Article WMC001540

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Article ID: WMC001540 ISSN 2046-1690

Predicting The Equilibrium pH Of An Aqueous

Solution: A New Approach Based On The

Mechanistic Chemistry Of Proton Transfer

ReactionsCorresponding Author:

Dr. Minhtri K Nguyen,Associate Clinical Professor, UCLA Medical Center, Department of Medicine, 757 Westwood Blvd, Rm 7501B,

90095-7417 - United States of America

Submitting Author:

Dr. Minhtri K Nguyen,

Associate Clinical Professor, UCLA Medical Center, Department of Medicine, 90095-7417 - United States of

America

Article ID: WMC001540

Article Type: My opinion

Submitted on:10-Feb-2011, 04:05:25 AM GMT Published on: 13-Feb-2011, 07:53:47 PM GMTArticle URL:http://www.webmedcentral.com/article_view/1540

Subject Categories:NEPHROLOGY

Keywords:Acid, Base, Stewart, Henderson-Hasselbalch, Proton, Bicarbonate

How to cite the article:Nguyen M K, Kurtz I . Predicting The Equilibrium pH Of An Aqueous Solution: A New

Approach Based On The Mechanistic Chemistry Of Proton Transfer Reactions . WebmedCentral NEPHROLOGY

2011;2(2):WMC001540

Source(s) of Funding:

None

Competing Interests:

None

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Predicting The Equilibrium pH Of An Aqueous

Solution: A New Approach Based On The

Mechanistic Chemistry Of Proton Transfer

ReactionsAuthor(s):Nguyen M K, Kurtz I

My opinion

Previous quantitative approaches that model acid-base physiology and

used to predict the equilibrium pH rely on the mathematical

convenience of electroneutrality/charge balance considerations (1-4).

This fact has caused confusion in the literature, and has led to the

assumption that charge balance/electroneutrality is a causal factor in

modulating proton buffering (Stewart formulation). In our recent

study, we reported the derivation of a new mathematical model to

predict the equilibrium pH based on the partitioning of H+ buffering in

a multiple-buffered aqueous solution (5). The goal of our study was to

determine whether it is possible to derive a mathematical model that

is both predictive and mechanistic. Specifically, the goal of our paper

was not to derive a predictive formula per se, but to derive a

predictive formula based on the underlying physical chemistry

involved (partitioning of H+ buffering) without utilizing the

mathematical convenience of electroneutrality/charge balance

considerations as had previous authors. Our reasoning was based on

the consideration that if a derivation based only on partitioning of H+

buffering was indeed possible, this would demonstrate convincinglythat electroneutrality/charge balance considerations are not only

mathematically not required, but are de facto not fundamental in

determining the pH from a chemical standpoint.

We were motivated to pursue this approach because we had

previously shown that although charge balance is a convenient

mathematical tool that can be utilized to calculate and predict the

equilibrium pH, charge balance (electroneutrality considerations) is

not a fundamental physicochemical parameter that is mechanistically

involved in predicting or determining the equilibrium pH value of a

solution (5,6). Indeed, if strong ion difference (SID, a term used in

the Stewart formulation which is based on electroneutrality and

charge balance considerations) were to have a mechanistic role indetermining the equilibrium pH, it must do so by imparting a fixed

macroscopic charge to the solution which will in turn cause the [H+]

to attain a given value in order to maintain macroscopic

electroneutrality. However, we demonstrated that for a given change

in SID due to the addition of HCl to a NaCl-containing solution,

electroneutrality is maintained (i.e. [Na+] + [H+] - [Cl-] - [OH-] = 0) at

all pre-equilibrium and equilibrium pH values, and that the

equilibrium pH is only determined by the dissociation constant of

water, Kw (5).

The significance and novelty of our model and certain technical

aspects of our pH measurements have recently been questioned (7-9).

These authors referred to previous formulas derived by otherinvestigators to predict and analyze the equilibrium pH of an aqueous

solution. These include the predictive formula published by Rang and

Herman et al, Charlot equation, Guenthers n-bar equation, de Levie

equation, Morels tableau method, and the quantitative approaches

discussed in the classic texts by Bjerrum, Ricci, Stumm, Ramette and

Butler (4,10,11-20). However, we stress that none of these authors has

achieved the goal of basing their derivations solely on the underlying

mechanistic chemistry involved, i.e. proton partitioning among

various buffers.

