1275
Anal. Chem. 1083, 55, 1275-1280
Determination of Mean Activity Coefficients with I on-Selective Electrodes Roger G. Bates* and Andrew G. Dlckson Department of Chemistry, University of Florlda, Gainesville, Florida 326 1 1
Mlklos Gratzl, Andrea Hrab6cry-P$XI, Ern0 Llndner, and Ern0 Pungor Institute for General and Analytical Chemistry, Technical Llniversity, Budapest, Hungary
Electromotive force (emf) measurements of cells without liquid Junction have been used to shed light on the factors influencing the accuracy of mean ion activity coefficients determlned with ion-selective electrodes. For this study, NaNO,, KNO,, and Ca(NO,), were used as model compounds. Their activity coefficients In aqueous solution at 25 "C were determlned both in dilute solutions and at molalities as hlgh as 2 to 4 mol kg-' by using electrodes responsive to Ida+, K+, Ca2+, and NO3- and different accepted evaiuatlon teclhniques. The results were compared wlth the literature data for these salts. A general nonllnear regression method Incorporating the extended Debye-Huckel equation was developed. By use of experlmentai and simulated data, It was found that reliable activlty coefflcients can only be obtained when randm errors do not exceed f0.5 mV, even though the electrode function is strictly linear. Otherwise a strong correlation between the various parameters renders the results inaccurate.
--
Ag; AgClIKCl (0.1 mol dm-3)11MN03(m) c
double junction reference electrode IMNO,(m)lNO,-ISE ( 2 ) The corresponding values of the cell emf (E) are as follows: E M = E o M + S1 log a, - Eref+ Ej (3) (4) where Ej is the potential across the liquid-liquid boundaries in the double-junction reference electrode, Eo is the standard potential of the cell, S is the Nernst slope factor (in theory 59.2 mV), Erefis the potential of the reference electrode, y+ is the mean activity coefficient of the electrolyte, and m is the molality in mol kg-l. The difference between the emf of the two cells is equal to the emf of the following cell without liquid junction: Me-[SE IMNO,(m)l N03--ISE
(5)
Then Mean activity coefficients of electrolyte solutions are often determined by measuring the emf of suitably chosen galvanic cells, with or without liquid junction ( I ) . The introduction of ion-selective electrodes (ISEs) has remarkably broadened the range of electrolytes which can be studied by this potentiometric technique. The purpose of the present work was t o establish how and under what conditions ion-selective electrodes can be used to determine mean ion activity coefficients with adequate accuracy. Among others, sodium-sensitive glass electrodes combined with a silver-silver chloride electrode or a fluoride electrode have been applied successfully to deteirmine mean activity coefficients for sodium chloride in the molality range 0.13-2.18 mol kg-l and for sodium fluoride between 0.001 and 1 mol kg-l (2). It appears, however, that the ion-exchanger-based nitrate ISE has not yet been used for this purpose. In the course of our present work, we have employed a sodium-sensitive glass electrode combined with a nitrate ISE to determine mean ion activity coefficients for sodium nitrate in aqueous solution. In addition, the nitrate electrode was combined with other cation-sensing electrodes in attempts to determine similar data for other alkali and alkaline-earth nitrates.
