Use of a Modification of the Debye-Huckel Equation to Calculate Activity Coefficients from Measured Activities in Electrolyte Solutions G.W . Net€‘ International Business Machines Corporation, Advanced Systems Development Division, 2651 Strang BIvd., Yorktown Heights, N. Y. 10598 With increased use of ion-selective electrodes for the measurement of ionic activities i n complex solutions such as blood, it is desirable to be able to calculate the various ionic activity coefficients from measured activities. These coefficients are important, both in the calculation of ionic concentrations from the activities and in the study of the interaction of ions with their environment. I n this paper the Debye-Hiickel equation is modified to relate the activity coefficient of one ionic species to the sum of the activities of all the ionic species in a solution rather than to the more usual ionic strength term.
surements of H+ and pCOe), and perhaps Ca2+ (6). The , a activity coefficient, y, of a particular ionic species, say, Y N ~ is function of the total ion population, as expressed by the Debye-Huckle theory on electrostatic interaction (8,9). Thus it is erroneous to assume that the Na+ concentration alone determines the for a particular blood specimen. It is important to consider the concentrations of all ionic species when a particular y is being estimated. This is indeed an extremely difficult problem; however, because of the potential importance of blood activity coefficients, it is worthwhile to attempt an approximation based on what we can measure.
INTHE
PAST FEW YEARS, there has been renewed research in the use of ion-selective electrodes for the analysis of body fluids (1-6). Ofparticular interest have been the glass electrodes for the measurement of H+, Na+, and KC. Also the ion-exchange-membrane electrode for the measurement of Ca2+has received much work (6). Eisenman and others have established that the potential appearing across the membrane surfaces of these potentiometric electrodes is a function of the types and relative activities of those ions to which the electrode is sensitive (7). Many workers using ion-selective electrodes for measurements of body fluids have expressed the belief that ionic activity may have more physiological significance than ionic concentration. Others feel that ion-selective electrodes will be useful only when the resultant measured activity can be accurately related to concentration as measured by conventional techniques. This relationship requires a knowledge of the activity coefficient of the ion in question. The activity coefficient of a particular ion gives valuable information, not only concerning the concentration of that ionic species whose activity is known but also concerning the relationship of that ionic species to its environment. This paper deals with initial work done to allow evaluation of the activity coefficient of an ion in the presence of other ionic species, where the various ionic activities are known. In the particular case of whole blood, activities can be measured for H+, Na+, K+, C1-, HC03- (derived from meal Present address, Responsive Data Processing Corporation, Radio Circle, Mount Kisco, N. Y. 10549.
(1) H. Dahms, Clin. Chem., 13,437-450 (1967). (2) H. Dahms, R. Rock, and D. Seligson, ibid., 14, 859-870 (1968). (3) S. M. Friedman, in “Methods of Biochemical Analysis,” D. Glick, Ed., John Wiley & Sons, New York, 1962, pp 71-106. (4) S.M. Friedman, S.L. Wong, and J. H. Walton, J . Appl. Physiol., 18, 933-954 (1963). ( 5 ) G. W. Neff and C. J. Sambucetti, “A Computer-Based Blood Chemistry Monitoring System-The Automatic Blood Analyzer,’’ IBM Advanced Systems Development Division, Technical Report No. 17-244, Yorktown Heights, N. Y., January 1970. (6) E. W. Moore, Ann. N . Y. Acad. Sci., 148,93-109 (1968). (7) G. Eisenman, R. Bates, G. Maddock, and S. M. Friedman, “The Glass Electrode,” Interscience, New York, 1965.
