Anal. Chem. 1997, 69, 5037-5044
Potentiometric Acid-Base Titration of a Colloidal Solution Birgitta Mo 1 rnstam,† Karl-Gustav Wahlund,*,† and Bengt Jo 1 nsson‡
Technical Analytical Chemistry and Physical Chemistry 1, Chemical Center, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
An improved method for evaluating acid-base titration data obtained from the potentiometric titration of solutions containing highly charged colloidal particles, such as proteins or ionic micelles, is presented. The use of the method is demonstrated in the determination of the charge number, as a function of pH, of a colloid, in this case the protein lysozyme titrated in differing sodium chloride concentrations. The method involves an improved calculation of the electrostatic potential and the concentration profiles of ions, e.g., the hydrogen ion, surrounding the charged colloidal particles. It uses a numerical program that solves the Poisson-Boltzmann (PB) equation in a cell model. Knowledge of the concentration profile of hydrogen ions enables a correct calculation of the amounts of hydrogen ions that have protonated the protein, thus the charge number at differing pH. The PB calculation provides an estimation of the solution concentration of hydrogen ions at the surface of the protein. It is shown that this surface concentration correlates well with the charge number on the protein over the wide pH range of 1.5-12, thus also at highly acidic pH, where the colloid is highly charged. This has been difficult to achieve before. The physical and chemical properties of solutions containing large ionic molecules or aggregates, such as proteins or ionic colloidal particles, are strongly influenced by the number of charges on these particles.1-3 Thus, it is highly important to be able to accurately determine the charge on the macromolecules and colloidal particles as a function of different parameters, e.g., of pH and of ionic strength. Potentiometric acid-base titration experiments and electrophoretic measurements were the two dominant experimental methods earlier for determining the charge characteristics of proteins and colloidal particles.4 Today, different NMR methods are becoming increasingly important in this field,5,6 although sensitivity problems still have an impeding †
Technical Analytical Chemistry. Physical Chemistry 1. (1) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, U.K., 1994. (2) Evans, D. I.; Wennerstro ¨m, H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet; VCH Publishers Inc: New York, 1994. (3) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1991. (4) Tanford, C. In Electrochemistry in Biology and Medicine; Shedlovsky, T., Ed.; John Wiley and Sons, Inc.: New York, 1955; pp 249-265. (5) Dwek, R. A. Nuclear Magnetic Resonance in Biochemistry; Clarendon Press: Oxford, U.K., 1973.
effect on them. However, potentiometric titration in combination with determination of the isoelectric point of a particle is an inexpensive method for measuring the charge,7 a method that in most cases is rather accurate if the data are evaluated properly. For ions of low charge, such as monovalent acids or bases, it is easy to estimate the charge from the consumption of the titrant and from the pH. However, for an ionic colloidal solution, this is a rather difficult problem, due to the hydrogen ion concentration gradient in such systems.3 The hydrogen ion concentration gradient originates in the electrostatic field outside a highly charged surface. This has been treated by Tanford,4,8,9 but limitations in the possibilities to calculate the electrostatic potential at various distances from the surface did require simplifications in the calculations. These make the determination of the charge difficult when it has a high value such as at low pH where a protein may be highly protonated. In this paper, a new model for facilitating the evaluation of potentiometric acid-base titration data obtained from an ionic colloidal solution is presented, a method that yields improved accuracy in estimation of the charge number on the colloidal particle, especially when the particle carries a high charge. This approach is based on the fact that the concentration profile of ions outside a charged colloidal aggregate can be estimated by use of a Poisson-Boltzmann cell model calculation.10,11 The concentration profile allows the relationship between the pH in the colloidal system and the total number of H+ and OH- ions in the solution surrounding the charged colloidal particles to be obtained. When this relationship has been established, the change in charges on the particles that occurs during titration is easy to evaluate. Other data the model requires, in addition to titration data, are the isoelectric point and an estimation of the size and geometry of the colloidal particle. The method is generally applicable to multiply charged colloidal particles. Here we use the protein lysozyme as an example. We discuss the advantages of determining the concentration of H+ and OH- ions at the surface of the titrated particle. This allows the well-known Henderson-Hasselbalch equation1 to be used to describe the titration curve of a protein. The surface
‡
S0003-2700(97)00451-4 CCC: $14.00
© 1997 American Chemical Society
(6) Gurd, F. R. N.; Wittebort, R. J.; Rothgeb, T. M.; Neireiter, G. In Motions of Aliphatic Residues in Proteins in Biochemical Structure Determination by NMR; Bothner-By, A. A., Glickson, J. D., Sykes, B. D., Eds.; Marcel Dekker: New York, 1982; p 1. (7) Matthew, J.-B.; Gurd, F. R. N.; Garcia-Moreno, B.; Flanagan, M. A.; March, K. L.; Shire, S. J. CRC Crit. Rev. Biochem. 1985, 18, 91-197. (8) Tanford, C.; Wagner, M. L. J. Am. Chem. Soc. 1954, 76, 3331-3336. (9) Tanford, C.; Roxby, R. Biochemistry 1972, 11, 2192-2198. (10) Jo ¨nsson, B.; Wennerstro ¨m, H.; Halle, B. J. Phys. Chem. 1980, 84, 2179. (11) Gunnarsson, G.; Jo¨nsson, B.; Wennerstro ¨m, H. J. Phys. Chem. 1980, 84, 3114.