EXPERIMENTAL SECTION (A) Cell Arrangement. The mean alctivity coefficients of NaN03, KNOB,and Ca(NO& in aqueous solution were calculated from the emf of suitable galvanic cells. For a univalent cation M+, the following cells were studied: Ag; AgClIKCl(O.1 mol d m ~ 3 ) 1 1 M N 0 , ( ~ ~ ) v
I'
double junction reference electrode IMNO,(m)IM+-ISE (1) 0003-2700/83/0355-1275$01.50/0
-
EMN03=EN0a EM =EoN03 - E o M
- (Sl + S2) log my3 =EDmo3- Slzlog m - S12log y3
(6)
where E 0 ~ ~=oEoNOs a - EoMand S12= SI + S2. For bivalent cations the equations are to be modified accordingly. (B) Electrodes. As a reference electrode, a home-made double-junction Ag,AgCl reference electrode was used (3). For sensing nitrate, three Orion nitrate electrodes, Model 93-07, were employed throughout the work; for sodium, a Corning sodiumselective glass electrode (Catalog No. 476 210) was used. For the measurement of potassium, home-made PVC membranes containing valinomycin as active material and different plasticizers (dipentyl phthalate, dioctyl sebacate) (3.0% valinomycin, 30.1% PVC, and 66.9% plasticizer ( 4 ) )were used. These membranes were mounted in a Philips IS-560 electrode body. In addition, use was made of an Orion potassium-selectiveelectrode (Model 93-19) and a Beckman cation-selective glass electrode (Catalog No. 39 137). For measurements in Ca(N03)zsolutions, a neutral-carrier based electrode was prepared from 0.9% ETH 1001 ligand, 64.3% o-nitrophenyl octyl ether, 0.4% sodium tetraphenyl borate, and 34.4% PVC (5). A PVC membrane electrode containing Orion 92-20-02 ion exchanger for calcium (6), as well as an Orion calcium electrode (Model 93-20-01)and an Orion divalent cation electrode (Model 93-02) were also employed. (C)Reagents. All reagents were of analytical reagent grade. Solutions were prepared with water distilled in Pyrex glass. Potassium nitrate and sodium nitrate crystals were pulverized in a porcelain mortar and then dried at 120-125 OC under vacuum (1-2 torr) and stored over silica gel in a vacuum desiccator. Owing to the hygroscopic nature of the salt, calcium nitrate solutions were prepared as follows. Ca(N03)z.4H20(400-5001 g) was dried in an oven at 120-140 O C for 24 h (incongruentmelting at 40 OC (7)) and then pulverized in a porcelain mortar and heated in a muffle furnace at 250 "C for at least 24 h. Then the material was allowed to cool in a vacuum desiccator over Mg(C104)2, weighed immediately after cooling, and dissolved in a weighed 0 1983 Amerlcan Chemical Soclety
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 8, JULY 1983
Table I. Emf Data for NaNO, no.
mNaNO,
1
0.002 0.005 0.008
2 3 4 5 6 7
8
9 10 11
12 13 14 15 16 17 18 19
0.010
0.020 0.050 0.070 0.10
0.20 0.30 0.40 0.60 0.80 1.00
1.20 1.40 1.60 1.80 2.00
Table 11. Emf Data for KNO, from Three Separate Cells emf (mV) Corning Na+ vs. Orion NO,--1SE 14.8 -30.4 -52.8 -63.7 -96.0 -136.9 -161.8 -173.0 -203.9 -220.9 -233.3 -250.0 -263.2 -273.7 -280.7 -286.5 -292.3 -296.3 -300.9
amount of water. The solution thus obtained was used as a stock solution for preparing solutions of lower concentration. (D) Apparatus. Emf measurements were carried out with a Corning Model 130 digital pH meter with an Orion Research Inc. (Model 605) manual electrode switch. A Thermovac (Model CTB 50) thermostat, a YSI (Model 72) proportional controller, and a Hewlett Packaxd (Model 2801A) quartz thermometer were used for control and measurement of temperature. All measurements were performed at 25.00 f 0.01 "C. (E) Calculations. The data were analyzed with a Radio Shack TRS-80and an HP-85 type desk-top computer. Least-squares minimizations were carried out by using a modified Marquardt algorithm (8). The procedure was checked in several cases for the absolute minimum by the grid method.
