Open access peer-reviewed chapter

The Conditions Needed for a Buffer to Set the pH in a System

By Norma Rodríguez‐Laguna, Alberto Rojas‐Hernández, María T. Ramírez‐Silva, Rosario Moya‐Hernández, Rodolfo Gómez‐Balderas and Mario A. Romero‐Romo

Submitted: November 24th 2016Reviewed: April 5th 2017Published: September 27th 2017

DOI: 10.5772/intechopen.69003

Downloaded: 1365

Abstract

It is a known fact that buffer systems are widely used in industry and in diverse laboratories to maintain the pH of a system within desired limits, occasionally narrow. Hence, the aim of the present work is to study the buffer capacity and buffer efficacy in order to determine the useful conditions to impose the pH on a given system. This study is based on the electroneutrality and component balance equations for a mixture of protons polyreceptors. The added volume equations were established, V, for strong acids or bases, as well as the buffer capacity equations with dilution effect, β dil, and the buffer efficacy, ε, considering that the analyte contains a mixture of the species of the same polyacid system or of various polyacid systems. The ε index is introduced to define the performance of a buffer solution and to find out for certain, whether the buffer is adequate or not to set the pH of a system, given the proper conditions and characteristics.

Keywords

  • buffer
  • buffer capacity
  • buffer efficacy
  • polyacid systems
  • electroneutrality equation.

1. Introduction

Currently, there are studies that examine the progress of an acid‐base titration for one or various polydonor systems, extending sometimes this study to the theme of buffer capacity [116]. In the scientific literature, there are algorithms and simulators to construct acid‐base titration curves, even considering a wide range of different mixtures of different polydonor systems [1720].

The buffer solutions have a certain buffering capacity that is used to maintain constant the pH of a system, having only a small uncertainty. The buffercapacity, β, has been defined as the quantity of strong acid or strong base (in the buffer solution) that gives rise to a one unit pH change in 1 L of solution, as an intensive property of the system [15]. This involves using directly the concentration of either a strong base or an acid in the buffer solution, without considering the dilution effect, as King and Kester [2], Segurado [3], Urbansky and Schock [4], De Levie [8] do, among others. Urbansky and Schock also mentioned the use of concentration to simplify the maths. Nevertheless, the dilution effect on buffer capacity was first considered by Michałowski, as Asuero and Michałowski have established in its thorough and holistic review [6].

The buffer capacity considering the effect of dilution, βdil, is defined as the added amount of strong base or strong acid required to change in one unit the pH of an initial Vo volume of the buffer solution formed by species of only one polydonor system [7]. By their definition, βis an intensive property by considering the concentration, while βdil is an extensive property to include the amount of substance.

A buffer solution is used to impose the pH in a given system; generally speaking, therefore, the buffer is added to a working system selected to impose a given pH, thus giving rise to a mixture between both systems. It would be convenient to know the minimum concentration of the buffer components in the mixture, as well as the minimum volume that must be added in order for it to fulfill its function. From the existing works in the literature, an evaluation of the buffer performance was attempted, in general a βwas provided, although up to this moment this problem has not been dealt with quantitatively.

Figure 1 shows the scheme, in which a buffer solution is used to buffer the pH of a given system. It is observed that a mixture of both systems exists: the buffer system (BS) and the original system (OS). In general, it is necessary to define which would be the BS and the study system.

Figure 1.

Schematization of the manner in which the pH is generally set in an original system (OS) with a buffer system (BS).CBSois the initial concentration of the buffer system before mixing,VBSis the volume of BS added to OS,COSoandVOSare the initial concentration and initial volume of the original system before mixing, respectively;Vois the mixture volume formed by the solutions with volumesVBSandVOS,Cmixis the overall concentration of the mixture,CBSmixis the buffer components’ concentration in the mixture; andCOSmixis the original system’s solutes concentration in the mixture.

In many applications, it is convenient that the volume of the buffer solution that is added to the system of interest be VBS < VOS, in order not to alter much the composition of the study system.

The present work extends the study of βdil that is contained in a mixture of polydonor systems. Moreover, a new concept is introduced, buffer efficacy(ε), as an index to estimate the performance of a buffer. Finally, it is shown how these indexes allow the determination of the useful conditions (minimum concentration and volume), for a buffer to enable imposing the pH in a system of interest.

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2. Theoretical background

2.1. Description of the components, species, equilibria, and fractions in a mixture of polydonor systems

In order to evaluate the useful performance of a buffer to impose the pH in a given system, it is necessary to establish the expressions of βor of βdil that consider mixtures of various polydonor systems.

