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APPENDIX D Historical Correction Factors for Alkalinity and Acid Status of Surface Waters Assumptions Regarding ProtolyteS The assumptions made in the following derivations are (1) that t*he protolytes consist only of carbonate species (e.g., CO2, HCO3, CO]~) and acid and base (H+, OH-) and (2) that the system is closed; that is, that total carbonate (Ct) is constant (Ct = [CO21 + [HCO3] + [Cog ], where CO2 = CO2(aq) + H2CO3). Assumption (1) is assessed later in this discussion. Assumption (2) has to be applied in the derivations (using Ct) in order to be able to relate alkalinity, pH, and CO2 acidity. Actually the condition that must be met for assumption (2) to be valid is that pH and acidity be measured for the same Ct. This condition is not required for alkalinity because changes in Ct would not alter alkalinity. Fundamental Definitions Alkalinity (lAlk]) is defined for the ion balance condition for the Volubility of CO2; thus for all the protolyte species, HCO3, Cog ~, H+, and OH-, the ion balance condition is [H+] = [OH-] + [HCO3] + 2[COi~ , (la) and for this condition [Ark] = 0. base, n is then defined as *Authored by James R. Kramer. 471 [Alk], the "excess
412 [Ark] = [OH~ - [H+] + [HCO3] + 2 [Cat] . (lb) CO2 acidity ([Acy] ) is defined for the ion balance condition of the "neutral" salt, MHCO3. For this stoichiometry, the ion species are M+, HCO3, CO3-, H+, OHr; in addition the stoichiometry of MHCO3 results in [M+] = Ct = [CO2] + [HCO3] + [Cat-]. Substituting the stoichiometric condition into the ion balance expression ([M+] + [H+] = [HCO3] + 2[C~3 ] + [OH ]) results in [CO2] + [H+] = [CO] ] + [OH~ . CO2 acidity, ([Acy]), = 0 for the condition of Eq. (2a' Thus [Acy] is defined as the "excess acid" relative to the MHCO3 stoichiometry: (2a) [Acy] = [CO2] + [H+] - [OH ] - [COME . (2b) Examination of Eqs. (lb) and (2b) shows that [Ark] and [Acy] can be positive or negative and are precisely zero at the endpoints as defined for the specific solution conditions (e.g., Eqs. (la) and (2a)). An alternative way of viewing the stoichiometric condition for alkalinity is to imagine a titration of strong acids, defined by their anions, PA, to strong bases, defined by their cations, [C, in the presence of CO2. The ion balance expressions, rearranged, are £C - ZA = [HCO3] + 2[CO3 ] + [OH ] - [H+] (3a) or TIC - ZA = [Alk]. Thus the alkalinity endpoint is the condition in which strong acids equal strong bases with variable amounts of Cot in solution. If there were no dissolved CO2 in solution, then Eq. (3a) would reduce to PA - £C = [OHS] - [H+], (3b) (3c) where the endpoint would be [H+] = [OH-] = 10-7 eq/L, the more common endpoint for titration of a strong acid with a strong base. By examining Eq. (3a) and comparing it with Eq. (3c) one can qualitatively deduce that the amount of dissolved
473 CO2 present will change the alkalinity endpoint. The quantitative expressions for the pHS of endpoints for alkalinity and CO2 acidity are obtained by solution for H+ ion concentration in Eqs. (la) and (2a). The following expressions or approximations are used in the following derivations: [CO2] = Ct[H+]2/D, [HOCK] = Ct[H+]Kl/D, [ Con ] = CtKlK2/D, where where and D = [H+]2 + [H+]K1 + K1K2, K1 = [Ht [HCO3]/[CO2], K2 = [H+][C~ ]/[HC~], Kw = [Ho [OH ]. For pHs less than 7 (and usually less than 8.3), Eq.(la) reduces to [H+] = [HOCK] (4a) (4b) (4c) (4d) (Sa) because [OH ] and [Cog ~] are negligible relative to the other terms. Substitution of the approximation of Eq. (4b) (i.e., [Cog ~] is negligible so [HCO3] = CtKl/([H+] + K1)) into Eq. (5a), rearrangement, and solution by the quadratic equation give [H+] = {(Ki + 4CtKl)l/2 - K1}/2. For -log K1 = 6.34, Kin is maximally only 1 percent of 4CtK1 for Ct > 10- eq/L. Thus Eq. (5b) becomes [H+] = (CtKl)l/2 _ K1/2. (5b) (5c) From Eq. (5c), it is obvious that the alkalinity endpoint is not fixed but varies with total carbonate (Ct). For example, the alkalinity endpoint pH is 5.18 for Ct = 100 peq/L (typical of a dilute unbuffered water),
