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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
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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] = 107
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
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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),
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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):
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[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] + 4.987x103Ct  9.120x105, 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)
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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
vpH 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)
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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 lowalkalinity 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).
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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 (103 to 104 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)
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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
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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 strongacid 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
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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: 20312035.