J. Phys. Chem. B 1999, 103, 6809-6814
6809
pK Determination. A Mean Field, Poisson-Boltzmann Approach Augusto Agostinho Neto,* Elso Drigo Filho, Marcelo Andres Fossey, and Joa˜ o Ruggiero Neto Departamento de Fisica Instituto de Biocieˆ ncias, Letras e Cieˆ ncias Exatas UNESP, 15054-000-Rua CristoVa˜ o Colombo, 2265, Jd. Nazareth, Caixa Postal 136, Sa˜ o Jose´ do Rio Preto, SP, Brazil ReceiVed: September 10, 1998; In Final Form: January 8, 1999
A model describing dissociation of monoprotonic acid and a method for the determination of its pK value are presented. The model is based on a mean field approximation. The Poisson-Boltzmann equation, adopting spherical symmetry, is numerically solved, and the solution of its linearized form is written. By use of the pH values of a dilution experiment of galacturonic acid as the entry data, the proposed method allowed estimation of the value of pK ) 3.25 at a temperature of 25 °C. Values for the complex dimensions and dissociation degree are calculated using experimental pH values for solution concentration values ranging from 0.1 to 60 mM. The present analysis leads to the conclusion that the Poisson-Boltzmann equation or its linear form is equally suited for the description of such systems.
I. Introduction The purpose of the present work is to discuss a theoretical model associated with an experimental methodology to obtain the value of the equilibrium dissociation constant of an ionizable monoprotic acid molecule in a dielectric medium. The value of the pH and the concentration of the dissociable molecule are the required data. The model system under consideration, the ensemble representative system, is a complex with a total effective electric charge proportional to the ionization degree within a cell whose dimensions depend on the acid concentration values. The cell volume is subdivided into two regions: an inner region extending from the center of the complex to a radius Rm, in which the proton (hydronium ion) is bound to the complex, and an outer region extending from the distance Rm to the cell border with radius Rc. In the outer region the proton is driven by the mean electrostatic potential (mean force potential). The mean electrostatic potential and the mean electrostatic field are required to vanish in the cell boundary. The DebyeHu¨ckel1 model adopts different boundary conditions: field and potential vanish in a region “far” from a central ion where the ionic concentrations are the bulk ionic concentrations. The concepts underlining the model are derived under the following considerations. The dissociable molecule in solution is surrounded by an ordered structure of solvent molecules, extending for some layers. In this innermost region, the assumption for the solvent as a continuous dielectric is not a good approximation. In a protic solvent as water, there are a number of places for a proton (hydronium ion) to be chemically bounded, and it is accepted that the protons may jump from molecule to molecule (hydrogen bridges). The minimum free energy value in an equilibrium state is related to a finite probability for the available places being occupied. A qualitative picture for the situation of a dissociable solute molecule in a protic solvent as water is of a solute molecule hydrogen-bonded to a water molecule in close contact * To whom correspondence should be addressed. E-mail: augusto@ df.ibilce.unesp.br.
and indirectly bonded to others near water molecules. The evidence showing that a true chemical bonding is present with the proton in carboxylic acids says that the hydrogen bridge proton between the carboxyl and the water molecule spent an appreciable fraction of the time in the solute molecule moiety. Such events cannot be described using a mean field approach with the water as a continuous dielectric. At larger distances from the solute molecule it is expected to find bulk water. The solvent can be described as a continuous dielectric, and a mean electrostatic field can be used to describe the ionic behavior from a distance, denoted Rm, that is concentration-dependent. The present article is divided into sections as specified below. In section II the description of the experimental procedures and conditions to obtain the experimental data are reported. Section III presents the statistical thermodynamics basis and the mathematical equations for the description of the ensemble representative system. An iterative numerical method for the solution of the nonlinear Poisson-Boltzmann equation is presented. The solution of the linearized form of the equation is also given therein. In section IV the experimental data of galacturonic acid solutions in the range of concentrations from 0.1 to 62 mM are analyzed. The pK value of the acid, the average dissociation degree, the values of the (concentrationdependent) radius Rm of the complex, and the electrostatic potential value at Rm are calculated. Section V is devoted to discussion and conclusions. II. Experimental Procedure Samples. D-Galacturonic acid monohydrate purchased from Sigma Co. was used without any further purification. The sample was dissolved in a twice-distilled water, and the concentration in equiv/L was determined by titration using NaOH solution standardized by Merck. The water used in the present work was of conductometric degree: deionized and twice-distilled in quartz with electrical conductance of