Although Ref. [8] presented equations that describe the behavior of βfor polydonor systems, the nomenclature, which has shown in Ref. [7] to study βdil, is considered here to generalize its equations for the case of buffer solutions of mixtures of different polydonor systems.

A polyprotic system [6, 7] can be represented as follows:

HnL(na)//HjL(ja)/ La/H+,where j{0,1,,a,,n}E1

H nL(na) is the polyprotic acid (weak acid in general), L a−is the base of the system, and the neutral species is H aL; H+ is the exchanged particle in the reaction, nis the number of protons of the polyprotic acid, a− is the charge of the base (expressed in elementary charge units).

The species that go from H(n1)L(na)up to HL(a‐1) are the system’s formal ampholytes.

The global formation equilibria of the species of a polydonor system are represented according to Eq. (2).

La+jH+HjL(ja)withβj=[HjL(ja)][La][H+]j.wherej{0,1,,a,,n}E2

By definition βo = 1.

When there is a mixture of cpolydonor systems in aqueous solution with (c+ 1) components, a general representation of the set of polyprotic systems is given as:

Hnk(Lk)(nkak)//Hjk(Lk)(jkak)//(Lk)ak/H+E3

where k∈ {1, 2,…, c}, jk∈{0, 1,.., ak,…, nk}.

Hnk(Lk)(nkak)is the polyprotic acid of the kth polydonor system, (Lk)akis its polybase, nkis the number of protons of the kth polyprotic acid, and akis the charge of the kth polybase. The species that go from H(nk1)(Lk)(nkak1)up to H(Lk)(1ak)are the system’s ampholytes.

A representation of the kth polydonor system’s global formation equilibria in a mixture is given as:

(Lk)ak+jkH+Hjk(Lk)(jkak)withβjk=[Hjk(Lk)(jkak)][(Lk)ak][H+]jkE4

where k∈ {1, 2,…, c}, jk∈ {0, 1,.., ak,…, nk}.

Also, in this case β0k1needs to be considered.

It can be demonstrated that the molar fraction to describe each of the cdistributions of the species of each of the polydonor systems in the mixture with respect to H+ is given by Eq. (5):

fjk=[Hjk(Lk)(jkak)][Lk]T=βjk[H+]jkjk=0nkβjk[H+]jkE5

where k∈ {1, 2,…, c}, jk∈ {0, 1,.., ak,…, nk}.

where [Lk]T is the total concentration of the kth component in the mixture. As can be observed, the molar fractions only depend on pH and on the equilibrium constants βjk.

2.2. Description of the mixture to be titrated

There are Nsolutions, each containing one Hjk(Lk)(jkak)species in a Cojkmolar concentration, assuming a volume Vojkis taken from each solution to form only one mixture with an overall volume Vo, then: N=k=1c{jk=0nk(1)}=k=1c{nk+1}and Vo=k=1c{jk=0nkVojk}. This mixture is titrated with a strong MOH base at Cbconcentration or with a strong MX acid at Caconcentration, measuring the pH.

Each species has associated countercations or counteranions (Mzjk+or Zzjk) depending on whether (jkak) they are negative or positive, which lack the acid‐base properties.

2.3. Expressions for the titration plots of polydonor systems mixtures

Although Asuero and Michałowski [6] and De Levie [9] have presented some mathematical representations of added volume as a function of pH for these systems, we have preferred to follow the same procedure and notation used to deduce the added volume equations, proposed by Rojas‐Hernández et al. [7]. Then, for the case of the mixtures of species of various polydonor systems, the following expressions can be deduced:

The added volume expression for a strong base, Vb, is

Vb=k=1c{jk=0nk{(jkak)(VojkCojk)}[jk=0nk(VojkCojk)][jk=0nk{(jkak)fjk}]}Vo([H+][OH])Cb+[H+][OH]E6

If one considers now that a strong acid is added to the mixture, then the Vaexpression becomes:

Va=k=1c{jk=0nk{(jkak)(VojkCojk)}+[jk=0nk(VojkCojk)][jk=0nk{(jkak)fjk}]}+Vo([H+][OH])Ca[H+]+[OH]E7

Eqs. (6) and (7), [OH]=Kw[H+]agree with the water self‐protolysis equilibrium.

As can be observed, the added volume equations obtained for a strong base or a strong acid bear the same mathematical form. It is relevant to note which comes from the component’s balance of each polyprotic system must be independently added, thus giving rise to the double summations appearing in Eqs. (6) and (7).

