Abstract
The generalized concept of ‘dynamic’ buffer capacity β V is related to electrolytic systems of different complexity where acid–base equilibria are involved. The resulting formulas are presented in a uniform and consistent form. The detailed calculations are related to two Britton–Robinson buffers, taken as examples.
Similar content being viewed by others
1 Introduction
Buffer solutions are commonly applied in many branches of classical and instrumental analyses [1, 2], e.g. in capillary electrophoresis, CE [3–5], and polarography [6]. The effectiveness of a buffering action at a given pH is governed mainly by its buffer capacity (β), defined primarily by Van Slyke [7]. The β-concept refers usually to electrolytic systems where only one proton/acceptor pair exists. A more general (and elegant) formula for β was provided by Hesse and Olin [8] for the system containing a n–protic weak acid H n L together with strong acid, HB, and strong base, MOH; it was an extension of the β-concept from [9]. The formula for β found in the literature is usually referred to the ‘static’ case, based on an assumption that total concentration of the species forming a buffering system is unchanged. The dilution effects, resulting from addition of finite volume of an acid or base to such dynamic systems during titrations, was considered in the papers [2, 10], where finite changes (ΔpH) in pH, affected by addition of the strong acid or base, were closely related to the formulas for the acid–base titration curves. The ΔpH values, called ‘windows’, were considered later [11] for a mixture of monoprotic acids titrated with MOH; the dynamic version of this concept was presented first in [10].
Buffering action is involved with mixing of two (usually aqueous) solutions. The mixing can be performed according to the titrimetric mode. In the present paper, the formula for dynamic buffer capacity, \( \beta_{V} = \left| {\frac{{{\text{d}}c}}{\text{dpH}}} \right| \) related to the systems where V 0 mL of the solution being titrated (titrand, D) of different complexity, with concentrations [mol·L−1] of component(s) denoted by C 0 or C 0k , is titrated with V mL of C mol·L−1 solution of: MOH (e.g. NaOH), HB (e.g. HCl), or a weak polyprotic acid H n L or its salt of M m H n−m L (m = 1,…,n), or H n+m LB m type as a reagent in titrant (T) are considered. This way, the D + T mixture of volume V 0 + V mL, is obtained, if the assumption of additivity of the volumes is valid. It is assumed that, at any stage of the titration, D + T is a mono-phase system where only acid-base reactions occur. The formation function \( \bar{n} = \bar{n}(pH) \) [12, 13] was incorporated, as a very useful concept, into formulas for acid-base titration curves, obtained on the basis of charge and concentration balances, referred to polyprotic acids.
2 Definition of Dynamic Buffer Capacity
In this work, the buffer capacity is defined as follows:
where
denotes the current concentration of a reagent R in a D + T mixture obtained after addition of V mL of C mol·L−1 solution of the reagent R (considered as titrant, T) into V 0 mL of a solution named as titrand (D). From Eqs. 1 and 2 we have:
The buffer capacity β V is an intensive property, expressed in terms of molar concentrations, i.e., intensive variable. The expressions for \( \frac{{{\text{d}}V}}{\text{dpH}} \) in Eq. 3 will be formulated below.
3 Formulation of Dynamic Buffer Capacity
Some particular systems can be distinguished. For the sake of simplicity in notation, the charges of particular species \( X_{i}^{{z_{i} }} \) will can be omitted when put in square brackets, expressing molar concentration \( \left[ {X_{i} } \right] \).
System 1A: V mL of MOH (C, mol·L−1) is added, as reagent R, into V 0 mL of K m H n−m L (C 0, mol·L−1). The concentration balances are as follows:
Denoting:
and applying the formula for mean number of protons attached to L−n [2]
in the charge balance equation
we get, by turns,
Differentiating Eq. 10 gives:
Applying the relation:
for z = α (Eq. 5) and \( \bar{n} \) (Eq. 6), we get [2, 12]:
and then from Eq. 11 we have:
Note that [H] + [OH] = (α 2 + 4K W)1/2 [12] (see Eq. 5), where K W = [H][OH].