474 whereas the endpoint pH = 4.52 for Ct is 2000 peq/L (typical of a solution in equilibrium with CaCO3). The endpoint pH for CO2 acidity is determined by solving for H+ ion concentration in Eq. (2a). For pH > 7, and [OH ] > [H+], Bq. (2a) becomes [CO2] = [CON ] + 1OH-]. Substituting the valid approximations for the pH range 7 to 9 of [C~] - Ct[H+]/([~+ + K1) and 1C~ ] = CtK2/([H+] + K2), into Eq. (6a), rearranging, and eliminating negligible terms give [H+] = [K1(K2 + KW/ct)]l/2 (6a) (fib) Thus the endpoint pH for CO2 acidity varies with total carbonate; for example, the endpoint pH is 8.09 for Ct = 100 AM (molal), typical of poorly buffered waters, and is 8.31 for Ct = 2000 AM, typical of water in equilibrium with CaCO3. There are many calorimetric and electrometric recipes for titration to a fixed pH. If the endpoint pH of the specific technique (defined as pHx) is known, the value obtained for the incorrect endpoint can be adjusted to the value for the definitions of alkalinity and CO2 acidity by the following equations: [Ark] = [Alk]X + [HCO3]X + 2[CO] 1x and + [OH ]x ~ [H+]x (7a) [Acy] = [Acy]x + [CO2]x + [H+]x - [OH ]x - [CONE , (7b) where the subscript x refers to concentrations of the various constituents at the titration endpoint of the pH - x. Thus, the correct expression is merely the measured value plus the difference from zero of the alkalinity or of the Cog acidity concentration terms at the titration endpoint. For example, this can readily be ascertained by writing sums for alkalinity. The value of [Alk]X is equal to the change in its constituents to get to the correct endpoint (e) from initial conditions (i) plus that required to get to the titration endpoint (x); or (assuming the case [Ark] = [HCO3] - [H+] for simplicity of illustration only):
475 [Alk]X = [initial conditions] - [correct endpoint condition] + [correct endpoint condition! - [titration endpoint condition] or [Alk]X = ([HCO3] i - [H+] i) - ( [HCO3] e - [H+] e) + ( [HCO3] e - [H+] e) ( [HCO3] X [H+] x) - Remembering that [Ark] = 0 at the correct endpoint, then for this example [HCO3] e = [H+] e, and Eq. (7c) becomes [Alk]x = [Ark] - [HCO3] X + [H ]x, or, on rearranging (7c) [Ark] = [Alk]X + [HCO3] X - [H ]X- (7d) The corrections for the endpoint using Eqs. (7a) and (7b) are carried out by reformulation in terms of Ct. Thus Eqs. (7a) and (7b) become: [Ark] = lAlk] + Ct([H+]XKl - 2KlK2)/Dx + KW/[~1X - 1~+] and [Acy] = [Acy]x + (Ct[H ]X - KlK2)/DX + [H+]X - ~ /[H ]x' where Dx = D for [ H+] = [Hx+]. If [MO] = [Alk]X for methyl orange titration (PHX = 4.04 or 4.19), and if [AC] = [Acy]x for phenolphthalien titration (PHX = 8.25), then Eqs. (7e) and (7f) become [Ark] = [MO] + 4.987x10-3Ct - 9.120x10-5, PHX = 4.04 [Ark] = [MO] + 7. 030x10 3Ct - 6.457xlO 5, PHX = 4.19 and (7e) (7f) (7g) (7h) [Acy] = [AC] + 3.905x10 3Ct 1.773x10 6 (7i)
476 where [Alk], [Acy], loo], and [AC] are in equivalents per liter and Ct is in molal units; other values used to determine the coefficients are -log K1 = 6.34, -log K2 = 10.33, and -log Kw = 14. For a given set of data, [Ark] and [Acy] are determined in an iterative fashion. First the approximate values of [Ark] and [Acy] are obtained for Ct = 0; then convergence to a constant value of Ct is obtained by substituting Ct = [Ark] + [Acy] back into Eqs. (7g)-(7i). If Gran functions or modified Gran functions are used to determine [Ark] and [Acy], the correct endpoint is automatically obtained because of the nature of the function. For any region of an acid/base titration curve, [Alk1 and [Acy] are valid because they are statements of ion balance for the entire system. Thus for a titration of acid with volume (v) and concentration (Ca) relative to the correct endpoint volume (va) for sample volume (Vs), the expression for titration is obtained from ion balance considerations as (Van + v)([HCO~] + 2[C~ ~] + [OH-] - [H+1) , ~ . . _ A _ = (V - Va)Ca (A similar expression can be written for a base titration.) The Gran function employs the more acidic titration region beyond the endpoint where [H+] >> ([HCO3] + 2[C~ ~] + [OHT]). Thus Eq. (8a) becomes (Vs + v)[H ] = (Va ~ V)Ca (8a) Equation (8b) may be rearranged into F1 = aO + alV' where F1 = (Vs + v)[H+], and aO and al are linear- regression coefficients for a titration data subset of v-pH values that do not violate the condition of excess [H ]. The equivalence point volume is then obtained from Va = -a ~al, because va = v when F1 = 0; and [Ark] = vaC revs. If correctly used, the Gran function gives the true alkalinity. However, close attention must be paid to certain details because Eq. (8b) is valid only at some distance from the equivalence point; thus, small errors in the function can produce large errors in va on large extrapolation. First, the titration should be carried out in a constant ionic strength medium owing to the large changes in acid, which would alter the activity (8b)