Analytical Chemistry, Vol. 69, No. 24, December 15, 1997 5037
concentrations are obtained through numerical solution of the Poisson-Boltzmann equation in the cell model. EXPERIMENTAL SECTION Chemicals. Hydrogen chloride, 1 M, sodium hydroxide, 1 M, sodium chloride, and sodium acetate were purchased from Merck (Darmstadt, Germany). The standard buffers of pH 9.180 and pH 13.00 were from Riedel de Hae¨n (Seelze, Germany), those of pH 4.01 and pH 7.00 were from Radiometer Analytical (Copenhagen, Denmark), and that of pH 2 was from P H Tamm Laboratories (Uppsala, Sweden). All the chemicals were pro analysis. Lysozyme chicken egg white (L6876) was purchased from Sigma (St. Louis, MO). Doubly distilled water was filtrated (0.45 µm) and was boiled for at least 15 min prior to storage under helium. Apparatus. The titrations were performed using an automatic burette, Methrom 665 Dosimat, in combination with a Methrom 633 pH meter, both from Methrom (Herisau, Switzerland). The equipment was controlled by a computer. The pH glass electrode, with an Ag/AgCl internal reference electrode and a KCl bridge, was obtained from Radiometer Analytical (pHC 2411); according to the manufacturer, the electrode was especially designed to minimize the alkaline error. Titrations. The titrant (1 M hydrogen chloride or 1 M sodium hydroxide) was added in 0.01 mL portions every 40th second, the pH being recorded immediately, prior to the subsequent addition of acid or base. In order to obtain pH curves for pH 1.5-12, one base titration and one acid titration were performed for each of two separate solutions, the two curves being combined. All titrations were performed at 25 °C. Standards. Prior to titration of the protein, the pH meter was calibrated using standard pH buffers. Since the ionic strength in standard buffer solutions differs from that of the protein solutions used in our experiments, calibration measurements were also performed by means of blank titrations of non-protein salt solutions of the same salt concentrations as the protein solutions. This was carried out partly to determine the difference in liquid junction potential, ∆E(Cs), in the pH meter between the salt solution and the standard pH buffer4,12 and partly to control the linearity of the pH response. The following relationship was used in the calibration:
Figure 1. Experimentally obtained values for ∆JP(Cs) as a function of pH in the HCl titration of a salt solution at 25 °C. The salt concentrations given refer to those present before the addition of HCl. The initial pH is ∼7.
∆JP(Cs) is the effect on the pH meter response of the difference in liquid junction potential between the salt solution and the standard buffer solution. pH symbolizes the pH value obtained after correction for the liquid junction potential. Since the value of γH+ was not available, it was approximated by the mean activity coefficient γ(,HCl. The tabulated values of γ(,HCl for different concentrations of HCl1,13 were used to construct an empirical equation for the activity coefficient γ(,HCl measured over the range of concentrations used in the experiments:
γ(,HCl ≈ 1 -
1.17xCs 1 + 2.05xCs
+ 0.19Cs
(2a)
The same procedure, carried out for γ(,NaOH gave the equation
γ(,NaOH ≈ 1 -
1.21xCs 1 + 2.00xCs
+ 0.09Cs
(2b)
and where pHread is the reading from the pH meter, e is the unit charge, k is the Boltzmann constant, T is the absolute temperature, CH+ and γH+ are the concentration and the activity coefficient, respectively, of H+, in a solution having the concentration Cs of electrolytes carrying ions of charge number (1, i.e., in this work Cs is the sum of the concentrations of HCl (or NaOH) and NaCl.