INDIVIDUAL cs. MEAN ACTIVITY COEFFICIENTS In attempts to apply analytical expressions to determine activity coefficients, the classical approach has been to devise expressions that contain salt concentration as an independent variable, because concentration is what is usually known. The various expressions so derived are applicable to solutions of a single salt, and the activity coefficient generated is the mean activity coefficient for the salt. These activity coefficients are usually represented as yh,Yh, orfi, depending on whether one uses the molal, molar, or mole fraction scale, respectively. In the following discussion we shall use the molal scale and yk. The definition of y&is (9): y& =
(YIY’
. y2Y*)””,
v
=
Y1
+
v2
(1)
where one mole of electrolyte gives v1 moles of cations and v2 moles of anions and where y1and yzare the individual activity coefficients of the component ions. The quantity yk is useful, for it can be measured thermodynamically and, indeed, is the quantity that must be considered in attempts to correlate equations with existing tables of “activity coefficients.” However, in work with ion-selective electrodes, there is evidence that the measured voltage is truly a measure of the individual ionic activity coefficients (IO). We shall proceed to devise an expression for estimating blood electrolyte activity coefficients from these measured ionic activities on the basis of the Debye-Huckel theory. MODIFYING THE DEBYE-HUCKEL EQUATION
In the case of strong electrolytes, the Gibbs function consists of two major additive parts (11),the electrostatic energy, Gel, and the statistical, or entropy, term, G’. The corre(8) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,”
Butterworth, London, 1959. (9) J. T. Edsall and J. Wyman, “Biophysical Chemistry,” Vol. I, Academic Press, New York, 1958, pp 241-275. (10) A. Shatkay and A. Lerman, ANAL.CHEM., 41,514-517 (1969). (11) E. Glueckauf, in “The Structure of Electrolytic Solutions,” W. J. Hamer, Ed., John Wiley & Sons, New York, 1959, pp 97-113.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
1579
(4ae2/ekT)x n i Z i 2 ,6 is the radial distance from
sponding y terms derived from the Gibbs function are:
where
Following Glueckauf (11)and using volume fraction statistics, we can express the contribution of the entropy term to the activity coefficient as :
the central ion, r is the radius of the sphere in which no other charge exists, and E is the dielectric constant of the medium. It is from this potential function (Equation 5 ) that an expression for activity coefficients is derived. From this latter expression the individual ionic activity coefficients are derived and then transformed to the mean activity coefficient of the salt. Finally, the term x n i Z i 2in K~ is replaced by a term
K~
=
i
h-v
In y s =
~
In (1
V
+ 0.018rnp)- hV In (1 -
+
- 0.018hrn)
+
0.018~$3(/3 h - V) (3) v(l 0.018mp)
+
where h is the hydration number for the ion, v is the number of ions of electrolyte per molecule, m is the molal concentration, and p is the apparent molar volume of the electrolyte divided by the molar volume of water. We shall say little more about this entropy term. The ionic strength of human blood is in a range where the entropy term is small, and the variation of this term over the range of physiologically possible ionic strengths is such that there should be little variation in any particular y+ due to changes of y’. Nevertheless, we shall be using the term because we are going to introduce some rather severe changes in the ye’term, and, to test these changes, we shall be evaluating pure solutions over extremely broad ranges of concentration. With regard to ye’, the electrostatic contribution to a nonunity activity coefficient, the original expression, derived by Peter Debye, was an expression of the work done in populating a medium of uniform dielectric constant with point charges distributed about a central ion (12). The resulting change in the ion’s electrical energy compared with its electrical energy in a charge-free space gives rise to a change in the Gibbs function and a nonunity ye’ term. Two basic and critical assumptions are made in the Debye theory. The first deals with the distribution of the charges around the central ion. The distribution of charge concentration is assumed to follow a Boltzmann distribution as a function of electrical energy, e*, and the resulting charge density about a central ion of typej is (8): (4)
of i ions is performed over all ionic species in the solu-
where i
tion and where n i is the bulk concentration of the ith ionic species, .Tiis the valence of the ith ionic species, e is the charge of the electron, #, is the electrical potential with respect to the jth ionic species, k is the Boltzmann constant, and T is absolute temperature. Assuming this charge distribution, one attempts the solution, for #, of the boundary value problem, which consists of the simultaneous solution of the Laplace equation for a spherical area around the j ion in which no other charge resides and of the Poisson equation for all space around the sphere in which the charge density is p. The second basic assumption in the Debye theory is that the exponential function for p j can be approximated by the first term of the expansion of the exponential. With this assumption the solution for fi follows (8):
.~
(12) P. Debye and E. Huckel, Phys. Z . , 24, Part I, 185-206, Part I1
305-325 (1923). 1580
e