RESULTS AND DISCUSSION Values of the emf of cell 5 for solutions of NaN03, KNOB, and Ca(N0J2 are presented in Tables I, 11, and 111, respectively. Emf data in Table I are rounded averages of three parallel values and those in Tables I1 and I11 are rounded averages of values measured in a given time interval (5 points within 1 h). It was found during the measurements that neutral carrier cation electrodes cannot be used in concentrated nitrate solutions, due to anion interference (4, 9, 10). At molalities exceeding 0.1 mol kg-', the absolute value of the slope of the calibration curve decreases as the concentration increases, and even its sign can change. The anion interference could be reduced, but to a limited extent only, by incorporating tetraphenyl borate anions into the membrane, as suggested in the literature ( 4 , I I ) . A similar, although smaller, anion interference was observed for calcium electrodes containing phosphoric acid esters as the active component (12). (A) Method. Methods widely used for determining mean activity coefficients from emf data for concentration cells without transference have been summarized by Robinson and Stokes ( 1 ) . The essence of these methods is that the ratio of the activity coefficients at two different molalities can be determined from the emf data and, provided that one activity coefficient is known, that for the other solution can be calculated. In some cases the potentials of the two half cells are measured separately vs. a common reference electrode, and the mean activity coefficient is calculated by eq 6. The standard emf of the cell (E")and the slope of the calibration curve for the cell (SI2)must be known or determined separately. In a method widely used for determining the standard emf, cell emf data are measured at relatively low molalities where
no. 1 2
3 4
~ K N O ,
0.001
0.002 0.005 0.008
5
0.01
6 7 8 9
0.02 0.05 0.07 0.10 0.20 0.30 0.40 0.60 0.80
10 11
12 13 14 15 16 17 18
19 20 21 22
emf (mV) Beckman cation electrode vs. Orion NO,- electrode no. 2 no. 1 no. 3
1.00
1.20 1.40 1.60 1.80 2.00 2.50 3.00
110.4 88.4 37.3 14.7 4.8 -27.5 -69.3 -83.2 -97.8 -126.1 -142.8 -153.9 -169.0 -178.9 -186.2 -191.8 -195.6 -198.8 -202.6 -204.3 -209.6 -216.0
109.1 88.6 37.9 15.5 5.8 -26.2 -68.0 -82.1 -96.7 -126.1 -141.7 -152.8 -167.7 177.8 -185.1 -190.7 - 194.5 -197.7 -201.3 -203.1 -208.2 -214.9
110.3 89.5 38.6 16.3 6.5 -25.5 -67.2 -81.1 -95.9 -124.9 -140.6 -151.6 -166.7 176.7 -183.9 -189.5 -193.5 -196.6 -200.5 -202.2 -207.5 -214.0
the values of log y+ can be estimated with sufficient accuracy by the Debye-Huckel equation. From the emf data measured and log ya calculated, the values of E' = E S12log my+ (7)
+
are determined. From this, it follows from eq 6 that lim E' = E"
(8)
r n 4
Thus E" can be obtained by extrapolating E' data to zero molality. The extrapolation is usually done graphically. For a given set of points, the better the ya values are approximated by the Debye-Huckel equation, the greater is the accuracy of the value of the standard potential thus determined. When the Debye-Huckel equation
A~z+z-JP/~ log y* = -
= log YDH (9) 1+ B a N 2 is used for calculating log y+,the E' vs. m relationship is often approximately linear, and hence it is easy to carry out the extrapolation. In eq 9, A and B have the values 0.509 and 0.328, respectively, for aqueous solutions at 25 "C,I is the ionic strength, z+ and z- are the charges of the cation and anion, respectively, and a is the ion-size parameter in A. For approximate calculations, the latter is often taken to be 4 A for univalent ions. For the calculation of mean activity coefficients in a higher concentration range, an extended form of the Debye-Huckel equation may be used log y+ = log YDH + CI + DP (10)
where C and D are constants, usually determined by fitting eq 10 to experimental data. In this way the reason for the linearity of the E' vs. m relationship often observed can be seen. Thus if log ya, calculated by using the first two terms on the right, is inserted into eq 6, one finds E = E" - S12log m - SI2log YDH - SI2CI (11) On the other hand, since I = m for 1:l electrolytes, and
E'=a + om = E + S12log m
+ S12log YDH
(12)
ANALYTICAL CHEMISTRY, VOL. 55, NO. 8 , JULY 1983
1277 .-
Table 111. Emf Data for Ca(NO,), with Different Electrode Combinations of Two Designs of Calcium ISE and Two Different Nitrate ]Electrodes emf (mV) Orion divalent Orion divalent rn CaW 0 ,)* calcium ISE' cation electrode calcium ISEa cation electrode no. 2 3 4 5 6
7 8 9
10 1l 1%
13 14 15 16 17 18
146.5 121.5 89.6 73.7 67.1 15.6 5.8 -5.6 -23.2 -34.1 -40.3 -49.9 -55.3 -57.3 -60.4 -66.0 -69.1 -71.9
-80.6 -105.5 -139.3 - 15'7.6 -168.2 -226.5 -24 1.9 -25 1.8 -264.5 -280.6 -290.0 -29!3.5 -306.5 -31 1.7 -315.9 -32 2.3 -32'7.9
0.001 0.002 0.005 0.008 0.01 0.05 0.07 0.10 0.20 0.30 0.40 0.60 0.80 1.00 1.20 1.60 2.00 4.00
3.
