J Solution Chem (2015) 44:1256–1266 DOI 10.1007/s10953-015-0342-0

Dynamic Buffer Capacity in Acid–Base Systems Anna M. Michałowska-Kaczmarczyk1 Tadeusz Michałowski2

•

Received: 2 October 2014 / Accepted: 29 December 2014 / Published online: 16 June 2015  The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract The generalized concept of ‘dynamic’ buffer capacity bV 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. Keywords

Acid–base equilibria  Buffer capacity  Titration

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 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 HnL 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

& Tadeusz Michałowski [email protected] 1

Department of Oncology, The University Hospital in Cracow, Cracow, Poland

2

Faculty of Chemical Engineering and Technology, Cracow University of Technology, Warszawska 24, 31-155 Cracow, Poland

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‘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   dc  formula for dynamic buffer capacity, bV ¼ dpH related to the systems where V0 mL of the solution being titrated (titrand, D) of different complexity, with concentrations [molL-1] of component(s) denoted by C0 or C0k, is titrated with V mL of C molL-1 solution of: MOH (e.g. NaOH), HB (e.g. HCl), or a weak polyprotic acid HnL or its salt of MmHn-mL (m = 1,…,n), or Hn?mLBm 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 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.

2 Definition of Dynamic Buffer Capacity In this work, the buffer capacity is defined as follows:    dc   bV ¼  dpH

ð1Þ

where c¼C

V C  V0 C V0 þ V V0 þ V

ð2Þ

denotes the current concentration of a reagent R in a D ? T mixture obtained after addition of V mL of C molL-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:       dc dV    ¼ C  V0   dV  ð3Þ bV ¼      2 dV dpH ðV0 þ VÞ dpH The buffer capacity bV is an intensive property, expressed in terms of molar concentrations, dV i.e., intensive variable. The expressions for 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 Xizi will can be omitted when put in square brackets, expressing molar concentration ½Xi . System 1A: V mL of MOH (C, molL-1) is added, as reagent R, into V0 mL of KmHn-mL (C0, molL-1). The concentration balances are as follows:

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½M ¼ CV=ðV0 þ V Þ; ½K ¼ m  C0 V0 =ðV0 þ V Þ;

q X

½Hi L ¼

i¼0

C0 V0 V0 þ V

ð4Þ

Denoting: ½Hi L ¼ KiH  ½Hi  ½L; bi ¼ KiH  ½Hi ; fi ¼

bi q P bj j¼0

a ¼ ½H  ½OH ¼ 10pH  10pHpKW

ð5Þ

and applying the formula for mean number of protons attached to L-n [2] q P

q P

i  ½Hi L

n ¼ i¼1q P

½Hi L

¼ i¼0q P

i¼0

q P

i  KiH  ½Hi

j¼0

i  bi X q q X ¼ i¼0q ¼ i  fi ¼ i  fi P i¼0 i¼1 bj

KjH  ½H j

ð6Þ

j¼0

in the charge balance equation a þ ½M þ ½K þ

q X

ði  nÞ½Hi L ¼ 0

ð7Þ

i¼0

we get, by turns, aþ

CV C0  V0 C0  V0 þm ¼ ðn  nÞ  V0 þ V V0 þ V V0 þ V V ¼ V0 

V0 þ V ¼ V0 

ð8Þ

ðn  m  nÞ  C0  a Cþa

ð9Þ

ðn  m  nÞ  C0 þ C Cþa

¼ ððn  mÞ  C0 þ CÞ  V0 

1 n  C 0  V0  Cþa Cþa

ð10Þ

Differentiating Eq. 10 gives: dðV0 þ VÞ dV ¼ dpH dpH ¼ ððn  mÞC0 þ CÞ  V0 

dn

1 2

ðC þ aÞ



da

  dpH da dpH  ðC þ aÞ  n  C 0  V0  2 dpH ðC þ aÞ ð11Þ

Applying the relation: dz dz d½H dz ¼  ¼  ln 10  ½H]  dpH d[H] dpH d[H] for z = a (Eq. 5) and n (Eq. 6), we get [2, 12]:

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ð12Þ

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da ¼  ln 10  ð½H þ ½OHÞ dpH

ð13Þ

q X d n ¼  ln 10  ðj  iÞ2  fi fj dpH j [ i¼0

ð14Þ

and then from Eq. 11 we have: dV V0  ln 10 ¼  dpH ðC þ aÞ2

ðn  mÞ  C0 þ C  C0 

þ C0  ðC þ aÞ 

q X

!

