Dynamic Buffer Capacity in Acid-Base Systems.

The generalized concept of 'dynamic' buffer capacity βV 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.

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 HL 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, related to the systems where V0 mL of the solution being titrated (titrand, D) of different complexity, with concentrations [mol·L−1] of component(s) denoted by C0 or C0, 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 HL or its salt of MHL (m = 1,…,n), or HLB type as a reagent in titrant (T) are considered. This way, the D + T mixture of volume V0 + 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 [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.

Definition of Dynamic Buffer Capacity

In this work, the buffer capacity is defined as follows:wheredenotes 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 V0 mL of a solution named as titrand (D). From Eqs. 1 and 2 we have:The buffer capacity β is an intensive property, expressed in terms of molar concentrations, i.e., intensive variable. The expressions for in Eq. 3 will be formulated below.

Formulation of Dynamic Buffer Capacity

Some particular systems can be distinguished. For the sake of simplicity in notation, the charges of particular species will can be omitted when put in square brackets, expressing molar concentration .

System 1A: V mL of MOH (C, mol·L−1) is added, as reagent R, into V0 mL of KHL (C0, mol·L−1). The concentration balances are as follows: Denoting:and applying the formula for mean number of protons attached to L− [2]in the charge balance equationwe get, by turns, Differentiating Eq. 10 gives:Applying the relation:for z = α (Eq. 5) and (Eq. 6), we get [2, 12]:and then from Eq. 11 we have:Note that [H] + [OH] = (α2 + 4KW)1/2 [12] (see Eq. 5), where KW = [H][OH].

System 1B: When V mL of HB (C, mol·L−1) is added into V0 mL of KHL (C0, mol·L−1), we have [B] = CV/(V0+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 V0 mL of the mixture: (C0; m = 0,…,n; k = 1,…,P); (C0; m = 0,…,q − n; k = P+1,…,Q), HB (C0a) and MOH (C0b). Denoting −n—charge of , we have the charge balance equation:where:The presence of strong acid HB (C0a) and MOH (C0b) in the titrand D can be perceived as a kind of pre-assumed/intentional “mess” done in stoichiometric composition of the salts. Denoting: [HL(] = KH·[H]·[L(]; b = KH·[H], andwe 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 V0 mL of the mixture: (C0; m = 0,…,n; k = 1,…,P); (C0; m = 0,…,q − n; k = P+1,…,Q), HB (C0a) and MOH (C0b). 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 is added into V0 mL of the mixture: (C0; m = 0,…,n; k = 1,…,P); (C0; m = 0,…,q − n; k = P+1,…,Q), HB (C0a) and MOH (C0b). From chargeand concentration balances, Eqs. 19 and 21 andafter 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 is added into V0 mL of the mixture: (C0; m = 0,…,n; k = 1,…,P); (C0; m = 0,…,q − n; k = P+1,…,Q), HB (C0a) and MOH (C0b). Applying Eqs. 19, 27, 31 andin 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 ≥ 0; for this purpose, the absolute value (modulus) was introduced in Eq. 1. An analogous assumption was made for the static buffer capacity (β).

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 (C01) + H3PO4 (C02) + CH3COOH (C03), is titrated to the desired pH with NaOH (C) as T; in this case, C01 = C02 = C03 = 0.04 mol·L−1, and C = 0.2 mol·L−1. The D in BRB-II, consisting of H3BO3 (C01) + KH2PO4 (C02) + citric acid H3L(3) (C03) + veronal HL(4) + HCl (C0a), is titrated to the desired pH with NaOH (C) as T; in this case C01 = C02 = C03 = C04 = C0a = 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 (Eq. 40) and (Eq. 41) and: The formulas for (i = 1,…,4) and in Eqs. 39 and 43 were obtained on the basis of pK values found in [16-20].

Fig. 1

Curves of titration of BRB-I and BRB-II with NaOH. For details see the text

Fig. 2

The plots of a β vs. V and b β vs. pH relationships obtained for BRB-I and BRB-II. For details see the text

Note that

Final Comments

The mathematical formulation of the dynamic buffer capacity β 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 or type. However, in D, (NH4)2HPO4 (C0) is equivalent to a mixture of NH3 (2C0) and H3PO4 (C0), whereas KNaL (C0) is equivalent to a mixture of NaOH (C0), KOH (C0) and H2L (C0).