-82.2 -106.1 -139.5 -157.9 -168.6 -226.6 -242.2 -252.0 -264.4 - 280.0 -289.2 -298.6 -305.6 -311.3 -315.5 -321.8 -327.4
144.9 120.9 89.4 73.4 66.7 15.4 5.5 -4.8 -23.1 -33.4 -39.6 -49.0 -54.4 -56.6 -59.9 -65.5 -68.6 -71.3
' Orion ion exchanger 92-20-02 in PVC membrane. Table IV. Deviation of Mean Molal Activity Coeffiicientsa
mo,
NaNO,
-
data set 1 data set 2 datar set 3 s(Y+)
0.007
0.072
0.115
0.1099
Calculated by using eq 16, from literature data '(1)for NaNO, (Table I ) and KNO, (Table 11, data sets 1, 2, and 3) with extrapolation using points 1-7 (NaNO,) and 1-8 (KNO,). a
it follows that the parameters cy and p of linear regrefmionare as follows: a = Eo
1V
M
RSS = G C [Ei(measd) - Eij(calcd)12 i:;lj=1
(16)
where for each cell j
(13)
p = -S& This means that the extrapolation procedure outlined earlier ( I ) implies the explicit determination of Eo and implicit determination of C. A quadratic extrapolation-whic h should yield a result of greater accuracy-means an implicit determination of the constants of the extended Debye-Huckel equation containing also a quadratic term. (B) Statistical Analysis. Instead of performing a graphical extrapolation, the coefficients of eq 11can be obtained by a least-squares technique. The least-square81 solution chosen is that set of parameters ( E O , C, ) for which the residual sum of squares
...
N
[Ei(measd) - Ei(calcd)12
RSS =
as an adjustable parameter, in addition to the electrolyte dependent parameters (a, C, ...) of the extended Debye-Huckel equation. Furthermore, there exists the possibility of a slow drift of Eo with time. A variety of computational and experimental procedures have been suggested in the literature to deal with this (II,I3-16).We prefer to make a series of measurements using M independent cells &e,, several indicator electrodes in one solution). In this case it is necessary to minimize
i=l
is a minimum. N is the number of experimental points, and Ei(calcd) is obtained by using eq 11with a quadratilc term if desired. When ion-selective electrodes are used for these (determinations, it is often found that the use of a theoretical (Nernstian) value of the slope (SI*)is not justified. The slope then has to be determined experimentally, e.g., at a constant ionic strength. This often involves difficulties, as tlhe interference from the species used to adjust the ionic strength is usually not negligible. One advantage of a computerized least-squares approach over a graphical approach is that it provides a statistically valid solution. A further advantage is gained in that it is possible to treat the electrode sllope (&)
and log y+(rni)is given by eq 10. It is possible to reduce the chance of serious convergence problems by decomposing this minimization problem in the following manner (17,18). (i) Assume reasonable values for a, C, (D)and calculate log y+(rni). (ii) Obtain estimates of E,"and S, by a simple linear regression procedure. (iii) Hold these values of E; and S, constant and adjust a, C, (D)so as to minimize RSS. (iv) Calculate revised values of log y+(rnJ. Steps (ii), (iii), and (iv) are repeated until the solution converges to within the required tolerance. The accuracy of the results yielded by the minimization procedure (eq 16) can be characterized by the standard deviation from literature values of mean activity coefficient data calculated by using the parameters determined by this procedure, as follows:
where yi(ca1cd) is the mean activity coefficient determined by calculation for the ith solution and yi(lit.) is the mean activity coefficient for the ith solution, taken from the literature ( I ) . Some s(ya) data calculated for NaN03 and KN09are shown in Table IV. Only the data obtained for NaN03 are satisfactory. We have assumed that this may be due to our use in the minimization procedure of data for dilute solutions alone. This means that the calibration curve determined in
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 8, JULY 1983 EMF [mV]
Table V. Deviation of Mean Molal Activity Coefficients Calculated by Equation 1 6 over the Full Range of Molalities
NaNO, KNO, Ca(NO,),
data set 1 dataset 2 dataset 3 data set 1 dataset 2
s(r*)
RMS D,a mV
0.069
1.12
0.084
1.13
0.022 0.017 0.008
1.91 2.09 2.05
0.014 0.014 0.005
1.90 2.05 2.03
0.127 0.128
2.25 2.40
0.128
2.39
I
RMSD,~ mV
S(Y,)
L
y* estimated by using the Debye-Huckel equation extended by a linear term. y* estimated as before but
1
\
a
-3
-2
-1
0
+ '
KNO, NaNO,
including also a quadratic term.