q X

! i  fi

 ð½H þ ½OHÞ

i¼1

ð15Þ

2

ðj  iÞ  fi fj

j [ i¼0

Note that [H] ? [OH] = (a2 ? 4KW)1/2 [12] (see Eq. 5), where KW = [H][OH]. System 1B: When V mL of HB (C, molL-1) is added into V0 mL of KmHn-mL (C0, molL-1), we have [B] = CV/(V0?V). Then C is replaced by -C in the related formulas, and we have: ð n þ m  nÞ  C0 þ a Ca ð n þ m  nÞ  C0  a ðn  m  nÞ  C0  a ¼ V0  ¼ V0  C þ a ðC  aÞ

V ¼ V0 

ð16Þ

As we see, Eq. 16 can be obtained by setting -C for C in the related formula. Applying it to Eq. 15, we get ! q X dV V0  ln 10 ¼  ðn  mÞ  C0  C  C0  i  f i  ð½H þ ½OHÞ dpH ðC  aÞ2 i¼1 ! ð17Þ q X 2 ðj  iÞ  fi fj  C0  ðC  aÞ  j [ i¼0

System 2A: V mL of C molL-1 MOH is added into V0 mL of the mixture: Kmk Hnk mk LðkÞ (C0k; mk = 0,…,nk; k = 1,…,P); Hnk þmk LðkÞ Bmk (C0k; mk = 0,…,qk - nk; k k = P?1,…,Q), HB (C0a) and MOH (C0b). Denoting -nk—charge of Ln ðkÞ , we have the charge balance equation: a þ ½K þ ½M  ½B þ

qk Q X X

ðj  nk Þ½Hj Lk  ¼ 0

ð18Þ

k¼1 j¼0

where: qk X j¼0

P P mk C0k V0 C0k V0 ðk ¼ 1; . . .; P; P þ 1; . . .; QÞ; ½K ¼ k¼1 ½Hj LðkÞ  ¼ V0 þ V V0 þ V

ð19Þ

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½M ¼ Q P

½B ¼

CV þ C0b V0 V0 þ V

ð20Þ

mk C0k V0 þ C0a V0

k¼Pþ1

ð21Þ

V0 þ V

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. H i i Denoting: [HiL(k)] = KH ki[H] [L(k)]; bki = Kki[H] , and qk P

fki ¼

i  ½Hi LðkÞ 

qk P

iKkiH  ½Hi

qk P

ibki X qk bki i¼0 i¼0 i¼0 ; n  ¼ ¼ ¼ i  fki ¼ k qk qk qk qk P P P P i¼1 bkj ½Hj LðkÞ  KkjH  ½H j bkj

j¼0

j¼0

j¼0

ð22Þ

j¼0

we have: qk X d nk ¼  ln 10  ðj  iÞ2  fki fkj dpH j [ i¼0

ð23Þ

Introducing Eqs. 19–23 into Eq. 18 we get, by turns: Q p P P mk C0k V0 þ C0a V0 mk C0k V0 qk Q X X CV þ C0b V0 k¼pþ1 a þ k¼1 þ  þ ði  nk Þ½Hi LðkÞ  ¼ 0 V0 þ V V0 þ V V0 þ V k¼1 i¼0

aV0 þ aV þ

p X

mk C0k V0 þ CV þ D0 V0 

k¼1

þ

Q X

Q X

mk C0k V0 

k¼pþ1

Q X

nk  C0k V0

k¼1

nk  C0k V0 ¼ 0

k¼1

V0 þ V ¼ V0 

P X

Q X

ðnk  mk Þ  C0k þ

! ðnk þ mk Þ  C0k  D0 þ C

k¼Pþ1

k¼1



1 Cþa

Q P

nk  C0k  V0  k¼1 Cþa

ð24Þ

Q P X dV V0  ln 10 X  ðnk  mk Þ  C0k þ ðnk þ mk Þ  C0k ¼ 2 dpH ðC þ aÞ k¼Pþ1 k¼1



Q X

C0k 

k¼1

þ ðC þ aÞ 

qk X i¼1

Q X k¼1

where

123

 i  fki  D0 þ C  ð½H þ ½OHÞ C0k 

qk X j [ i¼0

! 2

ðj  iÞ  fki fkj :