When c= 1 hence giving k= 1, the equations are the same as those shown in reference [7], to determine the volume that is added to a strong base and a strong acid (Vband Va, respectively) in a system formed by species of the same polydonor system.

Eqs. (6) and (7) are exact analytic solutions to obtain titration plots pH = f(V) (estimating the volume from the pH values). These equations also allow obtaining exact equations of dpH/dV, hence the expressions for βdil will be shown in the next section. βdil is the first index used to explore quantitatively the application conditions for a buffer.

2.4. General expressions of dpH/dVband −dpH/dVa

Eqs. (6) and (7) are functions of the pH, thus it becomes possible to obtain analytic expressions for their first derivatives (dV/dpH). With the reciprocals of the first derivatives, exact algebraic expressions of the first derivative of the titration plot are obtained. This is to say dpH/dVband −dpH/dVa, which are used to detect the volumes at the titration points when the reactions are quantitative.

Extending the expressions for dpH/dVband −dpH/dVaconsidering a mixture of the species of various polydonor systems, the expressions obtained are as follows:

dpHdVb=Cb+10pH10pHpKw2.303k=1c([jk=0nk(VojkCojk)][jk=0nk{jkfjkik=0nk[(ikjk)fik]}])+2.303(Vo+Vb)[10pH+10pHpKw]E8
dpHdVa=Ca10pH+10pHpKw2.303k=1c([jk=0nk(VojkCojk)][jk=0nk{jkfjkik=0nk[(ikjk)fik]}])+2.303(Vo+Vb)[10pH+10pHpKw]E9

where k∈ {1, 2, …, c}, ik∈ {0, 1, …, nk} and jk∈ {0, 1, …, ak, …, nk}.

Equally, if c= 1 and k= 1, Eqs. (8) and (9) are the same as those shown in Ref. [7] to determine dpH/dVband −dpH/dVafor a mixture of the species of only one polydonor system.

2.5. General expressions of βdil

In order to determine the buffer capacity considering the dilution, βdil, the derivative is applied to the quantity of strong base or strong acid added as follows:

βdilb=dVbCbdpH=CbdVbdpHorβdila=dVaCadpH=CadVadpHE10

where βdilband βdilaare units of quantity of substance. The analytic mathematical expressions are shown in Eqs. (11) and (12).

βdilb=dVbCbdpH=2.303Cb{k=1c([jk=0nk(VojkCojk)][jk=0nk{jkfjkik=0nk[(ikjk)fik]}])+(Vo+Vb)[10pH+10pHpKw]}Cb+10pH10pHpKwE11
βdila=dVaCadpH=2.303Ca{k=1c([jk=0nk(VojkCojk)][jk=0nk{jkfjkik=0nk[(ikjk)fik]}])+(Vo+Vb)[10pH+10pHpKw]}Ca10pH+10pHpKwE12

Furthermore, it is worth noting that the different plots presented along this work were constructed from spreadsheets done through Excel 2007 in Microsoft Office using the equations heretofore presented.

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3. Some case studies

3.1. Application of expressions for the titration curves fitting experimental data of the Britton‐Robinson buffer

Figure 2 shows the pH = f(Vb) and curve of dpH/dVb= f(pH) retaking experimental data from Lange’s Handbook of Chemistry[21] (markers • and ▲, respectively). The experimental curve dpH/dVbwas calculated as the finite differences quotient of the pH values and the volumes measured during the titration (ΔpH/ΔVb) using the average volumes for each interval. Also shown are the pH = f(Vb) and dpH/dVb= f(pH) curves obtained using Eqs. (6) and (7) (solid and segmented lines, respectively) [22].

Figure 2.

Titration of 100 mL of equimolar solution of Britton‐Robinson 0.04 M ([AcO′] = [PO4′] = [BO3′] = CSB = 0.04 M) with NaOH 0.2M.—represents the calculated curve of pH = f(Vb), • denotes the experimental curve of pH = f(Vb), ‐‐‐‐‐ denotes the calculated curve ofdpH/dVb= f(Vb), and ▲ denotes the experimental curve of ΔpH/ΔVb. The pKa values are as follows: pKa = 4.66 for acetic acid [23]; pKa1 = 2.1 [24], pKa2 = 6.75 [25], and pKa3 = 11.71 [26] for phosphoric acid; and pKa = 9.15 for boric acid [27]. pKw = 13.73 [28,29].