477 coefficient of H+ ion and would also probably change the pH electrode response. Second, the pH response must be linear with respect to the logic+] for the subset of data used (i.e., pH = a + b log[H ]). Quite often lack of attention to these details and/or violation of the condition for use of Eq. (8b) leads to an underestimation of alkalinity. The modified Gran function uses data points on both sides of the equivalence point by considering Eq. (8a), which has no titration conditions. To carry out the analysis, Eq. (8b) is modified so that the carbonate terms are expressed as a function of Ct. Once again Ct is obtained iteratively from Ct = [Ark] + [Acy] (Kramer 1982). Colorimetric pH Correction The colorimetric pH correction can be carried out quantitatively only if the acid/base properties of the solution (e.g., alkalinity), the kind of colorimetric dye needed to obtain its concentration and dissociation constant, and the respective indicator and sample volumes are known. Statistical correlations of electrometric pa and colorimetric pH have limited value because they do not incorporate the above factors. For example, a water sample of high alkalinity would not be altered by the pH indicator solution; hence the colorimetric pa would read as the correct pH; whereas a low-alkalinity solution might be greatly modified by the colorimetric indicator. Furthermore, the pH effect of one dye would vary compared to the effect of another indicator. Since indicators are (were) used for specified pH intervals, different colorimetric pH readings must take into account the use of different indicators. In this report colorimetric pH, MO alkalinity, and phenolphthalein (AC) free CO2 were determined for historical lake data from New York, New Hampshire, and Wisconsin. Alkalinity and acidity (and Ct) were obtained by using Eqs. (7g) and (7h).* pH was determined *The free CO2 titration for historical Wisconsin data used Na2CO3 instead of NaOH, requiring a modification of Eqs. (7e)-(7h).
478 three different ways by using the following pairs of data: [Alk]/[Acy], [Alk]/pHC, and 1Acy]/pHc, where PHC refers to the calorimetric pH measurement. Derivation of the three equations considers the ion balance condition before and after addition of indicator dye. Before the addition of dye, we may invoke Eq. (3a): £C - ZA = [Ark] = B. On addition of indicator of volume VI, concentration CI, and acid dissociation constant KI to a sample of volume Vs. a new ion balance condition occurs because of the interaction of the dye protolyte with the solution and dilution owing to mixing of dye and solution. Indicator dyes historically were added as the "neutral" salt, MId, so that the stoichiometric condition gives [M+] = [Id-] + [HId]. The ion balance that results is (9a) [M+] + [B]' = [Id ] + [Alkc], where [B]' is the diluted difference of strong cation, [AlkC] is the alkalinity altered by dilution and reaction with the indicator, and the subscript c refers to the solution condition after addition of colori- metric pH indicator. On substitution for [M+] and [B]' and considering dilution, Eq. (9a) becomes lHId] + [Alk]' = [AlkC]. (9b) (9c) Inspection of Eq. (9c) reveals the conditions under which the calorimetric indicator will change the system. If alkalinity is much larger than the dye concentration, there will be no alteration of the solution by the addition of dye. Additionally, if [H+] or [OH ] is large compared with the dye concentration, then the HE ion will be marginally affected. Dee indicator solution concen- _ _ _ ~ ~ A _ ~ · "rations range from about 10 ~ to 10 ~ M in sample solution. Thus alkalinity concentrations of about absolute (10-3 to 10-4 eq/L) or less would be altered by the indicator solution. For pH less than about 8, [Ark] is approximated as [HCO3] - [H+], and Ct ~ [CO2] + [HCO3]. Thus, the relationship between Ct and [Ark] becomes [Ark] = CtK ~ ([H ] + K1) - [Ho , and rearrangement gives a quadratic equation for [Ho : (9d)