dilute solutions was applied for more concentrated solutions; thus in effect an extrapolation was involved. This extrapolation may introduce significant errors since SI2for ion-selective electrodes may not be considered as strictly constant over a wide concentration range. Consequently, in later attempts minimization according to eq 16 was attempted for all the experimental points, those measured in concentrated as well as in dilute solutions. In this way no extrapolation is implied and, furthermore, the statistical reliability of the procedure is greatly increased as the number of points is greater. The minimization procedure embodied in eq 16 has been applied by use of the Debye-Huckel equation extended by a linear term or by both a linear and a quadratic term (eq 10). The results obtained for NaN03, KN03, and Ca(N03)2with the extended Debye-Huckel equation are shown in Table V. The data were also analyzed by the Pitzer equation (19). The results were inferior to those obtained with the DebyeHuckel equation and consequently are not included in Table V. In addition to s(ya) values, it is possible to characterize the deviation of the measured emf data from the calibration curve taken by using y+ values determined by the minimization procedure using the root square deviation
RMSD=
d -
(19)
where p is the number of parameters adjusted in the minimization procedure. As shown by the data in Table V, a calibration extended over the whole range does not necessarily result in a reduction in s(y+). A further problem is that experimental data apparently similar in RMSD (similar minimum values of RSS), yield results highly different in s(y+) (see, e.g., data series 1, 2, and 3), and in some cases lower RMSD data are coupled
ioga
Ca(NO,), Figure 1. Calibration curves for the cells Na+-ISE INaNO,I NO,--ISE, K+-ISE IKNO,INO,--ISE, and Ca2+-ISEICa(NO,),INO,--ISE. with larger values of s(yf) (e.g., for NaN03, RMSD is about half that for KNOB,whereas s(y+) is much smaller for KN03 than for NaN03). As the results thus obtained were not satisfactory, we have attempted to separate experimental and calculation errors. As a first step, the linearity of the calibration curves determined by using the emf values measured and activity coefficient data taken from the literature was studied (Figure 1) and the residual error (RMSD with p = 2) of the experimental points around the calibration curve determined by the method of least squares was calculated for different concentration intervals. The results are summarized in Table VI. For comparison, some results calculated by using literature data are also included in the table (13, 15). As the results show, the residual error is 1-2 mV when calibration is extended to the whole concentration range studied, except for calcium nitrate, for which the calibration curve deviates from linearity at high concentrations due to anion interference (Figure 1). A comparison of the data in Tables V and VI reveals that the error (RMSD) is somewhat smaller if the activities are calculated by using y+ values determined in the minimization procedure (Table V). This result indicates that a t the level of experimental errors (systematic and random) inevitably encountered (Table V), the minimization procedure is not suitable for determining y+ values with adequate accuracy, that is, a simple search for the minimum in RSS is not an appropriate criterion. The systematic nature of the experimental errors is suggested by the fact that the results obtained by using activity coefficients calculated from experimental emf values have a smaller error
Table VI. Root Mean Square Deviations (RMSD, eq 20) and Correlation Coefficients Determined for Measured Potential Data by using Literature Activity Coefficients data points data points RMSD, mV useda RMSD, mV r2 used a 1.000 1-19 1.29 0.30 1-8 NaNO, 1-22 2.04 3.33 1-8 KNO, data set 1 1.000 1-22 2.14 1-8 3.60 data set 2 1-22 2.09 3.55 1-8 data set 3 1-16 1-8 6.92 2.83 data set 1 CaWO,), 0.994 1-16 1-8 6.92 2.64 data set 2 data set 3 data set 4
a
0.48 0.24
CaClJShatkay ( 1 5 ) NaCUShatkay (13) Numbers identify solutions in Tables I, 11, and 111.
1-8 1-8
6.66 6.60 1.27 0.57
0.974
1-18 1-16 1-16
1,000
16-23 6-14
1.000
ANALYTICAL CHEMISTRY, VOL. 55, NO. 8, JULY 1983
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Table VII. Comparison of ~ ( 7Data ~ )(for Full Range Calibration)