3.2. Effect of the quantity of a buffer solution on βdil

Intuitively, it is known that the performance of a buffer solution is better whenever a larger volume is taken to set the pH. The βshown in the scientific literature [114, 20] does not consider this feature, for which it is necessary to have an index that evaluates the effect of the size of the buffer solution to impose the pH. For that purpose, the definition of βdil is used to include in its mathematic expression, the term Vo, as observed in Eqs. (11) and (12). Subsequently, it is shown how βdil takes this effect into account.

Figure 3 shows a series of βdil = f(pH) plots for a 1000 and 10 mL buffer solutions containing the species H3PO4 and H2PO4 at different concentrations of the PO4’ (PO4’ = H3PO4/H2PO4 /HPO4 2−/PO4 3−/H+) system. In this case, these concentrations can be represented as CBSo=Cmix=CBSmixbecause the buffer system is a mixture, in agreement with Figure 1 . It is necessary to underline that the axis βdil is log, just as Urbansky and Schok [4] do, in order to compare βdil within an ample PO4’ concentrations range.

Figure 3.

Calculated plots ofβdil =f(pH) of systems initially containing the H3PO4 and H2PO4 at PO4’ species at overall concentrations of 10−1, 10−3, 10−5 and 10−6 M. The broken line represents the plot ofβdil(H2O) = f(pH) for water and its basic and acid particles.Cb=Ca=0.5 M. pKa1 = 2.1 [24], pKa2 = 6.75 [25], and pKa3 = 11.71 [26]. (a)Vo= 1000 mL. (b)Vo= 10 mL.

Figure 3a represents a larger buffer system than that represented in Figure 3b , because the initial volumes were 1000 and 10 mL, respectively. As can be observed in Figure 3 , βdil increases with increasing quantity of the system (Vo). Hence, βdil indicates well the expected behavior for a buffer system: the pH in a system is better imposed when the buffer amount is larger.

Figure 3 also shows the plot for water, βdil(H2O) = f(pH), titrated with strong base and strong acid. It sets the lower limit given by this solvent and its acid and basic particles (broken line), with respect to all aqueous solutions. Therefore, it is established that any solution, including that of the same solvent, has certain βdil. It can be observed that the concentration diminution of the PO4’ system provokes that βdil diminishes and that the width of the pH interval also decreases where the PO4’ system contributes more to βdil than the solvent.

There is one minimum concentration of the buffer system(CBSmin), small enough, where the PO4’ system almost does not contribute to βdil,(CBSmin ≈ 10−6 M), so that the plot of βdil of the PO4’ system can be discerned from the plot of βdil(H2O). Just as a minimum concentration is shown for the PO4’ system; whenever there is an acid‐base pair (HL/L) with pKa ≈ 7.0, the CBSmin will be the same (10−6 M). Although in other cases, when pKa < 7.0 or pKa > 7.0, it must be expected that the CBSmin be larger; this is to say, for pKa = 5 or pKa = 9, the CBSmin ≈ 10−5 M and for pKa = 3 or pKa = 11, the CBSmin ≈ 10−3 M.

3.3. Buffer efficacy (ε) of a buffer system

The previous section showed that the βdil has advantages over βin order to evaluate the buffer performance. However, the shortcomings of this situation refer to βdil, which by definition is the quantity of strong base or strong acid added to change by one unit the system’s pH, which is a fairly large change. Therefore, it is necessary to define a new index having a smaller change than βdil.

Following the idea proposed by Christian in his textbook [30], it is possible to approximate the derivative by means of a finite difference quotient. Even when the pH change is acceptable for a buffer system, it depends on the application or on the system to be considered, a ΔpH ≤ 0.1 is sufficiently small to comply with the approximation established through Eq. (13).

βdil=d(VC)dpHΔ(VC)ΔpHE13

Then

Δ(VC)βdilΔpHE14

In this work, the buffer performance will be assessed considering a ΔpH = 0.1 [30]. The buffer efficacy, ε, is defined as the quantity of strong base or strong acid that provokes a pH change of only one‐tenth in a system. The expression of εis as follows:

εΔ(VC)βdilΔpH = 0.1βdilE15

3.4. Application of ε: buffer system’s concentration threshold

A buffer system is used to set the pH, therefore, it is necessary to know its useful conditions to fulfill its function. Then, for the sake of a deeper understanding it is relevant to establish first a limit to determine the moment in which the buffer system’s concentration sets pH conditions over those of the water and of its acid and basic particles, just as those of the system of interest.