containing the ionic strength, I , of the solution. These last two steps are what we wish to investigate. If indeed ionsensitive electrodes give a measure of individual ionic activity (IO),then it is that form of the Debye-Huckel equation for y that we should use, not the mean coefficient, yi. The interaction between the measuring electrode surface and the ion being measured is most likely associated with the chemical potential of the ion and is therefore most likely associated with the activity of that ion (7). Thus, the potential at the surface-liquid interface is a measure of the interaction of the surface and those ions that are at or above a certain freeenergy level. Making a distinction between the mean activity coefficient and the individual activity mefficients in blood is important because of the mixed nature of the ions making up the salts in blood. We have evidence that y ~ and * ~ C are I not equal in blood (13). The use of ionic strength as equal to ionic concentration, or more particularly point-charge concentration, in the expression of y+ is usually justified in pure solutions of univalent salts on the basis that a strong electrolyte completely dissociates and all molecules can be considered to split into “ions” and thus into point charges. Here a critical condition is that the “ion” be free to move, restricted only by electrostatic interaction. The resulting charge distribution and charge density, p, so critical in the Debye-Huckel theory, are dependent on the “ion’s” freedom of movement. However, results reported on ion pair formation, even in strong electrolytes, suggest that some fraction of the charged particles are not free to move as point charges but are bound to some degree to oppositely charged particles (14). This unit now acts more as a dipole than a point charge. The dipole moment, of course, is a function of the energy bond between these two charges, but the important point is that the charges distribute themselves as a pair, not as point charges. As a first approximation to adjusting the Debye-Huckel equation for ion pair formation, we should like to be able to use a better approximation for the number of ions that can act as point charges. The assumption introduced here is that the activity of a particular ion is a better measure of the number of ions free enough to act as point charges than is the ionic concentration, or total number of atoms. The difference between these two quantities represents ions that are participants in bonds of various levels. These bound ions can be represented electrostatically only as dipoles, quadrapoles, etc. Therefore, the term x n , Z i 2 in ~2 is evaluated by 1
using x a i Z Z 2 instead of crniZL2,where ai is the activity of z
1
the ith ionic species and m,, its concentration. Thus, the expression for the electrostatic contribution to the activity coefficient for a particular ion, say, the jth ion, in a set of i
(13) G. W. Neff, W. A. Radke, C . J. Sambucetti, and G. M. Widdowson, Cliti. Clzem., 16, 566-572 (1970). (14) J. F. Duncan and D. L. Kepert, in “The Structure of Electrolytic Solutions,” W. H. Hamer, Ed., John Wiley & Sons, New York, 1959, pp 380-400.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
Table I. The Activity Coefficient, y, for NaCl Calculated with Equation 7. P = 2.61 A, h = 3.3, /3 = 1.03, Y = 2.0 m a log Y 6 -1% Yoslodel -log yoalcd -1% Yobsd Ycalod Yobsd 0.001 0.000965 ... 0.0157 0.0157 0.0155 0.965 0.965 0.01 0.00902 0.0447 0.0447 0,0447 0.902 0.902 0.05 0.04096 0.6030 0.0874 0.0870 0.0866 0.818 0.819 0.1 0.0778 0.0057 0.1143 0.1086 0.109 0.779 0.778 0.3 0.2133 0.0172 0.1677 0.1505 0.148 0.707 0.711 0.5 0.3405 0.0288 0.1970 0.1682 0.167 0,679 0.681 0.7 0.4669 0.0405 0.2181 0.1776 0.176 0.664 0.667 1 .o 0.6580 0.0583 0.2412 0.1838 0.182 0.655 0.658 2.0 1.336 0.1189 0.2934 0.1745 0.175 0.669 0.668 3.0 2.142 0.1823 0.3237 0.1454 0.146 0.715 0.714 4.0 3.132 0.2493 0.3546 0.1053 0.106 0.784 0.783 5.0 4.365 0.3203 0.3775 0,0572 0,059 0.876 0.873 Part of the data in columns 1 and 8 was taken from E. Glueckauf, Chapter 7 in “The Structure of Electrolytic Solutions,” W. J. Hamer, Ed., John Wiley & Sons, New York, 1959, Table 2, p 100. The rest of the data in those columns came from R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” Butterworth Scientific Publications, London, 1959, Table 9.3, p 236. *At m = 0.0001, -log y = 0.0214, and u = 19.04 X 0.5094; - (-logy) r = 0.3294; (-log 7) = 2.61. 0
Table 11. The Activity Coefficient, y, for HCI Calculated with Equation 7. I = 2.80 A, h = 5.57, /3 = 1.05, Y = 2.0 m U 1% YE -log Yoalcdel -log Yoalob -1% Yobsd Yoalod Yobsd 0.0109 0.975 0,975 0.0001 0.0110 0.0109 0.0005 0.000488 0,966 0.966 0.0152 0.0152 0.0154 0.000966 0.0002 0.001 0,930 0.929 0.0315 0.0322 0.0008 0.0323 0.005 0.00464 0.906 0.905 0,0430 0.0435 0.0015 0.0445 0.01 0.00905 0.0797 0.0807 0.832 0.830 0.0077 0.0874 0.05 0.0415 0.0986 0.0989 0.797 0.0154 0.1140 0.796 0.1 0.7964 0.0308 0.1465 0.1157 0.1154 0.767 0.767 0.2 0.1533 0.1225 0.122 0.754 0.755 0.046 0.1685 0.3 0.2265 0.1220 0.121 0.755 0.757 0.078 0.2000 0.5 0.3784 0.1133 0.112 0,770 0.7 0.5409 0.110 0.2233 0.773 1.o 0.8091 0.158 0.2505 0.925 0.092 0,808 0.809 1.2 0.190 0.0756 0.076 0.840 0.840 1.007 0.2656 1.5 0.241 0.895 1.3430 0.2855 0.0415 0.048 0.908 2.0 2.0186 0.332 0.3135 -0.0185 -0.004 1.043 1.009 The data in columns 1, 2, and 3 were taken from E. Glueckauf, in “The Structure of Electrolytic Solutions,” W. J. Hamer, Ed., John Wiley & Sons, New York, 1959, Table 4, p 105.
