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][4][5], and polarography [6]. The effectiveness of a buffer at a given pH is governed mainly by its buffer capacity (b), defined primarily by Van Slyke [7]. The b-concept refers usually to electrolytic systems where only one proton/acceptor pair exists. A more general (and elegant) formula for b 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 b-concept from [9]. The formula for b 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 (DpH) in pH, affected by addition of the strong acid or base, were closely related to the formulas for the acid-base titration curves. The DpH 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, b V ¼ dc dpH related to the systems where V 0 mL of the solution being titrated (titrand, D) of different complexity, with concentrations [molÁL -1 ] of component(s) denoted by C 0 or C 0k , is titrated with V mL of C molÁL -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 monophase system where only acid-base reactions occur. The formation function n ¼ nðpHÞ [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: denotes the current concentration of a reagent R in a D ? T mixture obtained after addition of V mL of C molÁL -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 b V is an intensive property, expressed in terms of molar concentrations, i.e., intensive variable. The expressions for dV dpH 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 X zi i will can be omitted when put in square brackets, expressing molar concentration X i ½ . System 1A: V mL of MOH (C, molÁL -1 ) is added, as reagent R, into V 0 mL of K m-H n-m L (C 0 , molÁL -1 ). The concentration balances are as follows: Denoting: and applying the formula for mean number of protons attached to L -n [2] n ¼ in the charge balance equation we get, by turns, Differentiating Eq. 10 gives: Applying the relation: for z = a (Eq. 5) and n (Eq. 6), we get [2,12]: and then from Eq. 11 we have: Note . 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 molÁL -1 MOH is added into V 0 mL of the mixture: K m k H n k Àm k L k ð Þ (C 0k ; m k = 0,…,n k ; k = 1,…,P); H n k þm k L ðkÞ B m k (C 0k ; m k = 0,…,q kn k ; k = P?1,…,Q), HB (C 0a ) and MOH (C 0b ). Denoting -n k -charge of L Àn k ðkÞ , 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: we have: Introducing Eqs. 19-23 into Eq. 18 we get, by turns: where System 2B: V mL of C molÁL -1 HB is added into V 0 mL of the mixture: K mk H nkÀmk L k ð Þ (C 0k ; m k = 0,…,n k ; k = 1,…,P); H n k þm k L k ð Þ B m k (C 0k ; m k = 0,…,q kn 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: i Á f ki :Þ Á ð½H þ ½OHÞ:: System 3A: V mL of C molÁL -1 M m H nÀm L is added into V 0 mL of the mixture: K mk H nkÀmk L k ð Þ (C 0k ; m k = 0,…,n k ; k = 1,…,P); H nkþmk L k ð Þ B mk (C 0k ; m k = 0,…,q kn k ; k = P?1,…,Q), HB (C 0a ) and MOH (C 0b ). From charge and concentration balances, Eqs. 19 and 21 and and then dV dpH System 3B: V mL of C molÁL -1 H nþm LB m is added into V 0 mL of the mixture: K m k H n k Àm k L ðkÞ (C 0k ; m k = 0,…,n k ; k = 1,…,P); H n k þm k L k ð Þ B m k (C 0k ; m k = 0,…,q kn 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 b V C 0; for this purpose, the absolute value (modulus) was introduced in Eq. 1. An analogous assumption was made for the static buffer capacity (b).
Note that

Final Comments
The mathematical formulation of the dynamic buffer capacity b 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. (NH 4 ) 2 HPO 4 or potassium sodium tartrate (KNaL) are not examples of the salts of K m k H n k Àm k L ðkÞ or H n k þm k L ðkÞ B m k type. However, in D, (NH 4 ) 2 HPO 4 (C 0i ) is equivalent to a mixture of NH 3 (2C 0i ) and H 3 PO 4 (C 0i ), whereas KNaL (C 0j ) is equivalent to a mixture of NaOH (C 0j ), KOH (C 0j ) and H 2 L (C 0j ).