3.4.1. Imposing the pH of the buffer system over the water and its acid and basic particles

Because the system’s pH needs to be imposed, the efficacy of the system’s buffer, the buffer (εBS)should be larger or at least equal than the efficacy of the buffer amplified ten times that of the water, ε10(H2O)= 10ε(H2O), not just the buffer efficacy of the water, ε(H2O)s. In the present work, a factor of 10 is considered sufficiently large to assess a buffer’s performance.

Figures 4a and b show the curves of ε(H2O)for Vo= 1000 mL and Vo= 10 mL of water, respectively, titrated with a strong acid and a strong base at 1 M concentration (marker…). Also, it is shown the plot of ε10(H2O)(marker…), which will be considered as the limit where the buffer concentration is useful to set the pH. It is also observed in Figures 4a and b that both ε(H2O)and ε10(H2O)depend, as expected, on the quantity of the system.

Figure 4.

Calculated plots ofε(H2O)andε10(H2O)forVo= 1000 mL andVo= 10 mL of water and its acid and basic particles (marker… and marker…, respectively). TheεBScurves of NH4 +/NH3 buffer solutions at different concentrations, [NH3']Tot =CBSo=Cmix=CBSmix: —10−1 M,———10−3 M, —10−5 M.Cb= Ca = 1 M. pKa = 9.25 [31]. (a) and (c) 1000 mL. (b) and (d) 10 mL.

Figure 4c and d shows, apart from the ε(H2O)and ε10(H2O)plots, those of εBSof the NH4 +/NH3 buffer solutions at different concentrations for systems with VBS = Vo= 1000 mL and VBS = Vo= 10 mL, respectively, as those shown in Figure 1 . In order to establish the limit in which a buffer works to set the pH, it is necessary to compare the εBScurve with that of the ε10(H2O), so that compliance with εBSε10(H2O)can be verified.

In this case, the lowest NH4 +/NH3 buffer concentration falls to a point that almost equals the ε10(H2O)plot. Therefore, this concentration is termed as threshold buffer concentration(TCBS) with a value of TCBS≈ 10−4 M. It can also be seen from Figure 4c and d that TCBSdoes not depend on the size of the system (Vo).

Then, whenever there is a buffer formed by an acid‐base pair HL/L with pKa ≈ 7.0 it must be expected a greater TCBS, of approximately 10−5 M; when pKa = 5 or pKa = 9, the TCBSwill also be approximately 10−4 M; and for pKa = 3.0 or pKa = 11.0, the TCBS≈ 10−2 M.

Furthermore, in Figure 4c and d it can also be noted that the useful interval for the BS to set the pH, when the CBSo=Cmix=CBSmix= 10−1 M, is pKa – 1 < pH < pKa + 1 [32]. Whereas for CBSo=Cmix=CBSmix= 10−3 M, the interval to set the pH is smaller because the εBSbecomes closer to ε10(H2O).

When the εBS< ε10(H2O)strong acid or strong base must be used also at adequate concentration to set the pH. The strong base and strong acid in these extremes is using because the acid and basic particles of the solvent contribute more to εthan the buffer system components.

Finally, from the analysis of the plots in Figure 4 , it can be determined that the pH limits of the buffer performance and the buffer system threshold concentration do not depend of the system’s size.

3.4.2. Setting the buffer’s system pH over that of the original system

It is necessary to set new limits (of pH and buffer concentration) whenever there is a mixture of the system of interest and the buffer system, because it is not sufficient to consider only the water effect.

The pH of a given system (OS) is set upon adding a buffer system (BS), hence, the buffer component concentration in the mixture (CBSmix) must be greater than the concentration of the solutes of the original system (COSmix), because both systems have their own ε, though how large is it? ( Figure 1 ).

An example of the use of ε, to evaluate setting the pH to a 9.0 value at 100 mL (VOS) of an acetylacetone solution (acac') (OS), is given next, at 10−3 M concentration (COSo1), at different NH4 +/NH3 buffer concentrations in the mixture (CBSmix).

Figure 5a shows the εof the acetylacetone solution (OS), stated in the previous paragraph (εOS, marker ×××××), the buffer efficacy amplified 10 times that of the original system(ε10OS, marker ‐‐‐‐), also presenting the plot of ε(H2O)(marker…).

Figure 5.

