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Understanding, Deriving, and Computing Buffer Capacity
Edward T. Urbansky and Michael R. Schock
Reprinted horn Journal of Chemical Education, Vol. 77, pp 1640-1644, December 2000.
Copyright ©2000 by the Division of Chemical Education of the American Chemical Society.
Reprinted by permission of the copyright owner.
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Understanding, Deriving, and Computing Buffer Capacity*
Edward T. Urbansky* and Michael R. Schock**
National Risk Management Research Laboratory, Water Supply and Water Resources Division U.S. Environmental
Protection Agency, Cincinnati, OH 45268-0001; •Urbansky.EdwardeEPA.gov; **SchockMchael@EPA.gov
We dedicate this paper to the memory of aquatic chemist
Werner Stumm and to all those who helped to lay the
foundations of aqueous equilibrium chemistry.
In the past two years more than 600 papers have been
published involving buffer capacity. It appears in varied
disciplines, including analytical, environmental, geo-, and
biochemistry, physiology, medicine, dentistry, and agriculture.
The following indicates the range of topics in the recent
literature: acid rain (1), pollution control (2), metal interactions
in fish (3), cellular metabolism (4), drug solubility (5), muscle
function (6-7), thyroid function (8), tooth decay (9-11), plant
nutrient uptake (12), soil lime application (13), potentiometry
(14), and corrosion in potable water systems (15-16). From
these examples, it is clear that buffer capacity finds application
throughout the sciences.
Despite this importance, a well-known environmental
chemistry text omits the topic (17), and a volume on corrosion
has limited coverage (18). Although mass balance and charge
balance arc covered in quantitative analysis textbooks (19-
21), buffer capacity is treated minimally. Texts that treat equi-
librium and buffer capacity more rigorously are often reserved
for upper-level or specialty chemistry courses (22-29).
Donald Van Slyke's classic authoritative work (30) was
published in 1922, several years after that of better known
Henderson and Hasselbalch. Van Slyke's specialty was blood
chemistry, where carbon dioxide equilibria dominate the acid-
base chemistry. Since that time, attempts have been made to
clarify the definition of buffer capacity (31) or provide an
approach for its computation (32). Others have expanded Van
Slyke's work, applying the concept of buffer capacity to various
areas (27,29,33-36). Fortunately, the advent of spreadsheets
has made short work of the lengthy, involved calculations re-
quired for effective use of buffer capacity (1, 37-39), and
JCE Software itself has produced a software package that
includes buffer capacity calculation (40).
Definition of pH
Understanding pH is a prerequisite for understanding
buffer capacity. The term pH is formally defined by eq 1:
(1)
where {H*} represents the activity of hydrogen ion, YH+
represents the activity coefficient of the hydrogen ion, mu+
represents the molality of the hydrogen ion in the solution
being measured, and m^0 represents the standard state molal-
ity of the hydrogen ion, normally defined as unity. For reasons
tThis paper is the work product of United States government
employees engaged in their official duties and is therefore in the
public domain and exempt from copyright.
beyond the scope of this article, die activity coefficient cannot
be determined. In practice, pH is defined operationally in
terms of aNernstian relationship (eq 2) between an electrode's
response (measured potential) in a solution of unknown pH
and its response in a solution of known pH. In both cases,
the potential is meas'ured relative to a standard reference half-
cell, for example Hg/Hg2Cl2 or Ag/AgCl electrodes:
(2)
where pHunk is the calculated pH of the unknown solution;
pHstj is the defined pH of the standard solution; £ut^ is the
measured potential of the unknown solution relative to the
reference electrode; Ead is the measured potential of the
standard solution; R is the universal gas constant (8.314 J
mol"1 K"1); and F is the Faraday constant, 96,485 C (mol e)"1.
In the United States, standard buffers of known pH are
established by the National Institute of Standards and
Technology (MIST), and NIST-traceable buffers are routinely
used in American laboratories.