different types of ions is:
“ I
1
where A and B are the usual constants in the Debye-Huckel equation (11) and r j is the average radius of closest approach of another charged particle to t h e j ion. VERIFICATION OF EQUATION 6 WITH NaCl AND HC1 To test the use of a’s instead of rn’s in the equation, we fitted data to observed y* values for NaCl and HCI. Since it is yEt that is given in the literature, Equation 6 was extended as usual to give yk. Thus, the final equation used to fit the experimental data is: -0.509
&-
+
log YE (7) 1 0.329r d a The results of applying this equation are given in Tables I and 11. The manner of applying Equation 7 is as follows. The capture radius, or radius of closest approach, r , is evaluated by solving Equation 7 for r with a known yi at a very
logy& =
+
dilute state in which yBis negligible and where yi is close to 1. This was done for NaCl at m = 0.0001 (Table I) and HC1 at m = 0.0005 (Table 11). With the calculated value of r and the values of a calculated from rn and the observed yi’s, we can solve Equation 7 for log yi. Making Equation 7 dependent on a (Le., y) makes it unsolvable in the case of solutions in which the concentrations are known and the activities are not. In these cases solutions can be found by iterative techniques. In the case of ion-selective electrodes, however, it is the activity that is measured, and, knowing the activities of the various species in a solution, we can calculate the respective activity coefficients and thus the concentration of each species. For NaCl, it was necessary to use a slightly different value of h than Glueckauf uses. We use h = 3.3 instead of 3.6. However, because h is always chosen for a best fit to the data, this does not detract from the theoretical basis for Equation 7. It is interesting that the capture radius, r, derived for NaCl in this manner, by using a’s instead of rn’s, is almost equal to the sum of the Pauling crystallographic radii for Na+ and C1-. Tables I and I1 show that the use of Equation 7 gives a smooth transition from the region of high dilution, where the electrostatic term is dominant, to the region of high concentration, where the statistical term
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
1581
dominates. This occurs without the need for an arbitrary curve-fitting term. APPLICATION OF EQUATION 6 TO COMPLEX ELECTROLYTE SOLUTIONS Equation 6 may be used in deriving values of concentration from measured values of activity for the various charged particles in complex electrolyte solutions. When ionselective electrodes are used t o measure the electrolytes in blood or plasma, it is necessary to be able to determine concentration since this is the quantity that has medical significance at present (13). The activity measured with electrodes has been shown to be accurate, at least for Na’, K+, C1-, Cazf, and HC0,- (5-7, 13, 14). The y measured is the individual y for the selected ion; thus, standardization can be accomplished accurately with pure electrolyte solutions. As discussed by Neff et al. (13), these standards (whose y’s are known accurately) can be used to standardize the electrodes and give accurate activity measurements in complex solutions. However, the subsequent calculation of concentration from the measured activity is complicated by the dependence of y on a . In most of the literature concerning the measurement of blood constituents with electrodes (mostly Na+ and K+) the y’s were evaluated by measuring the slope of a best-fit line through a series of points on a plot of activity 5s. concentration. Typically, the activities of a number of samples are measured with electrodes and their concentrations are measured by standard wet-chemical techniques. For each sample a point is plotted on the activity us. concentration diagram. A best-fit line through these points is determined, and its slope is a measure of y. This technique assumes that the particular y is either constant or a function only of the concentration of the ion being measured. The DebyeHiickle theory, however, shows that, at least for electrostatic effects, y should depend on all the charged particles in the solution. If one were able to measure the activities of all charged particles in blood, then Equation 6 would allow calculation of the corresponding y and the concentration could then be calculated from a / y . Presently in whole blood we can measure aNa, aK, acl, aH, ac,, and aHcoj. Although this is far from knowing the activities of all charged particles and although the equation is only an approximation, using this approach should give a better value for any desired yj than simply assuming that yj is the same as it would be in a pure solution containing an equal amount of ion j . To use Equation 6, we must know x a i Z i 2 . This can be considered as i i