Curves ofε= f(pH) for 100 mL solutions related toFigure 1. The line marked with ….in all cases represents the buffer efficacy of water,ε(H2O), and its acid and basic particles. (a) CurvesεOSandε10OSbelong to an acac 10−3 M ([acac']Tot =COSo=Cmix=COSmix) solution in the absence of the buffer system (markers ××××× and‐‐‐‐, respectively). (b) CurvesεBSbelong to the NH4 +/NH3 buffer at different concentrations, [NH3']Tot =CBSo=Cmix=CBSmix: —10−2 M,———10−3 M, —10−4 M. (c) Curvesεmix,εOS, andε10OSbelong to the solutions containing acac 10−3 M plus NH4 +/NH3 10−2 M buffer, 10−3 M, and 10−4 M in the mixture. 1:εOS, 2:COSmix=10‐3 M +CBSmix= 10−4 M, 3:COSmix= 10−3 M +CBSmix= 10−3 M, 4:ε10OS, and 5:COSmix=10‐3 M +CBSmix= 10−2 M. pKa = 9.0 for the system Hacac/acac and pKa = 9.25 for the NH4 +/NH3 system [31].Cb=Ca= 0.2 M.

Figure 5b shows, apart from the ε(H2O)curve, those corresponding to εBSfor 100 mL of the NH4 +/NH3 system solution at different concentrations. The εBSmagnitude depends on buffer’s concentration and decreases until it reaches a TCBS, which is given when the εBScurve almost becomes equal to that of ε10(H2O). In this case, TCBS≈ 10−4 M.

In order to impose the pH of the original system, the buffer efficacy of the mixture (εmix) should be greater or, at least equal to ε10OS. Now, if the buffer is added to the original system to set the pH, the εmixhas contributions of the original system and to the buffer system; thus, the buffer does not always set the pH in the system as shown in Figure 5c . The curve 1 that represents a εOSis practically identical to the curve 2 that corresponds to a mixture of the original system, with the buffer solution with concentration 10 times smaller than the solutes in the original system: in these cases, the pH of the system depends only on the original system, because εmixεOS. The curve 3 shows that the buffer with 10−3 M concentration does already contribute to εmixapart from the original system, but has not set the pH yet because εOS< εmix< ε10OS.

Finally, Figure 5c depicts the curve 4 as corresponding to ε10OS, whereas the curve 5 represents a εmixcorresponding to the buffer threshold concentration for the buffer system component of the mixture, TCBSmix(whenever it is required to set the pH at a value of 9.0), for which εmixεBS= ε10OS, and therefore, there is an adequate buffer performance to set the pH of the system within the 8.25 < pH < 10.0 interval. It must be noted that if it is required to set pH > 10.0 values, a strong base must be used, apart from the buffer system. To the extent that εmix>>ε10OSthe buffer performance to set the system pH becomes better. Approximately TCBSmixmust be 10 times larger than COSmix. Therefore, it is established that

CBSmixTCBSmix10COSmixE16

Eq. (16) becomes specific whenever the original system is set to a pH = 9.0 when this system bears one acid‐base pair with pKa = 9.0 (Hacac/acac) [31]. This example is the most difficult case because this OS acid‐base pair competes almost equally with the BS acid‐base pair (NH4 +/NH3, pKa = 9.25) to set the system’s pH.

If the pKa of some species in the original system moves away from the pH value that is desired to impose with the buffer system, the factor of 10 in Eq. (16) becomes smaller.

It must not be forgotten that the buffer system should have a conjugated acid‐base pair with a pKa value close to the pH that is desired to impose. As can be observed from Figure 5c , in this case it is difficult that the buffer system imposes the pH to a 9.0 value because both systems (BS and OS) have pKa values similar to that pH value. It is worth clarifying that the NH4 +/NH3 buffer imposes easier the pH to a 9.0 value to the extent that the pKa of the acid‐base pair of the original system drifts apart from pH = 9.0; then the TCBSmixdiminishes till it reaches a limit value given by ε10(H2O).

Consider now the case that a pH 9.0 shall be imposed to the 100 mL (VOS) of the acetylacetone solution (acac') (OS), with a 10−3 M (COSo) concentration, using now the Britton‐Robinson [12, 21] buffer at different concentrations. For this example, the εOS, ε10OS, and ε(H2O)are the same as those presented in Figure 5a .

Figure 6a shows, apart from the curve of ε(H2O), the curves of εBSfor 100 mL solution of the Britton‐Robinson buffer at different concentrations. In this case, the CBSo=Cmix=CBSmixdiminishes till reaching a TCBS, having a value of TCBS≈ 10−5 M because at this point the εBSalmost equals the ε10(H2O)curve.