It is also possible to directly measure the proton molar
concentration, represented as [H*]. This is done by the cali-
brating electrodes through titrations of strong acid and strong
base (e.g., HC1O4 and NaOH) at constant ionic strength1 in
a strong electrolyte (depending on the application, usually
KNO3, NaCl, LiClO4 or NaClO4) and while sparging with
a minimally soluble unreactive gas (e.g., Ar) to keep out CO2.
To prevent evaporative loss, the gas is presaturated with water
vapor. Proton concentrations are often used in kinetic studies
where it is desirable to accurately quantitate all species in
molar concentrations. Equilibrium constants are expressed in
molar concentrations rather dian activities, but they are valid
only for a specific ionic strength and medium, for example,
0.50 M NaClO4. Other than seawater, natural and potable
waters tend to have low ionic strength; thus, this approach is
limited in applicability even though it is always valid. Activity-
based equilibrium constants are generally measured using an
inert strong electrolyte to control ionic strength. After deter-
mination of the molar-concentration-based values, a plot of
TTagainst \i is used to extrapolate to zero ionic strength (infinite
dilution).
In common usage, a pH meter estimates {H*}. As solu-
tions become more dilute and ionic strength approaches zero,
the activity coefficients for all ions approach unity; that is,
lim
[i—»0
y=l
Unit activity coefficients are routinely assumed since y ~ 1
for u < 0.005 M. Nevertheless, this remains an assumption;
a particular situation may demand ionic strength corrections.
By assuming unit activity coefficients, we can interchange
molalities, molarities, and activities and dispense with activity
1640 Journal of Chemical Education • Vol.77 No. 12 December 2000 • JChemEd.chem.wisc.edu
-------
coefficients in subsequent equations. However, we must not
forget that activity coefficients are always implicitly present.
Definition of Buffer Capacity2
For an aqueous solution, the buffer capacity is defined
in terms of the concentration of acid or base that must be
added to influence pH. The formal definition of the buffer
capacity, 2ft, is given by eq 3:
, = dCb/d(PH) = -dCa/d(pH)
(3)
where Cb represents the concentration (normality) of added
base and Ca represents the concentration of added acid. Van
Slyke expressed the definition in this form because he •was
most interested in deducing A [COJ or A[HCO3~] foraA(pH)
that he could measure. We retain this form for historical
reasons and because it simplifies the mathematics.
Buffer capacity is always positive; every solution resists
pH change according to Le Chatelier's principle. Because pH
goes down upon the addition of acid, a minus sign is required
when considering acid. Buffer capacity is a continuous and
nonlinear function; therefore, it is a derivative. For a given
value of pH, it makes no difference whether we diink in terms
of a differential change in pH resulting from an infinitesimal
addition of acid versus an infinitesimal addition of base (other
than the sign of the differential).
The buffer capacity can be resolved into a series of terms,
with one term for each active component.
SS = 2&QH- + S&H* + 228,
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(17)
We note that the same equations apply to all diprotic acids
and bases, even zwitterions (e.g., glycine), for which the
initial charge balance equations are different. We leave these
derivations as exercises.
Triprotic Weak Acid and Its Conjugate Bases
This derivation is more complicated and some level of
detail is instructive.
mass balance-. [X]T = [X3-] + [HX2'] •*• [H2X1 + [HjX]
charg balance Cb^[H+]-[H1 +3P?-] +2[HX?1
(18)
(19)
Limiting ourselves to die contribution of the weak electrolyte
system alone (i.e., excluding 2SW), we use the equilibrium
expressions and execute similar mathematical operations:
= \X\TSID
(20)
where 5 «(1 + P,[H*] + P2[H+]2 + p3[H+]3)(2p1 + 2PJH+]) -
(3 + 2p,[H*] H- p2[H+]2)(p! + 2p2[H+] + 3p3[H+]2) and
£>=(!+ P,[H+] + P2[H+]2 + p3[H+]3)2. Multiplication of
cq 20 by (-In 10) [H*] produces the following result for the
buffer capacity contribution of this weak electrolyte system:
g&HjX = (In 10)[H+][X]TMJD (21)
where N= p, + 4p2[H+] + (P,p2 + 9P3)[H+]2 + WfoWP*
P2P3[H*]4. The function is plotted for phosphoric acid m
Figure 2.