ð25Þ

J Solution Chem (2015) 44:1256–1266

1261

D0 ¼ C0b  C0a

ð26Þ

System 2B: V mL of C molL-1 HB is added into V0 mL of the mixture: Kmk Hnk mk LðkÞ (C0k; mk = 0,…,nk; k = 1,…,P); Hnk þmk LðkÞ Bmk (C0k; mk = 0,…,qk - nk; k = P?1,…,Q), HB (C0a) and MOH (C0b). We have the balances Eqs. 18 and 19, and ½M ¼ Q P

C0b V0 V0 þ V

ð27Þ

mk C0k V0 þ C0a V0 þ CV

k¼Pþ1

½B ¼

ð28Þ

V0 þ V

Introducing Eqs. 19, 27, 28 into Eq. 18 and applying Eqs. 13, 22, 23, 26 we obtain: P X

dV V0  ln 10 ¼  dpH ðC  aÞ2

ðnk  mk Þ  C0k þ

Q X

C0k 

ðC  aÞ 

qk X

i  fki :Þ  ð½H þ ½OHÞ::

i¼1

k¼1 Q X

ðnk þ mk Þ  C0k

k¼Pþ1

k¼1

 D0  C 

Q X

qk X

C0k 

ð29Þ

! 2

ðj  iÞ  fki fkj

j [ i¼0

k¼1

System 3A: V mL of C molL-1 Mm Hnm L is added into V0 mL of the mixture: Kmk Hnk mk LðkÞ (C0k; mk = 0,…,nk; k = 1,…,P); Hnk þmk LðkÞ Bmk (C0k; mk = 0,…,qk - nk; k = P?1,…,Q), HB (C0a) and MOH (C0b). From charge a þ ½K þ ½M  ½B þ

qk Q X X

ðjnk Þ½Hj LðkÞ  þ

q X

ðj  nÞ½Hj L ¼ 0

ð30Þ

j¼0

k¼1 j¼0

and concentration balances, Eqs. 19 and 21 and q X

CV V0 þ V

ð31Þ

mCV þ C0b V0 V0 þ V

ð32Þ

½Hj L ¼

j¼0

½M ¼

after introducing Eqs. 19, 21, 31, 32 into Eq. 30 and applying Eqs. 6, 13, 14, 22, 23 and 26, we obtain: ! Q P X X V0 þ V ¼ V0   ðnk  mk ÞC0k  ðnk þ mk ÞC0k þ ðn  mÞ  C þ D0  k¼1

k¼Pþ1 Q P

ð33Þ nk  C0k  n  C

1 þ V0  k¼1 ðn  m  nÞ  C  a ðn  m  nÞ  C  a and then

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dV ¼  dpH

P X

V0  ln 10  2  q P n  m  i  fi  C  a



C0k 

qk X

q X

i  fki þ

i¼1

k¼1 q X

 C

i  fi  n þ m

 C  D0

!

2

ðj  iÞ  fi fj þ ½H þ ½OH 

nm

j [ i¼0



Q X

C0k 

q X

! i  fi

! Ca

i¼1 qk X

!!

q X

2

ðj  iÞ  fki fkj  C 

j [ i¼0

k¼1

ðnk þ mk Þ  C0k

!

!

i¼1

Q X k¼Pþ1

k¼1

i¼1 Q X

ðnk  mk Þ  C0k þ

2

ðj  iÞ  fi fj

j [ i¼0

ð34Þ System 3B: V mL of C molL-1 Hnþm LBm is added into V0 mL of the mixture: Kmk Hnk mk LðkÞ (C0k; mk = 0,…,nk; k = 1,…,P); Hnk þmk LðkÞ Bmk (C0k; mk = 0,…,qk - nk; k = P?1,…,Q), HB (C0a) and MOH (C0b). Applying Eqs. 19, 27, 31 and Q P

½B ¼

mk  C0k V0 þ C0a V0 þ m  CV

k¼Pþ1

ð35Þ

V0 þ V

in Eq. 30, we obtain: V0 þ V ¼ V0  

P P

ðnk  mk Þ  C0k 

Q P

ðnk þ mk Þ  C0k þ

k¼Pþ1

k¼1

Q P

nk  C0k þ D0 þ ðn þ m  nÞ  C

k¼1

ðn þ m  nÞ  C  a

V0 þ V ¼ V0 



P X k¼1

ðnk  mk Þ  C0k 

Q X

ð36Þ ! ðnk þ mk Þ  C0k þ ðn þ mÞ  C þ D0

k¼Pþ1 Q P



nk  C0k  n  C 1 þ V0  k¼1 ðn þ m  nÞ  C  a ðn þ m  nÞ  C  a ð37Þ

Then applying Eqs. 6, 13, 14, 23 and 24 in 37, we have:

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J Solution Chem (2015) 44:1256–1266

dV ¼  dpH

1263

V0  ln 10  2  q P n þ m  i  fi  C  a

P X

ðn þ mÞ  C  D0 

C0k 

qk X i¼1

k¼1



C

q X

i  fki þ C  !

q X

! i  fi

i¼1

2

ðj  iÞ  fi fj þ ½H þ ½OH

j [ i¼0



Q X



qk X

C0k 

k¼1

ðnk þ mk Þ  C0k

k¼Pþ1

k¼1

i¼1 Q X

Q X

ðnk  mk Þ  C0k þ

2

ðj  iÞ  fki fkj  C 

j [ i¼0

ðn þ m 

q X

!

q X

! 2

ðj  iÞ  fi fj

j [ i¼0

i  fi Þ  C  a

i¼1

ð38Þ In all cases it is assumed that bV 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).

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 (C01) ? H3PO4 (C02) ? CH3COOH (C03), is titrated to the desired pH with NaOH (C) as T; in this case, C01 = C02 = C03 = 0.04 molL-1, and C = 0.2 molL-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 molL-1, and C = 0.2 molL-1. For BRB-I we have the equation for the titration curve: V ¼ V0 

ð3  n1 Þ  C01 þ ð3  n2 Þ  C02 þ ð1  n3 Þ  C03  a Cþa

ð39Þ

(see Fig. 1), where: n1 ¼ ð3  1034:243pH þ 2  1025:72pH þ 1013:3pH Þ=ð1034:243pH þ 1025:72pH þ 1013:3pH þ 1Þ

ð40Þ

n2 ¼ ð3  102171pH þ 2  1019:592pH þ 1012:38pH Þ=ð1021:713pH þ 1019:592pH þ 1012:38pH þ 1Þ

ð41Þ

n3 ¼ 104:76pH =ð104:76pH þ 1Þ

ð42Þ

For the BRB-II buffer we have the equation for titration curve

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12 10

1

8 2

pH 6

1 - BRB-I 2 - BRB-II

4 2 0

0

4

2

6

8

10

V, mL Fig. 1 Curves of titration of BRB-I and BRB-II with NaOH. For details see the text

(a)

BRB-I, BRB-II

0.05 0.04

βV

0.03 1

0.01 0

1 - BRB-I 2 - BRB-II

2

0.02

1 2

0

2

4

6

8

10

V, mL

(b)

BRB-I, BRB-II

0.05 0.04

βV

0.03 0.02

1

1

0.01 0

1 - BRB-II 2 - BRB-II

2

2 2

4

6

8

10

12

pH Fig. 2 The plots of a bV vs. V and b bV vs. pH relationships obtained for BRB-I and BRB-II. For details see the text

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V ¼ V0 

1265

ð3  n1 Þ  C01 þ ð2  n2 Þ  C02 þ ð3  n3 Þ  C03 þ ð1  n4 Þ  C04 þ C0a  a Cþa ð43Þ

(see Figs. 1, 2), where n1 (Eq. 40) and n2 (Eq. 41) and: n3 ¼ ð3  1014:283pH þ 2  1011:152pH þ 106:39pH Þ=ð1014:283pH þ 1011:152pH þ 106:39pH þ 1Þ n4 ¼ 107:43pH =ð107:43pH þ 1Þ The formulas for ni (i = 1,…,4) and pKi values found in [16–20]. Note that 3 X

n3

ð45Þ

in Eqs. 39 and 43 were obtained on the basis of

ðj  iÞ2  fki fkj ¼ fk1 fk0 þ 4fk2 fk0 þ 9fk3 fk0 þ fk2 fk1 þ 4fk3 fk1 þ fk3 fk2 ;

j [ i¼0 2 X

ð44Þ

2

ðj  iÞ  fki fkj ¼ fk1 fk0 þ 4fk2 fk0 þ fk2 fk1 ;

j [ i¼0

1 X

ð46Þ 2

ðj  iÞ  fki fkj ¼ fk1 fk0 :

j [ i¼0

5 Final Comments The mathematical formulation of the dynamic buffer capacity bV 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 Kmk Hnk mk LðkÞ or Hnk þmk LðkÞ Bmk type. However, in D, (NH4)2HPO4 (C0i) is equivalent to a mixture of NH3 (2C0i) and H3PO4 (C0i), whereas KNaL (C0j) is equivalent to a mixture of NaOH (C0j), KOH (C0j) and H2L (C0j). 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.

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