Figure 6.

The curvesε= f(pH) are for the 100 mL of the solutions related toFigure 1. The broken line in all cases represents the buffer efficacy of the water,ε(H2O), and its acid and basic particles. (a) CurvesεBSfor equimolar Britton‐Robinson buffer solutions at different concentrations, [AcO']Tot = [PO4']Tot = [BO3']Tot =CBSo=CBSmix: ―10−1 M,———10−3 M, ―10−5 M. (b) Curvesεmix,εOS, andε10OSof solutions that contain 10−3 M acac plus 10−2 M, 10−3 M, and 10−5 M Britton‐Robinson’s buffer in the mixture. 1:εOS, 2: [acac'] =COSmix= 10−3 M +CBSmix= 10−5 M, 3: [acac'] =COSmix= 10−3 M +CBSmix=10−3 M, 4:ε10OS, and 5: [acac'] =COSmix= 10−3 M +CBSmix= 10−2 M. The pKa values used in the model are as follows: pKa = 9.0 for acac’ [31]; pKa = 4.66 for acetic acid [23]; pKa1 = 2.1 [24], pKa2 = 6.75 [25], and pKa3 = 11.71 [26] for phosphoric acid; and pKa = 9.15 for boric acid [27]. pKw = 13.73 [28,29].Cb=Ca= 0.2 M.

As stated, the εmixshould be larger or at least equal to ε10OS. Once again, the buffer efficacy of the mixture (εmix) contributes both the acac’ system and the Britton‐Robinson buffer; thus the buffer does not always set the pH in the system as shown in Figure 6b . Curve 1 represents the εOS, that is practically equal to curve 2 corresponding to a mixture of the original system with the buffer solution with a concentration 100 times smaller than the solutes in the original system: in these cases, the system’s pH depends only on the original system, because εmixεOS. Curve 3 shows that in the 4.5 < pH < 7.4 interval, the buffer with 10−3 M concentration contributes more to the εmixthan the original system, consequently, the buffer has the system’s capacity to set the pH in this interval but not at pH = 9.0 as required in this example. Curve 3 also has another region where both systems contribute almost the same as εmix(7.4 < pH < 10.7) and because εmix< ε10OS, the buffer is not capable of fulfilling its function in this interval.

Figure 6b also shows curve 4 that corresponds to ε10OSand curve 5 that represents a εmixcorresponding to a threshold concentration for the buffer system’s components in the mixture, TCBSmix(when the pH to be imposed is 9.0), for which εmixεBS= ε10OS. At this same concentration, the buffer system’s components show a good performance to impose the pH, not only at 9.0 but within the 3.8 < pH < 10.0 interval; which is large because it corresponds to a wide spectrum buffer. Approximately TCBSmixmust be 10 times larger than the COSmixas established by Eq. (16), because the acid‐base pair of the Britton‐Robinson buffer is the H3BO3/H2BO3 with a pKa = 9.15 value. It must be underlined that if it is intended to impose the pH to other value (pH < 8.1), the TCBSmixis smaller.

3.5. Application of ε: threshold volume of the buffer system

The concentration of the buffer components in the mixture (CBSmix) must be larger than the solute concentration in the system considered (COSmix) to set the pH of the system. However, it is also necessary to know the buffer minimum volume that must be added to the original system with the aim of fulfilling its function ( Figure 1 ). Next, an example is given to determine this minimum volume on the acetylacetone case using Britton‐Robinson buffer to impose the 9.0 pH in the system.

If there are 100 mL of an acac (OS1) solution at 10−3 M concentration (COSo1), and two buffer (BS1 y BS2) solutions with components 10−1 M (CBSo1) and 1 M (COSo2) concentration: which is the minimum volume that should be added to each buffer solution to set a pH = 9.0 in the system?

Figure 7a shows that the ε10OScan be attained when 10 mL of the buffer solution with 10−1 M (SB1) component concentration is added to the original system. Then, this minimum volume is the threshold volume(TVBS1) for this specific buffer solution. If now, the buffer solution of 1 M (SB2) component concentration is added to the system of interest, the threshold volume is different, as shown in Figure 7b , under these conditions the TVBS2= 1 mL.

Figure 7.