In response to an inquiry with regard to how the initial reactant

concentrations were calculated in our study (5,8), they were

calculated as follows: At each titration step, the reactant

concentrations of each sample containing the mixture of buffers were

first calculated based on the measuredpH of the samplepriorto the

addition of HCl as follows:

Since [H+]sample [A-]sample = Ka [HA]sample and [ATOT]sample = [A

-]sample +

[HA]sample :

[A-]sample = ([ATOT]sample x Ka)/([H+]sample + Ka) and [HA]sample = [ATOT]

sample [A-]sample

Based on the water association/dissociation equilibrium reaction:

[OH-

]sample = Kw/ [H+

]sample

After the addition of HCl, the initial reactant concentrations as

displayed in Table 1 were calculated by accounting for the amount of

H+ and OH- added and the dilutional effect of the added volume (5):

[ATOT]sample = (0.01 x 0.02)/Total Vol where Vol = volume

[A-]i = ([A-]sample x Volsample)/ Total Vol and [HA]i = ([HA]sample x Vol

sample)/ Total Vol

[H+]i = ([H+]sample x Volsample + [H

+]HCl x VolHCl) / Total Volume

where [H+]HCl = H+ concentration of HCl solution; VolHCl = volume of

HCl added; and HCl is assumed to be completely dissociated.

[OH-]i = ([OH-]sample x Vol sample + [OH

-]HCl x Vol HCl) / Total Volume

where [OH

-

]HCl = OH

-

concentration of the HCl solution = Kw/[H

+

]HCl

All the initial reactant concentrations displayed in Table 1 were

determined based on the above calculations, and the calculated values

were rounded to the fourth decimal place (5). There is no dilution

error in Experiment 4 as suggested (8). In Experiment 4, the total

volume of the solution for the initial data set was 20.04 ml because

more HCl was initially needed to titrate the pH of the solution to the

calibration range of our pH measurements. Thereafter, the total

volume changed in increment of 0.02 ml. Moreover, the suggestion

that buffer B is diluted by another method than buffer A is an

impossibility (8). The dilution is performed exactly as described in

the Methods section of our article. It is important to note that both

buffers A and B were mixed in the same solution and not in differentsolutions. Therefore, the same volume of HCl was added to the same

solution containing the mixture of buffers A and B. In reviewing the


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data in Experiment 2, there are typographical errors in the reported

[HB]. The correct [HB] in Experiment 2 should be: 1.0503E-03,

1.5365E-03, 2.0763E-03, 2.6531E-03, 3.3220E-03, 4.0649E-03,

4.9081E-03, 5.7530E-03, 6.6020E-03, 7.4804E-03, and 8.3978E-03

respectively. However, the reported predicted pH values are correct

as originally stated; this can be easily verified by entering the values

of [HB] listed here into our mathematical model and solving for pH.

It was also suggested that two buffers in the first experiment in our

article were exposed to different ionic strengths (8). First, in our

study (5), the apparent equilibrium constant K of each buffer was

calculated based on the thermodynamic equilibrium constant Kand

the ionic strength of the solution: pK = pK 0.51I (Eq. 23). The

ionic strength of the solution was calculated based on Eq. 24:I=

cZ2. Therefore, the same exact value for ionic strength was entered

into Eq. 23 to calculate the apparent equilibrium constant K of the

two buffers. Second, one needs to consider the temperature

dependence of the pKof any buffer pair. The pKof PIPES (Amresco,

Solon, OH) at 25C is 6.80 and not 6.76 (21). Using this pKvalue of6.80, the ionic strength is ((6.80 6.7559457)/0.51)2 = 0.00746,

which is the same ionic strength calculated for Buffer B. In reply to

the footnote regarding buffer B (8), the pKof HEPES (Amresco,

Solon, OH) at 25C is 7.55 and not 7.48 (21). Indeed, in our study,

the buffers and pH electrode were incubated at 25C in a temperature

regulated water bath to ensure that the pKs of the buffers used in our

experiments were the same as those reported by our supplier.

The validity of our data was questioned suggesting that the

equilibrium pH as calculated by the Henderson-Hasselbalch equation

is higher than the measured equilibrium pH (8). However, we note

that this author actually calculated the equilibrium pH by entering the

initial (pre-equilibrium) reactant concentrations, [A

-

]i and [HA]i, intot he Hender son-Hasse l ba l ch equat i on . S i nce t he

Henderson-Hasselbalch equation and any equation modeling

acid-base equilibrium reactions only holds true for reactant

concentrations at equilibrium, one has to enter the equilibrium

reactant concentrations into the Henderson-Hasselbalch equation to

calculate the equilibrium pH. In our study, the equilibrium reactant

concentrations were expressed in terms of the initial reactant

concentrations, i.e. [A-]e = [A-]i y and [HA]e = [HA]i + y. When

one enters the equilibrium reactant concentrations, [A-]i y and [HA]i

+ y, into the Henderson-Hasselbalch equation to calculate the

equilibrium pH, the measured and calculated pH values agree.