d
m
where m refers to those ionic species whose activities are measured, and d refers to those ionic species whose activities are derived from mean values of the remaining blood constituents that are present in charged form. Since we are dealing primarily with water as the solvent, A and B are known from the literature. Also, the rj.s can be calculated from data on pure solutions. Thus, instead of evaluating the yjs empirically for whole blood, we canevaluate z a d Z d z d
empirically and use this with the measured am’s to calculate the y;s and subsequently the mj’s. An average value of the term x a d Z d 2 can be evaluated d
empirically as follows. A number of blood samples are analyzed with electrodes to measure the activities mentioned above. For at least one of the ions, say, Na+, concentration also is measured by a standard technique (flame photometry). For 1582
this ion, y is determined from the measured activity and measured concentration. Since the electrolytes in blood are somewhat dominated by Na+ and C1-, a reasonable estimate of rj for sodium is the value determined for pure NaCl in Table I (rNn = 2.61 A). Using these values for y3 and r3,we can solve Equation 6 for ~ U , Z for,each ~ sample. Summing a
the measured activities for each sample yields a value for C a m Z m 2 ,and thus from Equation 8 we can obtain C a d Z d 2 m
d
for each sample. Then, in subsequent blood analyses where only the activities are measured, we can use this average value of with the measured values of activity to calculate d
c a Z Z t 2 . Knowing this, we can determine y j and then the a
required values of concentration. CONCLUSIONS It is obvious that the use of the activity of a particular ionic species as an estimate of the number of electrostatically free ions present is not valid in cases where the activity coefficient becomes greater than 1. The results presented in Tables I and I1 demonstrate that certainly not all of the atoms of a salt, even a strong electrolyte, act as point charges. For strong electrolytes, the degree of ion pairing should give a better value of the number of “free ions.” Certainly the dipole effect of the ion pairs should be considered as part of the overall electrostatic interaction. In analyzing the electrostatic effect of these dipoles, it will be necessary to know not only their number but also the distance between the two ions of the dipole. As a first approximation to such an analysis we could assume that all molecules in a pure electrolyte solution are either electrostatically free or bound in a pair. For NaCI, for instance, in the ranges considered in Table I, we can consider the activity to represent the number of free ions and the difference between concentration and activity t o represent the number of dipoles. Further, we could assume that the two ions making up the dipole are separated by a multiple of solvent molecules. This model (including dipoles as well as point charges) then could be used to look for a better fit of the data in Table I. In applying the equations to mixtures of electrolytes, we have to learn more of the characteristics of the capture radius, r 3 . This quantity will vary, of course, for different ions because of the different ionic radii. For any particular ion, the radius of closest approach of another charged particle is a function of the type of particle. In a mixture of many different carriers of the charge the resulting average r3 will be a function not only of the ion whose activity we wish to measure but also of the numbers and types of the other ionic species. It was encouraging to see that the capture radius of very dilute NaCl (Table I) is very nearly equal to the crystallographic radius. If, by proper choice of n, (the ion density), other electrolytic solutions could be shown to exhibit capture radii equal to their crystallographic radii, then the choice of r, would no longer be arbitrary, It is hoped that with the refinement of electrochemical techniques for measuring activities, a better understanding of activity coefficients and perhaps a clearer understanding of particle interaction in solutions can be gained. ACKNOWLEDGMENT The author gratefully acknowledges the assistance of Dr. Carlos J. Sambucetti of IBM, whose advice during discussions of this paper was very helpful.
RECEIVED for review February 6, 1970. Accepted August 3, 1970.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