The curves forε= f(pH) for 100 mL of different solutions. The broken line in all cases represents the buffer efficacy of water,ε(H2O), and its acid and basic particles. The curves ofεOSandε10OSof acac solutions in the absence of the buffer system (markers ××××× and‐‐‐‐, respectively). (a) CurvesεOSandε10OScontaining 100 mL of the acac solution with a 10−3 M solute concentration, curveεmixcontaining the same acac solution asεOSand different added volumes of the buffer solution with a concentrationCBS= 0.1 M. Volumes added:10 mL,———1 mL and, 0.1 mL. (b) The curvesεOSandε10OSare the same that in (a), curvesεmixcontaining the same acac solution and different added volumes of the buffer solution withCBS= 1 M. Added volumes:1 mL,0.5 mL and0.1 mL. (c) CurvesεOSandε10OScontaining 100 mL of a acac solution with 10−2 M solute concentration, curvesεmixcontaining 100 mL of the acac solution with a 10−2 M solute concentration and different volumes added to the buffer solution withCBS= 1 M. Added volumes:10 mL,5 mL and,1 mL.

The TVBSis related to TCBSmixin the mixtures, with total volumes of 110 mL for the first case and 101 mL for the second case, because in both cases TCBSmixare 10 times greater than the original system’s solute concentrations in the mixture (COSmix).

If now, a larger concentration of acac is used in the original system, for example 10−2 M (COSo2), it is necessary to use the more concentrated buffer to impose the pH, for example SB2, with 1 M (CBSo2) component concentration.

It is clear that if the original system solutes’ concentrations in the mixture (COSmix) grows, the TCBSmixalso grows, consequently TVBSdoes it too.

Observing the equations shown in Figure 1 , it is possible to demonstrate that the threshold volume, TVBS, can be determined from the initial working conditions using Eq. (15)

CBSmixTCBSmix10COSmixE19

which can be rewritten as

CBSoVBSVoCBSoTVBSVo=10COSoVOSVoE17

From Eq. (17), the following is obtained

VBSTVBS=10(COSoVOSCBSo)E18

Figure 7c shows that the TVBSis equal to 10 mL, even if it is a volume added to the buffer solution with 1 M component concentration, because the original system is composed of 100 mL with 10−2 M solute concentration (COSo2). It can be proved that the TCBSfulfills the condition of being 10 times larger than the original system’s solute concentration.

Just like in the case of the threshold concentration (TCBSmix) of Eq. (16), Eq. (18) also corresponds to the most difficult case. The factor of 10 can be smaller to the extent that the pKa values of the original system move away from the pH that is to be set.

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4. Conclusions

A model has been proposed to study the buffer capacity with dilution effect, βdil, of mixtures of various polydonor systems in aqueous solutions. From the model, exact algebraic expressions were obtained that describe the buffer capacity with dilution.

Through the study of βdil of solutions containing one or more polydonor systems with different conditions and characteristics, it is concluded that βdil decreases when the total concentration of the polydonor systems (or mixtures of polydonor systems) decreases, and conversely.

When a polydonor system attains a very small concentration, this does not contributes practically to the buffer capacity, only the solvent’s acid and basic particles determine this property. Then, it is stated that any solution, even the pure solvent, has a buffer capacity.

It is shown that the βdil depends on the size of the system, which is information that is not considered in the βknown and used in the common scientific literature.

A new index has been introduced, ε, to measure the buffer efficacy of a buffer solution such that the quantity added of strong base or strong acid causes a change of only one pH tenth instead of one unit as βdil.

From the construction of the different curves of ε= f(pH), it is possible to identify a buffer threshold concentration in the mixture (TCBSmix), which allows knowing the minimum buffer concentration to set the desired pH of the system of interest. This concentration TCBSmixmust be, at least 10 times greater than that of the original system’s solutes in the mixture (TCBSmix=COSmix)in the most difficult case.

Similarly, the different curves of ε= f(pH) also allow determining the minimum volume (buffer threshold volume, TVBS) that must be added to the system of interest to set its pH (TVBS=10(COSoVOSCBSo))in the most difficult case.

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Acknowledgments

NR‐L wants to acknowledge DGAPA‐UNAM for a postdoctoral fellowship. The authors also want to acknowledge for partial financial support from PRODEP‐SEP through RedNIQAE and CONACyT through 237997 project.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Norma Rodríguez‐Laguna, Alberto Rojas‐Hernández, María T. Ramírez‐Silva, Rosario Moya‐Hernández, Rodolfo Gómez‐Balderas and Mario A. Romero‐Romo (September 27th 2017). The Conditions Needed for a Buffer to Set the pH in a System, Advances in Titration Techniques, Vu Dang Hoang, IntechOpen, DOI: 10.5772/intechopen.69003. Available from:

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