where Cs is the concentration in molality units. Molalities were used consistently throughout this work. ∆JP(Cs) was obtained by calculating log CH+ and log γH+ after each addition of HCl or of NaOH to the salt solution and making use of eq 1a. Since the initial pH was ∼7, the hydrogen ion concentration, CH+, in the salt solution can be expressed as the number of moles HCl added, nt, divided by the mass of the water, mH2O, in the solution obtained. Figure 1 shows the values of ∆JP(Cs) obtained experimentally in three series of differing initial salt concentration for the electrode in question. As shown in Figure 1, the values of ∆JP(Cs) for the three series were determined as being 0.00, -0.06, and -0.18 for the salt concentrations 0.01, 0.1, and 1 m, respectively. In each series, the value of ∆JP(Cs) was virtually independent of pH within the pH range of 1.7-3, a small decrease being observed at lower pH values, especially for the series in which the sodium chloride was 0.01 m initially. Corresponding series were also performed for additions of NaOH to the salt solutions of 0.01, 0.1, and 1 m. The CH+ obtained was based on the added amount of NaOH after
(12) Bates, R. G. Determination of pH. Theory and Practice, 2nd ed.; John Wiley and Sons, Inc.: New York, 1965.
(13) Handbook of Chemistry and Physics, 71st ed.; Lide, D. R., Ed.; CRC Press: Boston, 1990-1991.
pHread ) -log CH+ - log γH+ + ∆JP(Cs) ) pH + ∆JP(Cs) (1a) where
∆JP(Cs) )
5038
e∆E(Cs) kT ln 10
(1b)
Analytical Chemistry, Vol. 69, No. 24, December 15, 1997
combination with the autoprotolysis constant of water. The ∆JP(Cs) values obtained, -0.01, -0.01, and -0.08, respectively, were constant in the pH range of 10-12.5. The constant or virtually constant ∆JP(Cs) values obtained in the pH range of 1.7-3 and of 10-12.5 indicate the liquid junction potential to have been constant and the pH meter to yield the correct slope for the response curve. In the pH range of 3-10, ∆JP(Cs) was obtained by linear interpolation, the maximum error in ∆JP(Cs) being 0.05. THEORETICAL MODEL Calculation of the Charge Number from Titration Data. The purpose of the potentiometric acid-base titration is to determine how the number of charges on a particular substance varies as a function of the pH of the solution. To be able to do this on the basis of experimental titration data, it is necessary to determine the relationship between the amount of acid or base added in the titration and the change in the number of charges on the substance in question. If the solution is electroneutral during the titration process the following relationship holds:
nPzP ) nOH- - nH+ -
∑n z
i i
(3)
i
where nP is the amount of the substance to be investigated and zP is the charge number on it. ni and zi are the amount of and charge number on all other ions in the solution, in this case sodium and chloride ions. Given the condition of electroneutrality and the assumption that nP does not change during the titration process, the following relationship holds:
nP∆zP ) ∆ntH - (∆nH - ∆n0H)
(4)
∆nH ) nH+ - nOH-
(5)
where
∆zP is the change in the number of charges on the colloidal particle during titration. The term ∆nH - ∆n0H indicates the titration of water. ∆ntH, ∆nH, and ∆n0H represent the differences between the number of moles of H+ and OH- in the titrant added, in the titrated solution during titration, and in the initial solution, respectively. The value of ∆ntH is well-known since the volume of the titrant added and the H+ and the OH- concentration of it are carefully determined. The determination of ∆nH and ∆n0H, however, is complex, especially for colloidal solutions. ∆nH is obviously related to the pH of the solution, but no simple and general relation to pH exists, e.g., due to both the activity coefficients and the concentration of ions changing during titration. Further complications are to be found in colloidal solutions due to the presence of ion concentration gradients that extend outward from the surface of the charged colloidal particles. ∆nH, along with pH and the size of the system, mH2O, thus depends as well on the amount of the substance, the number of moles of salt ns, and the value of zP:
Figure 2. Schematic model of a cell with the radius R and of a spherical aggregate with the radius b. The aggregate is positioned in the middle and is surrounded by a water layer containing co-ions and counterions of the charged aggregate.