Hydrolysis Reactions
Many metal cations, for example Al(III), are Lewis acids,
reacting with water to produce hydrogen ions. They can
precipitate as insoluble hydroxides and oxides; accordingly,
it is necessary to ensure that solubility products have not been
exceeded by the ion products. In this case, cumulative hydrolysis
constants, rather than protonation constants are used.
mat balance: [AF]T = [Al3*] + [AlOH2*] +
[A1(OH)2*] + [A1(OH)3] + [A1(OH)4-]
charge balance: [A1(OH)41 + [OH'] =
Cb + [H*] + [A1(OH)2+] H- 2[A10H2+] + 3[A131 k ^
We leave it to readers and their advanced students to show diat
where JV« ph.[H1<5+4phj[H+]5-h (9p_hj+ feJAJ [«*]*+ <
4ph,Ph^[H*]3 + (9phlPh4 " "
and£>=([H*]4 + phl[H+]3-
(22)
Computafion and Numerical Approximation
Computing the buffer capacity equations is easily accom-
plished with any commercial spreadsheet, and a text (20)
exists on spreadsheet use. The analytic solutions have been
the primary focus of this discussion; however, extremely com-
—T r
relative error
2
Q.
o% .E
o
o
CD
•-10%
Q
a:
•-20%
pH
Figure 1. Buffer capacity (left ordinate) for the acetic acid/acetate +
water system is calculated using eq 15. [HOAc]T = [HOAc] +
[OAc'] - 0.10 M and pK"a = 4.50. A relative maximum occurs at
pH = pKa = 4-50. If we approximate S3 using A(pH) = 0.14 in eq
26, an error is introduced. The relative error, (S8approx - 2&1/SS, is
also shown (right ordinate). SBW is not shown explicitly.
10°-
§.
O 10-«-
&
•3 io-5
m
io-6
^\
r^
u
\f
y
pH
Figure 2. Buffer capacity for the phosphate system is calculated
using eq 21. [PO4]T = 0.10 M, p/Ca, = 2.15, PKa2 = 7.20, pK03 =
12.15; log p, = 12.15, log P2 = 19.35, and log P3 = 21.50.
Contributions of H* and OH" are shown as S8W. Contribution of the
phosphate system is SS(H3PO4). ® = S8(H3PO4) + 2SW. Local maxima
occur where the pH reaches a p(Ca.
plicated cases may be approximated with reasonable accuracy.
Equilibrium constants generally have at least 1-5% uncer-
tainty. Accordingly, even die analytic solutions are limited
by experimental error. We recall the definition of buffer
capacity as a derivative:
lim
d(pH) ApH-M)A(pH)
(25)
As a continuous function, SS can be subjected to linear
approximation. The exactness obtained depends on how
closely A(pH) approaches d(pH). For 0.01 < A(pH) < 0.1,
the speciation changes are often (but not always) small
enough. Values chosen for A(pH) are best limited to > 0.001;
otherwise, too many computations may be required. Buffer
capacity is always positive. However, choice of Q or Cj, is
arbitrary, and Cis evaluated over the entire pH domain. Thus,
negative numbers correspond to changing from adding acid
to adding base or vice versa. For this reason, the absolute value
1642 Journal of Chemical Education • Vol.77 No. 12 December 2000 • JChemEd.chem.wisc.edu
-------
is required for the approximation:
approx
(26)
Note that A(pH) = 1.0, the value used in Chiriac and Baiea's
interpretation of the IUPAC definition (31), poorly approxi-
mates the analytic solution. Figure 1 includes a plot that
illustrates the relative error in the acetic acid buffer capacity-
curve when eq 26 is used for A(pH) = 0.14. The relative error
in the approximation reaches a maximum of about 20%, but
the approximation itself can fall below the analytic solution
by 10% as shown. The error can be reduced by choosing a
smaller value for A(pH), but at the cost of increasing the
number of computations.
