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A bilogarithmic hyperbolic cosine method for the evaluation of overlapping formation constants at varying (or fixed) ionic strength is devised in this paper and applied to data reported in the analytical literature, i.e. succinic acid system, Cu(II)-glycine system and Ag(I)-aminobutan-1-ol system. The method is based on the linearization of the formation function ? = f(pH) or ? = f(pL) data. A theoretical slope of unity should be obtained thus proving the correctness of the assumed equilibria. An additional advantage of the bilogarithmic method proposed is that it provides a closed scale representation of Y and X unlike other plots. This paper forms part of an investigation into the uses of bilogarithmic methods and hyperbolic functions in parameter estimation. Methods based on the application of spectrophotometric measurements have been the subject of recent studies.

The exact determination of the thermodynamic formation constants of many dibasic acids is complicated by the overlapping [

For a diprotic acid H_{2}R, the average proton number [

where charges have been omitted for convenience. The stepwise thermodynamic formation constants of the acid is defined by

where parenthesis indicate activities and braces concentrations; f_{2}, f_{1} and f_{0} being the activity coefficients of the species H_{2}R, HR and R, respectively.

By combining Equations ((1)-(3)) we get

On rearrangement Equation (4), we obtain

Two different situations will be considered in that follows depending whether the proton number values were lower or higher than the unity.

By dividing Equation (5) by (H)^{3/2}, a further rearrangement leads to

By multiplying and dividing the right hand of Equation (6) by

we get

Making

and taking into account that

Equation (8) may be converted into

where pH = −log(H). By taking logarithmic on both sides of Equation (11), on rearranging we finally get

Thus, a representation of the left term of Equation (12) against the term into brackets of the right hand should give a straight line (Y = a_{0} + a_{1}X), obtained by linear regression [

The application of Equations ((12) and (13)) requires, however, the previous knowledge of

where N is the number of data pairs, and ñ is calculated from Equation (4) once both logK values are known, This task is easily carried out with the aid of an Excel spreadsheet.

In those cases in which the ionic strength is held constant by addition of an inert salt, e.g. potassium chloride or potassium nitrate 0.1 M, Equation (12) is converted into

where

_{H} the activity factor of hydrogen ion. Note that the ñ values when ionic strength is held constant are given by

In these situations, by dividing Equation (5) by (H)^{1/2}, on rearrangement we get

By multiplying through

Equation (19) is converted into

Taking into account the definition of hyperbolic cosine first and taking decadic logarithms on both sides of the resulting equation then, a posterior rearrangement leads to

When the left term of Equation (22) is plotted against the term into brackets of the right hand, a straight line (Y = a_{0} + a_{1}X) of unity slope should be obtained, from which the value of

Nevertheless, before Equation (22) can be applied,

If the ionic strength is maintained constant during the titration then

The basis of this discussion has been protonation reactions, but the same principles apply for metal complexation reactions M + L = ML and ML + L = ML_{2}

being the formation function or Bjerrum index in this case

Taking into account that V_{0} millilitres of the diprotic acid H_{2}R at a concentration C_{A} moles/liter, haven been titrated with a volume V of titrant, e.g. a strong monoacid base BOH, of concentration C_{B} moles/liter, the computation of ionic strength may be made assuming the Speakman [_{B}V < C_{A}V_{0}

In those cases in which C_{B}V > C_{A}V_{0 }we get

The Debye-Hückel equation [_{2}R

where A and B are constants of the Debye-Hückel theory, and ä is the so-called ion-size parameter, or some extended form of the empirical Debye-Hückel equation as the Davies equation [_{i}.