cell model in which all the colloidal aggregates are assumed to be evenly distributed in the solution so that every aggregate is surrounded in a regular manner by other aggregates.11,14-16 In such a cell model, the properties of the system can be described in terms of a thermodynamic description of the aggregate and the surrounding solution (a cell). The size of the cells is so defined that all of the cells incorporate the total volume of the solution. The geometric form of a cell is often chosen so that it will agree with the form of the colloidal aggregate, a spherical cell being conceived for a globular protein and a cylindrical cell for a linear polymer. In a cell model, it is rather simple to find the relationship among such quantities as concentration, charge on the different components, and thermodynamic quantities such as chemical potential or chemical activity.11,15 These relations are used here to obtain an equation for ∆nH considered as a function of pH, of activity coefficients and of the electrostatic potential, allowing useful expression for the charge on the protein to be developed from eq 4. In the following, it is assumed that the system can be described by a spherical cell model, in which the colloidal aggregate is designated as being a sphere with the radius b and the cell as being a sphere with the radius R; see Figure 2. For a system comprised of these cells, ∆nH can be obtained if one can determine the total number of moles of H+ and OH- in the solution and then utilize eq 5. The number of moles can be determined on the basis of the molalities and of the total mass of the water. However, in a colloidal system, the concentration of ions is not homogenous throughout the solution but shows strong gradients. This is due to long-range electrostatic interaction between ions and the charged colloidal particle. As a result, the local ion concentrations in various parts of the solution differ. If one knows how the local concentrations vary in the system, i.e., what gradients are found, one can calculate the total number of moles of ions in the system. In combination with the total mass of the water, this gives the mean molality, C h i, of the ions in the solution. These various values are then used to obtain ∆nH:
∆nH ) nH+ - nOH- ) mH2O(C h H+ - C h OH-)
(7)
(6)
In terms of the Boltzmann distribution, the local concentrations of an ion with a charge number z at a distance r can be written as15
The function f (pH, mH2O, nP, ns, zP) can be evaluated by use of a
(14) Hill, T. L. Statistical Mechanics; Addison & Wesley: Reading, MA, 1960. (15) Marcus, R. A. J. Chem. Phys. 1955, 23, 1057. (16) Jo ¨nsson, B.; Wennerstro ¨m, H. J. Colloid Interface Sci. 1980, 80, 482-496.
∆nH ) f (pH, mH2O, nP, ns, zP)
Analytical Chemistry, Vol. 69, No. 24, December 15, 1997
5039
Cz(r) ) Cz(R)e-ezφ(r)/kT
(8)
where φ(r) is the electrostatic potential at the distance r from the cell center, Cz(R) being the concentration at the cell boundary. It should be noted that one can only quantify the change in potential at a particular distance and that, to simplify the treatment, the electrostatic potential at the cell boundary φ(R) has been set to zero, i.e., φ(R) ) 0. The mean molalities follow then as
C h z(r) ) Cz(R)e-ezφ(r)/kT
(9)
where z ) +1 or -1. Equation 9 applies to H+ and to OHyielding their mean molalities. To obtain these, one obviously has to know their concentrations at the cell boundary and the radial dependence of the electrostatic potential within the cell, φ(r). Assuming the system to be in equilibrium, so that the activity of a particular species is the same at any distance from the cell center, the hydrogen and the hydroxyl ion activities at the boundary can be obtained directly from the experimentally determined pH so that aH+(R) ) aH+ and aOH-(R) ) aOH-. The general relationship between activity and concentration is
aH+(R) ) CH+(R)γH+(R)
(10a)
aOH-(R) ) COH-(R)γOH-(R)
(10b)
from which the two boundary concentrations, CH+(R) and COH-(R), can be obtained if the activity coefficients are known. These can be approximated using the mean activity coefficients, γ(,HCl and γ(,NaOH, in a bulk solution having the same mean salt concentration as present at the cell boundary. This approximation is only valid if the cell boundary does not diverge much from electroneutrality, i.e., if the ion concentration gradients are close to zero. A more complicated relationship is called for at low salt concentrations and/or at high concentrations of a colloidal aggregate, since unequal amounts of monovalent co-ions and counterions then prevail at the cell boundary, which thus is no longer electroneutral. In such cases, therefore, the activity coefficients at the cell boundary, γH+(R) and γOH-(R), diverge from γ(,HCl and γ(,NaOH. The following relationship for ∆nH can now be expressed on the basis of eqs 7, 9, and 10:
(
∆nH ≈ mH2O
)
10-pH -eφ(r)/kT 10pH-pKw eφ(r)/kT e e γ(,HCl γ(,NaOH
(11)
where Kw is the autoprotolysis constant of water. To express e-eφ(r)/kT, the electrostatic potential φ(r) in the cell must be determined. To do this, we have utilized a computer program PBCell17 (A Macintosh version of PBCell is available at the internet address http://www.membfound.lth.se/ChemEng1/ Prog.html.), which solves the Poisson-Boltzmann equation numerically: (17) Johnson, I.; Olofsson, G.; Jo ¨nsson, B. J. Chem. Soc., Faraday Trans. 1 1987, 83, 3331-3344.