Buffer Capacity Extremes
Buffer capacity functions usually reach relative maxima
at pH = p.^ for an individual deprotonation reaction. How-
ever, when pK^ < 3 or >11, the maxima begin to converge and
a single relative maximum is eventually formed. One relative
minimum usually occurs at the intersection of the contribu-
tion of the most protonated acid form and the hydrogen ion
contribution. Anodier relative minimum usually occurs at the
intersection of the contribution of the most deprotonated alka-
line form and the hydroxide ion contribution. One of these
is usually, but not always, the absolute minimum. However,
it is possible for a species to have p^ values suitably far away
from 7 that the minimum between two forms coincides with
the minimum of the water contribution. The function always
has an absolute minimum, but there are no absolute maxima.
The buffer capacity function is strictly decreasing for pH < 0,
since dS6H»/d(pH) = -2(ln 10)[H+]. It is strictly increasing
for pH > 14, since dS&OH-/d(pH) = 2(ln 10)^TW/[H+]. As a
practical limitation, buffer capacity is limited to solubilities,
for example 50% w/w NaOH (ca. 19 M OH") or 98% w/w
H2SO4 (ca. 19 M H+). However, neither of these represents
a typical aqueous solution and the activity of the water is
not unity.
The explicit analytic functions for buffer capacity are
continuous functions and are continuously differentiable;
therefore, the first derivative test may be used to locate critical
points and thus possible relative minima and maxima. For
example, a monoprotic acid and its conjugate base have 95'
given by
a' = 2(ln 10)£-a[X]T[H1([H+] - JQ/([H1 + Kj (27)
Because <3l>' changes sign about p^Ta, a relative extremum must
occur at the critical point (p^, SS); in this case, it is a relative
maximum.
Applications
Suppose the concentration of the titrant is much larger
than the concentration of the species being titrated, that is,
Qttant » Qtrated- The slope of the titration curve, d(pH)/
dV, is ca. 1/2S. Buffer capacity is a guide to titration. This
occurs, for example, if 1.0 mL of 5.0 mM CH3CO2H(aq) is
titrated with 2.0 M NaOH(aq); such a mediod is used in the
determination of stability constants. Plots of buffer capacity
can be used to find regions of maximal buffering when a
mixture of species is present and to estimate the overall
amount of acid or base that may be added •without changing
solution pH. Such a need is common in aqueous studies and
can be encountered in many disciplines.
Notes
1. Ionic strength is defined as (j. = ^Sc^a,2 for all i, where c;
represents the concentration of an ion i and zt represents its charge
(+l,-2, etc.).
2. The terms buffer capacity, buffer intensity, and buffer index
are often used interchangeably. In general, analytical chemists have
favored the first, geochemists the second, and engineers the third.
In this paper, we will use the terra buffer capacity. We do not make
the distinction made by Chiriac and Balea between buffer capacity
and buffer index (14). "We feel this distinction is the result of the
ambiguous semantics of the IUPAC definition. IUPAC did not
specify a derivative explicidy, but rather described a change in mo-
lar concentration of strong acid or base relative to 1 pH unit. We
interpret this usage is to mean a change in molarity per pH unit
and thereby to represent the derivative conventionally thought of
as buffer capacity. Certainly, Chiriac and Balea are correct in as-
serting that the normality of added base or acid to cause a pH shift
of ±1 is not equal to the derivative, because the buffer capacity func-
tion is nonlinear over such a large interval of the pH domain.
3. Although carbonic acid is a real species, it is often excluded
from expressions because it behaves identically to carbon dioxide
in terms of overall acidity. Reference 19 has an excellent discussion
of diis matter.
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