing no other ions and only carbonate in amounts in equilibrium with the atmosphere, the buffering capacity is negligible near neutral pH. Natural solutions, however, can have substantial buffering capacity. “Hard water” is an example of water with a substantial buffering capacity due to the presence of dissolved carbonates. How adversely lakes and streams are impacted by “acid rain” depends upon their buffering intensity. β CT fixed total dissolved CO 2, β PCO2 water in equilibrium with atmospheric CO 2, β CaCO3(s) water in equilibrium with calcite, and β An-Kaol. water in equilibrium with anorthite and kaolinite.

Complexation and Speciation Ions in solution often associate with other ions, forming new species called complexes. Complex formation is important because it affects the solubility and reactivity of ions, as we will see in the following section. In some cases, complex formation is an intermediate step in the precipitation process. In other cases, ions form stable, soluble complexes that greatly enhance the solubility of the one or both of the ions.

Complexes & Ligands Complexation is usually described in terms of a central ion, generally a metal, and ion(s) or molecule(s) that bind to it, or coordinate it, referred to as ligands.

Aquo-Complexes The simplest and most common complexes are those formed between metals and water or its dissociation products. As we found earlier a solvation shell, typically consisting of 6 or so water molecules that are loosely bound to the ion through electrostatic forces, surrounds ions in aqueous solutions. Truly “free” ions do not exist. This solvation shell is referred to as an aquo-complex. Water molecules are the ligands in aquo-complexes. The existence of these complexes is implicitly assumed (and accounted for through the activity coefficient) and not usually explicitly considered.

Other Complexes Ion pairs, where ions of opposite charge associate with one another through electrostatic attraction, yet each ion retains part or all of its solvation sphere. Two possibilities: o the two solvation spheres are merely in contact o the water molecules are shared between the two solvation spheres. o Ion pairs are also called outer sphere complexes. Complexes (sensu stricto), where the two ions are in contact and a bond forms between them that is at least partly covalent in nature. These are often called inner sphere complexes.

Stability Constants In its simplest form, the reaction for the formation of an ion pair or complex between a metal cation M and an anion or ligand L may be written as: mM + + L ℓ – ⇋ M m L ℓ As with any other reaction, we may define an equilibrium constant as: For example, the equilibrium constant for the reaction: Zn(H 2 O) OH – ⇄ Zn(H 2 O) 5 OH + + H 2 O We would normally, however, express this as: Zn 2+ + OH – ⇄ ZnOH + where β is known as the stability constant. By convention, stability constants are written so that the complex appears in the numerator (i.e., as a product of the reaction).

Second Stability Constants The zinc ion might associate with a second hydroxyl: ZnOH – + OH – ⇄ Zn(OH) 2 We can write an equilibrium constant as: Here, however, the notation for the stability constant and the equilibrium constant differs. Whereas K 2 refers to the reaction above, β 2 refers to the reaction: Zn OH – ⇄ Zn(OH) 2 and is: The equilibrium constant can be related to the Gibbs free energy of the reaction. Interestingly, the free energy changes involved in complexation reactions result largely from entropy changes. Indeed, the enthalpy changes of many complexation reactions are unfavorable, and the reaction proceeds only because of large positive entropy changes. These entropy changes result from the displacement of water molecules from the solvation shell. The link between the equilibrium constant and the free energy change is important complexation reactions, because it is difficult to determine the concentrations of individual complexes analytically. Thus our knowledge of chemical speciation in natural waters derives largely from predictions based on equilibrium thermodynamics.