Our analysis was also criticized from the viewpoint that the goal ofacid-base equilibrium calculations in clinical medicine ought to be

aimed at quantifying and characterizing the metabolic component of

an acid-base disorder (e.g. base excess) rather than defining the

equilibrium pH (9,22). We disagree. If the goal is to quantify and

characterize the metabolic component of an acid-base disorder, then

we feel that quantification and characterization of the partitioning of

excess H+ among the various buffers as provided by our model can

provide important additional insight. However, in addition to the

metabolic component, the respiratory component is also important

clinically. Moreover, in clinical medicine, defining the equilibrium

pH in certain circumstances is more essential than defining the base

excess in guiding the treatmentof mixed acid-base disorders. For

example, in a patient with mixed metabolic acidosis and chronic

respiratory alkalosis, the goal of therapy is to normalize the systemicpH rather than to correct the base excess. Indeed, therapy aimed at

correction of the base excess will result in worsening systemic

alkalemia in this clinical setting. Other examples of this kind are

purposefully omitted for the sake of brevity.

Complexity is apparently also an issue in that it was suggested that

the classic analytical methods utilized to predict the equilibrium pH

are less complex and cumbersome than our current mathematical

model (9). Although our mathematical model may be more complex

than certain classic analytical methods, in our view complexity is not

a sufficient criterion for choosing between mathematical models.

Furthermore, proponents of the Stewart strong ion approach have long

argued that although the Stewart strong ion approach is more complex

and cumbersome than the Henderson-Hasselbalch approach, the

Stewart s t rong ion formulat ion i s super ior to the

Henderson-Hasselbalch approach since SID is purported to play a

mechanistic role in acid-base physiology. This contrasts with the

same authors previous analysis: However, like the BE approach

and like any other method derived from considerations involving the

calculation of interval change via the assessment of initial and finalequilibrium states, the Stewart method does not produce mechanistic

information. These are basically bookkeeping methods. To believe

otherwise risks falling prey to the computo, ergo est(I calculate it,

therefore it is) fallacy (23). We view the latter statement as a

just ification for the need for our study and model. Therefore,

although our mathematical model may be more complex than other

formulas, it is the first to be based solely on the underlying

mechanistic physical chemistry involved.

We also disagree that the Stewart strong ion formulation is predictive

in physiological fluids (9). There is no mathematical model that is

predictive in vivo (including our new model) since the equilibrium

partial pressure of CO2 (PCO2) in physiological fluids cannot be

predicted as a result of the modulation of alveolar ventilation in

various acid-base disorders. In this regard, neither the Stewart strong

ion model, Henderson-Hasselbalch equation nor any other

mathematical model is predictive in bicarbonate-buffered

physiological fluids in vivo. Specifically, both the Stewart strong ion

formulation and Henderson-Hasselbalch equation consist of an

equilibrium term, PCO2, and are therefore not predictive in

physiological fluids where the equilibrium PCO2 may differ from its

initial value in acid-base disorders.

Finally, we must disagree with the contention regarding the

superiori ty of the Stewart strong ion model over the

Hender son-Hasse l ba l ch equat i on (7 ) . Bo t h t he

Henderson-Hasselbalch and Stewart strong ion approaches are based

on equilibrium reactant concentrations. In a multiple buffered solution,

the isohydric principle (a well accepted principle in acid-base

chemistry) underscores the fact that any buffer pair (assuming the pK

is accurately known) can be utilized to calculate the equilibrium pH

value. This fact alone necessitates that the Henderson-Hasselbalch

equation and Stewart strong ion model are theoretically identical

quantitatively in terms of their accuracy in calculating the equilibrium

pH. Indeed, recent analysis has demonstrated that the

Henderson-Hasselbalch equation and Stewart strong ion model are

identical quantitatively in terms of their accuracy in calculating the

equilibrium pH in a multiple buffered solution (6). If the Stewart

s t rong ion model i s quant i tat ively super ior to the

Henderson-Hasselbalch equation as suggested (7), then in our view it


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is incumbent on those holding this view to provide a valid

mathematical explanation as to why the Stewart strong ion model can

be mathematically simplified to the Henderson-Hasselbalch equation

as shown previously (6). In addition, those who maintain that the

Stewart approach and by inference the SID calculation is

mechanistically superior in interpreting acid-base phenomenology

need to provide a validphysicochemical explanation as to why, for a

given change in SID, electroneutrality is maintained (i.e. [Na+] + [H+]

- [Cl-] - [OH-] = 0) regardless of the actual value of [H+] as

demonstrated in the example given in Table 3 of our study (5). This

example highlights the lack of a causal connection between changes

in SID and [H+].

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