System 1B: When V mL of HB (C, mol·L−1) is added into V 0 mL of K m H n−m L (C 0, mol·L−1), we have [B] = CV/(V 0+V). Then C is replaced by −C in the related formulas, and we have:
As we see, Eq. 16 can be obtained by setting −C for C in the related formula. Applying it to Eq. 15, we get
System 2A: V mL of C mol·L−1 MOH is added into V 0 mL of the mixture: \( {\text{K}}_{{m_{k} }} {\text{H}}_{{n_{k} - m_{k} }} {\text{L}}_{\left( k \right)} \)(C 0k ; m k = 0,…,n k ; k = 1,…,P); \( {\text{H}}_{{n_{k} + m_{k} }} {\text{L}}_{(k)} {\text{B}}_{{m_{k} }} \)(C 0k ; m k = 0,…,q k − n k ; k = P+1,…,Q), HB (C 0a) and MOH (C 0b). Denoting −n k —charge of \( {\text{L}}_{(k)}^{{ - n_{k} }} \), we have the charge balance equation:
where:
The presence of strong acid HB (C 0a) and MOH (C 0b) in the titrand D can be perceived as a kind of pre-assumed/intentional “mess” done in stoichiometric composition of the salts. Denoting: [H i L(k)] = K H ki ·[H]i·[L(k)]; b ki = K H ki ·[H]i, and
we have:
Introducing Eqs. 19–23 into Eq. 18 we get, by turns:
where
System 2B: V mL of C mol·L−1 HB is added into V 0 mL of the mixture: \( {\text{K}}_{{m_{k} }} {\text{H}}_{{n_{k} - m_{k} }} {\text{L}}_{\left( k \right)} \) (C 0k ; m k = 0,…,n k ; k = 1,…,P); \( {\text{H}}_{{n_{k} + m_{k} }} {\text{L}}_{\left( k \right)} {\text{B}}_{{m_{k} }} \)(C 0k ; m k = 0,…,q k − n k ; k = P+1,…,Q), HB (C 0a) and MOH (C 0b). We have the balances Eqs. 18 and 19, and
Introducing Eqs. 19, 27, 28 into Eq. 18 and applying Eqs. 13, 22, 23, 26 we obtain:
System 3A: V mL of C mol·L−1 \( {\text{M}}_{m} {\text{H}}_{n - m} {\text{L}} \) is added into V 0 mL of the mixture: \( {\text{K}}_{{m_{k} }} {\text{H}}_{{n_{k} - m_{k} }} {\text{L}}_{\left( k \right)} \)(C 0k ; m k = 0,…,n k ; k = 1,…,P); \( {\text{H}}_{{n_{k} + m_{k} }} {\text{L}}_{\left( k \right)} {\text{B}}_{{m_{k} }} \)(C 0k ; m k = 0,…,q k − n k ; k = P+1,…,Q), HB (C 0a) and MOH (C 0b). From charge
and concentration balances, Eqs. 19 and 21 and
after introducing Eqs. 19, 21, 31, 32 into Eq. 30 and applying Eqs. 6, 13, 14, 22, 23 and 26, we obtain:
and then
System 3B: V mL of C mol·L−1 \( {\text{H}}_{n + m} {\text{LB}}_{m} \) is added into V 0 mL of the mixture: \( {\text{K}}_{{m_{k} }} {\text{H}}_{{n_{k} - m_{k} }} {\text{L}}_{(k)} \)(C 0k ; m k = 0,…,n k ; k = 1,…,P); \( {\text{H}}_{{n_{k} + m_{k} }} {\text{L}}_{\left( k \right)} {\text{B}}_{{m_{k} }} \)(C 0k ; m k = 0,…,q k − n k ; k = P+1,…,Q), HB (C 0a) and MOH (C 0b). Applying Eqs. 19, 27, 31 and
in Eq. 30, we obtain:
Then applying Eqs. 6, 13, 14, 23 and 24 in 37, we have:
In all cases it is assumed that β V ≥ 0; for this purpose, the absolute value (modulus) was introduced in Eq. 1. An analogous assumption was made for the static buffer capacity (β).
4 Britton–Robinson Buffers (BRB)
Two buffers proposed by Britton and Robinson [14], marked as BRB-I and BRB-II, are obtained by titration to the desired pH value over the pH range 2–12 [15]. The D (V = 10 mL) in BRB-I, consisting of H3BO3 (C 01) + H3PO4 (C 02) + CH3COOH (C 03), is titrated to the desired pH with NaOH (C) as T; in this case, C 01 = C 02 = C 03 = 0.04 mol·L−1, and C = 0.2 mol·L−1. The D in BRB-II, consisting of H3BO3 (C 01) + KH2PO4 (C 02) + citric acid H3L(3) (C 03) + veronal HL(4) + HCl (C 0a), is titrated to the desired pH with NaOH (C) as T; in this case C 01 = C 02 = C 03 = C 04 = C 0a = 0.0286 mol·L−1, and C = 0.2 mol·L−1. For BRB-I we have the equation for the titration curve:
(see Fig. 1), where:
For the BRB-II buffer we have the equation for titration curve
(see Figs. 1, 2), where \( \bar{n}_{1} \)(Eq. 40) and \( \bar{n}_{2} \)(Eq. 41) and:
The formulas for \( \bar{n}_{i} \) (i = 1,…,4) and \( \bar{n}_{3}^{ \bullet } \) in Eqs. 39 and 43 were obtained on the basis of pK i values found in [16–20].
Note that
5 Final Comments
The mathematical formulation of the dynamic buffer capacity β V concept is presented in a general and elegant form, involving all soluble species formed in the system where only acid–base reactions are involved. This approach to buffer capacity is more general than one presented in the earlier study [2] and is correct from a mathematical viewpoint, in contrast to the one presented in [21]. It is also an extension of an earlier approach, presented for less complex acid–base static [8] and dynamic [10, 12] systems. The calculations were exemplified with two complex buffers, proposed by Britton and Robinson [14].