479 [H+]2 + [H+](K1 + [Alkl) + Kl([Alkl - Ct) = 0. (9e) Equation (9c), the ion balance statement, can be modified for dilution, concentrations, etc., to give CIvI[Hc]/([H ] + KI) + VS[Alk] = CtV' {Kl/([Hc] + K1)} - [HC](Vs + VI)' (9f) where Am] equals the calorimetrically determined H+ ion concentration. Equation (9f) is solved for Ct and substituted into Eq. (9d), which is then solved for the actual sample H+ ion concentration by using the quadratic formula [H+] = 0.5 (-([Alk] + K1) + {([Alk] + K1)2 + 4([~] + Kl)/Vs [[~](Vs + VI) + Vs([Alk] + CIVI[HC]/([HC] + K1) - [Alk]VSKl/([H+] + Kl)]})l/2 Thus Eq. (10) uses [Alk]/pHC data to determine sample H~ ion concentration. Equation (9e) can be modified by substituting [Ark] Ct - [Acy] so that pH may be determined from [Acy]/pHC data. Again Eq. (9f) is solved for Ct. which is substituted into modified Eq. (9e) to give (10) [H+] = [-B ~ (B2 + 4Kl[Acy])l/2]/2, (11) where B = Kl[Acy]/[Hc+] - (Vs + vI)([HC+] + Kl)/Vs - CIvI([Hc+] + Kl)/Vs([~] + Kc) + K1. The third means of determining actual solution pH is from [Alk]/[Acy1 data. Substitution of Ct = [Ark] + [Acy] into Eq. (9e), simplification, and solution of the quadratic equation give [H+] = {-(K1 + [Alki) + [([Alk] + K1) + 4Kl[Acy}]l/2}/2. (12) In addition to three independent estimates of pH using Eqs. (10 to 12), one may obtain an independent estimate of alkalinity from [Acy] and the actual solution pH Ct is obtained from the definition of acidity (Eq. (2b)), substitution of Eqs. (4a) and (4c), and rearrangement to give
480 Ct = D([Acy] - [H+] + KW/[H+])/([H+]2 - K1K2) (13) Then the second estimate of alkalinity is obtained by the difference of Ct (Eq. (13)) and the value for [Acy]. Other Protolyte Effects An important assumption for the above calculations and reconstructions is that only carbonate protolytes are present in significant amounts. There are no data available to permit assessment of the importance of other protolytes. Therefore some limiting conditions are considered. Imagine another (organic) protolyte of total concentration C0 and dissociation constant Ko such that 4 ~ -log Ko ~ 9. These limits are imposed because if -log Ko lies outside the pH titration bounds of 4 to 9 units it will behave as a strong-acid anion or a strong base. In this regard Oliver et al. (1983) have proposed an empirical relationship between -log Ko and solution pH for organic acid functional groups: -log Ko = 0.958 + 0.90 pa - 0.039(pH)2. If this relationship is used to estimate the other protolyte acid dissociation constants, the contributions of organic protolytes to alkalinity should be negligible in samples with pH less than about 4.6. There could be, however, protolytes other than organics that would be reactive in the titration ranges. For example, most aluminosilicates and metal oxides would be reactive. In this case the total amount of other reactive protolytes must be compared with alkalinity or acidity if we assume that the acid dissociation constant is intermediate in the titration range. The critical region then for other protolytes to have maximum effect and thus negate the carbonate assumption is when the reacting carbonates are minimal and at intermediate pH where H+ ion would have a negligible buffering effect. This condition would be near the equivalence point for alkalinity, which would be near a pH of 5. At this condition either organic protolytes or aluminum hydrolysis might be another important factor. Oliver et al. (1983) estimated the ratio of total organic protolyte (C0) to dissolved organic carbon tDOC) to be about 10 (units of AM C0/mg DOC). If all the organic protolytes were reactive at pH of 5, then about l mg/L DOC (a "clear n
481 lake) would be equal to the carbonate buffer intensity at its minimum condition. For the same condition, one would need about 550 ug/L total aluminum because one half of the aluminum would be in the form of the aqua ion and the other half in the form of A1OH (pK = 5 at pH of 5), and only one half equivalent per mole contributes to alkalinity. REFERENCES Kramer, J. R. 1982. Alkalinity and acidity. Pp. 128 ff. in Water Analysis, Vol. 1: Inorganic Species, R. A. Minear and L. H. Keith, eds. New York: Academic Press. Oliver, B. G., E. M. Thurman, and R. L. Malcolm. 1983. m e contributions of humic substances to the acidity of colored natural waters. Geochim. Cosmochim. Acta 47: 2031-2035.