In those cases in which ñ < 1, the straight line intersect the X-axis at the point

from which we may evaluate the value of

By applying the law of random error propagation [

Taking into account [

where

An estimate of the uncertainty of these calculations is given by

In those cases in which ñ > 1 then

and the application of the random error propagation law gives in this case

Two principal difficulties should be self-evident. Primarily the present analysis requires a prior estimate of the individual stability constants. On this respect, preliminary values of

where

Expressions (35) and (36) are only approximate because of the influence of varying ionic strength. In addition, it is always disadvantageous to calculate stability constant from a minimum amount of experimental data. As a matter of fact, however, even the pH values of ñ = 0.5 and ñ = 1.5 may be taken as starting point for

In order to check the usefulness of the method it has been applied to a variety of systems previously described in the literature. Systems chosen for study were representative of the most difficult experimental situation encountered in practice. All have log K values similar in magnitude thus being very suitable for the purpose of this work. Experimental details and [pH,V] and [pL,ñ] data employed are given in that follows:

I. Succinic acid [_{R} = 0.005 M; V_{0} = 100 mL, C_{B} = 0.1 M (KOH); T = 25˚. Data [V, pH]: [1.00, 3.677; 1.25, 3.767; 1.50, 3.853; 1.75, 3932; 2.00, 4.009; 2.25, 4.081; 2.50, 4.153; 2.75, 4.223; 3.00, 4.291; 3.25, 4.361; 3.75, 4.498, 4.00, 4.569; 6.00, 5.135; 6.25, 5.204; 6.50, 5.273; 6.75, 5.342; 7.00, 5.412; 7.25, 5.480; 7.50, 5.554; 7.75, 5.629; 8.00, 5.208; 8.25, 5.789; 8.50; 5.881; 8.75, 5.981; 9.00, 6.099].

II. Cu(II)-Glicine system [

III. Silver(I)-4-aminobutan-1-ol [

The ideal methodology devised for H_{2}R/HR/R systems may be applied to simultaneous complex systems ML_{2}/ML/M. In this case the data available are (pL, ñ). _{1} of 8.12 to 8.16 and log K_{2} of 6.73 to 6.78. The results obtained in this paper are [8.177 - 8.143] for log K_{1}, and [6.772 to 6.645] for log K_{2}. The values obtained for ñ < 1 and ñ > 1 differ in 0.034 and 0.127 log units, for log K_{1} and log K_{2}, respectively. The slopes of our method in both cases are close to 1 (0.9787 ± 0.0271 for ñ < 1 and 0.9878 ± 0.0134 for ñ > 1).

A well defined slope (1.0073 ± 0.0136) was obtained for the system Ag(I)-4-aminobutan-1-ol (ñ > 1) and values of log K_{1} and log K_{2} of 3.416 ± 0.002 and 3.896 (assumed), respectively. Lansbury et al. [

A major goal of scientific experimentation is the discovery of relationships [

System | Method | log K_{1} | log K_{2} | Ref. |
---|---|---|---|---|

Succinic acid | Computer FORTRAN method | 5.634 | 4.200 | [ |

BHCM (ñ < 1) | 5.643 ± 0.008 | 4.183 | This paper | |

BHCM (ñ > 1) | 5.634 | 4.199 ± 0.002 | This paper | |

Cu(II)-Glycine | Successive approximation method | 8.16 | 6.73 | [ |

Correction term method | 8.13 | 6.78 | [ | |

Least squares treatment | 8,12 | 6.77 | [ | |

BHCM (ñ < 1) | 8.143 ± 0.009 | 6.772 | This paper | |

BCHM (ñ > 1) | 8.177 | 6.672 ± 0.005 | This paper | |

Ag(I)-4-aminobutan-1-ol | Weighted least squares method | 3.41 | 3.89 | [ |

Computer FORTRAN technique | 3.41 | 3.89 | [ | |

BCHM (ñ < 1) | 3.416 ± 0.002 | 3.896 | This paper | |

BCHM (ñ > 1) | 3.495 | 3.818 ± 0.007 | This paper |

that by applying other least-squares procedures, it is not possible to determine whether a given pH against fraction titrated curve is characterized only by the assumed reactions. In this respect, when the independent and dependent variables are varied over a number of orders of magnitudes, the points tend usually [

Samuel Beaumont,Julia Martin,Agustin G. Asuero, (2016) A Potentiometric Evaluation of Stability Constants of Two-Step Overlapping Equilibria via a Bilogarithmic Hyperbolic Cosine Method. Journal of Analytical Sciences, Methods and Instrumentation,06,33-43. doi: 10.4236/jasmi.2016.62005