5040
Analytical Chemistry, Vol. 69, No. 24, December 15, 1997
Figure 3. Calculated ratio of the mean concentration to the cell boundary concentration for monovalent co-ions and counterions as a function of the charge number on the globular aggregate in a 0.000 65 m protein solution. The radius of the aggregate has been set to 19 Å and the salt concentrations to 1, 0.1, and 0.01 m, respectively. The protein concentration and the size of the protein were chosen to agree with those in the experiments.
∇2φ(r) ) -
eNA
∑z C (R)e
-ezjφ(b)/kT
j j
(12)
0 r j
where ∇2 is the Laplace operator, NA the Avogadro number, 0 the permittivity of vacuum, and r the dielectric constant of water. The program provides the electrostatic potentials from which the ion concentrations can then be obtained. A detailed account of the numerical solution to the Poisson-Boltzmann equation using the PBCell program has been presented.17 The input data needed are the cell shape (spherical, cylindrical, or lamellar), the radius b of the aggregate, the thickness R - b of the solution layer, the charge density of the aggregate, the valency of the salt, the activity (obtained from the mean concentration and the activity coefficient) of the salt in the solution, and the temperature. The output data the program provides are boundary concentrations and mean concentrations, which thus provide directly the mean potential expressions given in eq 9 and needed in eq 11. The curves in Figure 3, constructed from solutions to eqs 9 and 12, show how the ratio of the mean concentration to the cell boundary concentration of the co-ions and the counterions varies with salt concentration as a function of the charge on the colloidal particle. A complication in using the data from Figure 3 in the evaluation of titration data is that the charge on the colloidal particle must be known before the mean potential expressions (in eq 9 and/or eq 11) can be calculated. Thus, a successive approximation procedure is needed. To readily carry out the calculations for varying charge number, i.e., for each pH, a function was fitted to the curves shown in Figure 3 using the following second-order equation:
e-eφ(r)/kT ≈ 1 ( K1(Cs)zP + K2(Cs)zP2
(13)
For systems of high salt concentrations, the coefficients K1(Cs) and K2(Cs) can be assigned constant values, but for systems of low salt concentration must be written as a function of Cs, since the salt concentration varies during titration. We have found the following equation to provide a good fit for the various salt
concentrations:
(
e-eφ(r)/kT ≈ 1 ( -
) (
)
0.023 0.001 z + z 2 1 + 64Cs P 1 + 164Cs P
(14)
Using this equation, it is possible to calculate ∆nH (eq 11) and hence the charge number on the protein:
zP ) z0P +
∆ntH - ∆nH + ∆n0H ) nP z0P +
t V t(C tH+ - C OH) - ∆nH + ∆n0H (15) nP
zP0 is the charge number on the protein at the start of titration; V t is the volume of the titrant added. The concentration difference t (C tH+ - C OH) is the difference between the concentrations of + the H and the OH- ions in the titrant. The ∆nH in eq 15 is calculated as
(
0 t ∆nH ) (mH + mH ) 2O 2O
)
10-pH -eφ(r)/kT 10pH-pKw eφ(r)/kT e e γ(,HCl γ(,NaOH
(16) which is a slight modification of eq 11 where m0H2O is the mass of the water in the titrated solution at the start of the titration and mtH2O is the amount of water added by the titrant. The γ( are obtained using eqs 2a and 2b. The mean potential expressions in eq 16 are calculated according to the empirical eq 14. A convenient way of solving eq 15 is to use a spread sheet program, such as Excel, for which the following columns are recommended: (1) Vt; (2) pHread; (3) pH; (4) Cs; (5) zP. To be able to calculate zP, it is necessary to know the correct value of zP0. First zP0 is assigned an a priori value and zP is calculated at each titration point. However, eq 15 is an implicit function, since zP is included in the mean potential expression in eq 14. The calculation of zP must thus be carried out by successive approximation until a constant zP is obtained at each titration point. For the calculations that follow, it is necessary that the pH range includes pI. The initial value