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.
- buffer capacity
- buffer efficacy
- polyacid systems
- electroneutrality equation.
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 [1–16]. 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 [17–20].
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
The buffer capacity considering the effect of dilution,
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
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.
In many applications, it is convenient that the volume of the buffer solution that is added to the system of interest be
The present work extends the study of
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
Although Ref.  presented equations that describe the behavior of
The species that go from up to HL(
The global formation equilibria of the species of a polydonor system are represented according to Eq. (2).
When there is a mixture of
is the polyprotic acid of the
A representation of the
Also, in this case needs to be considered.
It can be demonstrated that the molar fraction to describe each of the
2.2. Description of the mixture to be titrated
Each species has associated countercations or counteranions (or ) depending on whether (
2.3. Expressions for the titration plots of polydonor systems mixtures
Although Asuero and Michałowski  and De Levie  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. . 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,
If one considers now that a strong acid is added to the mixture, then the
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).
Eqs. (6) and (7) are exact analytic solutions to obtain titration plots pH = f(
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 (
Extending the expressions for
2.5. General expressions of
In order to determine the buffer capacity considering the dilution,
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.
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(
3.2. Effect of the quantity of a buffer solution on
Intuitively, it is known that the performance of a buffer solution is better whenever a larger volume is taken to set the pH. The
Figure 3 shows a series of
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 ,
Figure 3 also shows the plot for water,
There is one
3.3. Buffer efficacy (
ε) of a buffer system
The previous section showed that the
Following the idea proposed by Christian in his textbook , 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).
In this work, the buffer performance will be assessed considering a ΔpH = 0.1 . The
3.4. Application of : 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 should be larger or at least equal than the
Figures 4a and b show the curves of for
Figure 4c and d shows, apart from the and plots, those of of the NH4 +/NH3 buffer solutions at different concentrations for systems with
In this case, the lowest NH4 +/NH3 buffer concentration falls to a point that almost equals the plot. Therefore, this concentration is termed as
Then, whenever there is a buffer formed by an acid‐base pair HL/L with pKa ≈ 7.0 it must be expected a greater , of approximately 10−5 M; when pKa = 5 or pKa = 9, the will also be approximately 10−4 M; and for pKa = 3.0 or pKa = 11.0, the ≈ 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 = 10−1 M, is pKa – 1 < pH < pKa + 1 . Whereas for = 10−3 M, the interval to set the pH is smaller because the becomes closer to .
When the < 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 than 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 () must be greater than the concentration of the solutes of the original system (), because both systems have their own , though how large is it? ( Figure 1 ).
An example of the use of , to evaluate setting the pH to a 9.0 value at 100 mL () of an acetylacetone solution (acac') (OS), is given next, at 10−3 M concentration (), at different NH4 +/NH3 buffer concentrations in the mixture ().
Figure 5a shows the of the acetylacetone solution (OS), stated in the previous paragraph (, marker ×××××), the
Figure 5b shows, apart from the curve, those corresponding to for 100 mL of the NH4 +/NH3 system solution at different concentrations. The magnitude depends on buffer’s concentration and decreases until it reaches a , which is given when the curve almost becomes equal to that of . In this case, ≈ 10−4 M.
In order to impose the pH of the original system, the buffer efficacy of the mixture () should be greater or, at least equal to . Now, if the buffer is added to the original system to set the pH, the has 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 is 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 ≈ . The curve 3 shows that the buffer with 10−3 M concentration does already contribute to apart from the original system, but has not set the pH yet because < < .
Finally, Figure 5c depicts the curve 4 as corresponding to , whereas the curve 5 represents a corresponding to the
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−) . 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 diminishes till it reaches a limit value given by .
Consider now the case that a pH 9.0 shall be imposed to the 100 mL () of the acetylacetone solution (acac') (OS), with a 10−3 M () concentration, using now the Britton‐Robinson [12, 21] buffer at different concentrations. For this example, the , , and are the same as those presented in Figure 5a .
Figure 6a shows, apart from the curve of , the curves of for 100 mL solution of the Britton‐Robinson buffer at different concentrations. In this case, the diminishes till reaching a , having a value of ≈ 10−5 M because at this point the almost equals the curve.
As stated, the should be larger or at least equal to . Once again, the buffer efficacy of the mixture () 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 , 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 ≈ . Curve 3 shows that in the 4.5 < pH < 7.4 interval, the buffer with 10−3 M concentration contributes more to the than 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 (7.4 < pH < 10.7) and because < , the buffer is not capable of fulfilling its function in this interval.
Figure 6b also shows curve 4 that corresponds to and curve 5 that represents a corresponding to a
3.5. Application of : threshold volume of the buffer system
The concentration of the buffer components in the mixture () must be larger than the solute concentration in the system considered () 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 (), and two buffer (BS1 y BS2) solutions with components 10−1 M () and 1 M () 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 can 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
The is related to in the mixtures, with total volumes of 110 mL for the first case and 101 mL for the second case, because in both cases are 10 times greater than the original system’s solute concentrations in the mixture ().
If now, a larger concentration of acac is used in the original system, for example 10−2 M (), it is necessary to use the more concentrated buffer to impose the pH, for example SB2, with 1 M () component concentration.
It is clear that if the original system solutes’ concentrations in the mixture () grows, the also grows, consequently does it too.
which can be rewritten as
From Eq. (17), the following is obtained
Figure 7c shows that the is 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 (). It can be proved that the fulfills the condition of being 10 times larger than the original system’s solute concentration.
Just like in the case of the threshold concentration () 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.
A model has been proposed to study the buffer capacity with dilution effect,
Through the study of
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
A new index has been introduced, , 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
From the construction of the different curves of = f(pH), it is possible to identify a buffer threshold concentration in the mixture (), which allows knowing the minimum buffer concentration to set the desired pH of the system of interest. This concentration must be, at least 10 times greater than that of the original system’s solutes in the mixture in the most difficult case.
Similarly, the different curves of = f(pH) also allow determining the minimum volume (buffer threshold volume, ) that must be added to the system of interest to set its pH in the most difficult case.
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.