The salts specified in particular systems considered above do not cover all possible types of the salts, e.g. (NH4)2HPO4 or potassium sodium tartrate (KNaL) are not examples of the salts of \( {\text{K}}_{{m_{k} }} {\text{H}}_{{n_{k} - m_{k} }} {\text{L}}_{(k)} \) or \( {\text{H}}_{{n_{k} + m_{k} }} {\text{L}}_{(k)} {\text{B}}_{{m_{k} }} \) type. However, in D, (NH4)2HPO4 (C 0i ) is equivalent to a mixture of NH3 (2C 0i ) and H3PO4 (C 0i ), whereas KNaL (C 0j ) is equivalent to a mixture of NaOH (C 0j ), KOH (C 0j ) and H2L (C 0j ).
References
Albert, A., Serjeant, E.P.: The Determination of Ionisation Constants. Chapman and Hall, London (1984)
Asuero, A.G., Michałowski, T.: Comprehensive formulation of titration curves referred to complex acid–base systems and its analytical implications. Crit. Rev. Anal. Chem. 41, 151–187 (2011)
He, J.-L., Li, H.-P., Li, X.-G.: Analysis of prostaglandins in SD rats by capillary zone electrophoresis with undirected UV detection. Talanta 46, 1–7 (1998)
Schneede, J., Ueland, P.M.: The formation in an aqueous matrix, properties and chromatographic behavior of 1-pyrenyldiazomethane derivatives of methylmalonic acid and other short chain dicarboxylic acids. Anal. Chem. 64, 315–319 (1992)
Lagane, B., Treilhou, M., Couderc, F.: Capillary electrophoresis: theory, teaching approach and separation of oligosaccharides using indirect UV detection. Biochem. Mol. Biol. Educ. 28, 251–255 (2000). http://www.sciencedirect.com/science/article/pii/S147081750000031X
Jordan, C.: Ionic strength and buffer capacity of wide-range buffers for polarography. Microchem. J. 25, 492–499 (1980)
Van Slyke, D.D.: On the measurement of buffer values and on the relationship of buffer value to the dissociation constant of the buffer and the concentration and reaction of the buffer solution. J. Biol. Chem. 52, 525–570 (1922). http://www.jbc.org/content/52/2/525.full.pdf+html. Accessed 24 May 2015
Hesse, R., Olin, Å.: A simple expression for the buffer index of a weak polyprotic acid. Talanta 24, 150 (1977)
Butler, J.N.: Solubility and pH Calculations. Addison-Wesley Publishing Company Inc., Reading Mass (1964)
Michałowski, T., Parczewski, A.: A new definition of buffer capacity. Chem. Anal. 23, 959–964 (1978)
Moisio, T., Heikonen, M.: A simple method for the titration of multicomponent acid–base mixtures. Fresenius’ J. Anal. Chem. 354, 271–277 (1996)
Michalowski, T.: Some remarks on acid–base titration curves. Chem. Anal. 26, 799–813 (1981)
Asuero, A.G., Jiménez-Trillo, J.L., Navas, M.J.: Mathematical treatment of absorbance versus pH graphs of polybasic acids. Talanta 33, 929–934 (1986)
Britton, H.T.K., Robinson, R.A.: Universal buffer solutions and the dissociation constant of veronal. J. Chem. Soc. 10, 1456–1462 (1931). http://www.oalib.com/references/13396293
http://en.wikipedia.org/wiki/Britton-Robinson_buffer. Accessed 24 May 2015
http://en.wikipedia.org/wiki/Boric_acid. Accessed 24 May 2015
http://en.wikipedia.org/wiki/Phosphoric_acid. Accessed 24 May 2015
http://en.wikipedia.org/wiki/Citric_acid. Accessed 24 May 2015
http://en.wikipedia.org/wiki/Acetic_acid. Accessed 24 May 2015
http://www.zirchrom.com/organic.htm. Accessed 24 May 2015
Rojas-Hernández, A., Rodríguez-Laguna, N., Ramírez-Silva, M.T., Moya-Hernández, R.: Distribution diagrams and graphical methods to determine or to use the stoichiometric coefficients of acid–base and complexation reactions. In: Innocenti, A. (ed.) Stoichiometry and Research—The Importance of Quantity in Biomedicine, InTech, Rijeka, Croatia, pp. 287–310. (2012). http://www.intechopen.com/books/stoichiometry-and-research-the-importance-of-quantity-in-biomedicine
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Michałowska-Kaczmarczyk, A.M., Michałowski, T. Dynamic Buffer Capacity in Acid–Base Systems. J Solution Chem 44, 1256–1266 (2015). https://doi.org/10.1007/s10953-015-0342-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10953-015-0342-0