of zP in eq 14 may conveniently be set to zero. Completing this process yields a first approximation of the charge curve. The charge must then be displaced to fulfill the condition zP ) 0 at pI through adjusting the zP values by a constant. In this way, a new zP0 is obtained and another successive approximation is performed until the constant zP is arrived at. If zP does not equal zero at the isoelectric point, zP0 must again be adjusted and the iteration repeated. Interpretation of Titration Curves. When the above process has been performed, the number of charges on the colloidal particle can be determined as a function of pH within the pH range investigated. The form of the charge curve often differs greatly from the theoretical form given by the Henderson-Hasselbalch equation:1
zP )
∑
i)bases1
ni M
+ 10pH-pKai
∑
j)acids1
nj M
(17)
+ 10-pH+pKaj
where ni and nj are the numbers of basic and acidic groups on
the particle that titrate with the “mixed” acid dissociation constants pKaiM and pKajM, respectively. The difference occurs when socalled “intrinsic” pKa values are employed. These are the pKa, which a certain group would show if the protein were without charge. The reason why the Henderson-Hasselbalch equation fails is that for a charged colloidal aggregate the many charged groups on the particle affect each other by means of long-range electrostatic interaction, an interaction that varies with differing charge, i.e. at different pH-values. Empirical fitting of pKa values to the charge curves is sometimes reported. Such pKa values reflect the influence of electrostatic interaction but have little to do with the inherent protolytic properties of the basic and acidic groups. The treatment introduced by Tanford and Wagner8 for the titration of proteins deals with the electrostatic effect by expressing the protein charge as a function, not only of pH and of the intrinsic pKa values but also of an electrostatic factor. The latter, based on thermodynamic considerations, expresses the electrostatic potential on the protein. Obtaining it involves estimating the difference in electrostatic free energy between the charged and the uncharged state of each individual titration site. The value of this obviously changes with the charge. Since Tanford and Wagner reported their work, many papers have appeared discussing the thermodynamics of charged protein molecules.4,7,9,18-20 Using the same formalism as that of Tanford and Wagner, one obtains
zP )
∑
i)bases1
ni M+pH+eφ(b)/kTln10
+ 10-pKai
∑
j)acids1
nj M-pH-eφ(b)/kTln10
(18)
+ 10pKaj
where φ(b) is the mean electrical potential calculated as being found on the surface of the colloidal particle if the potential at the cell boundary is set to zero. The expression eφ(b)/kT ln 10 replaces the electrostatic factor used by Tanford and Wagner. In their treatment, the electrostatic factor was typically modeled as a linear function of the charge on the protein whereas the relationship, in fact, is often nonlinear. This was a convenient and largely necessary approximation in view of the limited computer capacity available at the time for solving the PoissonBoltzmann equation. Today, one is not restricted to such approximations since the Poisson-Boltzmann equation can now be solved numerically. Improved accuracy in the estimation of the electrostatic factor can thus be obtained. This gives better accuracy in estimating the charge number on the colloidal aggregate, especially when it has high values, which should result in more correct charge curves for proteins over a very wide pH range. As the results below will indicate, use of the mean electrostatic potential on the surface of the protein provides a good approximation in calculating reasonable estimates of the local concentration of ions at the surface. Obtaining more accurate estimates, taking account of the individual positions of the titrating groups, would require knowledge of the local potential present on each group, (18) Bashford, D.; Karplus, M. Biochemistry 1990, 29, 10219-10225. (19) Bashford, D.; Karplus, M. J. Phys. Chem. 1991, 95, 9556-9561. (20) Oberoi, H.; Allewell, N. M. Biophys. J. 1993, 65, 48-55.
Analytical Chemistry, Vol. 69, No. 24, December 15, 1997
5041
Figure 4. Calculated surface potential of a globular aggregate as a function of the charge number on the aggregate. Conditions as in Figure 3.
which would require involved statistical-mechanical calculations or simulation techniques.21 Equation 18 represents a modification of the HendersonHasselbalch curve (eq 17) aimed at providing a correct value for the charge number. Since the pH + eφ(b)/kT ln 10 is a close approximation of the hydrogen ion concentration at the protein surface, CH+(R)e-ezφ(b)/kT (the activity coefficient γH+ being neglected), eq 18 can be said to express the degree of protonation of the protein as a function of the surface concentration of hydrogen ions rather than as the solution concentration of hydrogen ions. The latter is only meaningful to use when there are no concentration gradients for the hydrogen ions. To be able to use eq 18, the value of φ(b) must be calculated for each zP value. This is easily performed, however, using the PBCell program, as indicated earlier. A typical example of the dependence of eφ(b)/kT ln 10 on zP in different salt solutions is shown in Figure 4. The difference between p(CH+(R)) and p(CH+(b)), i.e., eφ(b)/ kT ln 10, is rather large for a highly charged particle in a solution with a low salt concentration. This is due to the ion concentration gradients around the colloidal aggregate being very pronounced under these conditions. RESULTS AND DISCUSSION To verify the correctness of the titration method as well as the calculations involving activity coefficients and the liquid junction potential, and hence the estimation of the titration of water, an acid-base titration of a noncolloidal substance, sodium acetate in 0.1 m sodium chloride, was made down to pH 1.5. The number of charges was calculated using eq 4 in which the term ∆nH - ∆n0H compensates for the titration of water. The calculation of ∆nH - ∆n0H is very simple in this case, since in a solution without concentration gradients C h i ) Ci. The ∆zP reached constant values 1.0 below pH 2.5 as expected for acetate, which shows that the compensation for titration of water was correctly made. A fit of the pKa value to all the experimental points according to the Henderson-Hasselbalch equation (eq 17) gave pKa ) 4.61. After correction for the activity coefficient of the acetate ion (γ( in 0.1 m NaCl ) 0.778),13 it provides a thermodynamic constant of 4.72, which is close to the value 4.76 at 25 °C given in the literature. This shows that accurate estimates of the activity coefficients and (21) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976.
5042 Analytical Chemistry, Vol. 69, No. 24, December 15, 1997
Figure 5. Charge number on lysozyme in 0.1 m NaCl as a function of the pH. In the calculations, the isoelectric point in the solutions was set to 11.222 and the radius of lysozyme was set to 19 Å as obtained from the molecular mass and the density in solution: (+) case 1; (×) case 2; (O) case 3.
the ∆JP have been made. This is necessary for the following experiments on a colloidal system. Next, the titration method was applied to a colloidal system where it was necessary to account for the differing electrostatic potentials at various distances from the surface. A globular protein, lysozyme, was chosen for study as an example of a colloidal particle. It is assumed to have a uniform surface charge density and a spherical shape, allowing the detailed structural properties of the protein molecule, such as variations in surface charge density, to be neglected. Charge curves from the titration of 0.000 65 m lysozyme in a 0.1 m sodium chloride solution are presented in Figure 5. The three different charge curves were calculated from the experimental titration data using three different models: (1) the full model presented in this paper (eqs 15 and 16); (2) a model where the concentration gradients have been neglected. This means that the mean molalities of H+ and OH- in eq 7 were obtained only from the pH and the activity coefficients. This should give erroneous results because the mean molalities obtained are then equal to the cell boundary concentration Cz(R) in eq 9; (3) a model in which both the concentration gradients and the water titration are neglected (eq 4 with the omission of ∆nH - ∆n0H). Case 1 shows the expected variation as a function of pH of the charge number on the protein because it would increase with decreasing pH. The number of charges should reach constant values in the case that no more protonation can occur, i.e., all accessible groups have been completely protonated on approaching the lowest pH. Thus it appears that the full model presented in this work gives a charge curve which follows the expected pattern and therefore is indicated to have good accuracy. Case 2 shows an erroneous pH dependence below pH 2 caused by not taking the concentration gradient of hydrogen ions into account in calculating ∆nH. Too low values for zP are observed, the deviation from case 1 increasing when pH decreases. This is because in calculating ∆nH from eq 7 one obtains too high values. The mean molal concentrations C h H+ then used is actually the cell boundary concentration Cz(R) in eq 9. However, the true mean (22) Anderson, E. A.; Alberty, R. A. J. Phys. Colloid Chem. 1948, 52, 13451364.
Figure 6. Charge number on lysozyme as a function of pH at three different salt concentrations. In the calculations, the isoelectric point of lysozyme was set to 11.2.
molal concentration, C h z(r), of H+ is lower than this due to that the electrostatic potential expression takes values
