373
Anal. Chem. 1985, 57,373-375
Quantization Error with the Single- and Double-Known Addition Method in Ion-Selective Potentiometry Sir: As a sequel to an article on classical error propagation in connection with the double-known addition method (DKAM) ( I ) , previously unpublished work concerning the single-known addition method (SKAM) as used with ion-selective electrodes (ISEs) was expanded to cover the DKAM case; artifacts due to the quantization of the measured electromotive forces (emfs) by a digital voltmeter (DVM) are investigated. An error-propagation analysis of the standard addition method using ISEs involves at least seven equations with the uncertainties of the associated parameters and variables, c.f. ref 2, namely (a) dilution of the sample (concentrations, volumes), (b) ionic strength (concentrations of all ionic species), (c) activity coefficients (e.g., the extended DebyeHuckel equation with two adjustable parameters and one constant (all temperature dependent) and the ionic strength), (d) liquid-junctionpotential (e.g., the Henderson equation with ionic mobilities, concentrations, and activity coefficients), (e) calibration function (e.g., the Nernst equation with the activity of the primary ion, the slope (temperature dependent), or the Nicolsky-Eisenman equation with-in addition-the activities and selectivity factors of the interferents), (f) the DVM equation (truncates the true emfs to digital readings in q = 0.001, 0.01, 0.1, or 1mV steps, depending on the instrument used, (9) the standard-addition equation (involves all of the above). As eq f is not a continuous function but a staircase function, the step height q cannot be interpreted as a standard deviation in the usual sense, especially if this error contribution is large relative to other contributions. The quantization effect can be singled out and visualized by means of numerical simulation, as will be shown below. If all error contributions are to be taken into account, the complexity of the system of equations necessitates the use of the Monte Carlo method (3, 4).
For a less-than-rigorous analysis, the following assumptions may be proposed: (h) the dilution effect of adding concentrated analyte is negligible (eq a), (i) the ionic strength hardly changes upon addition of analyte and thus the activity coefficient(s) remain(s) constant (eq b and c), (i) because of (i) and out of convenience, activities are set equal to the corresponding concentrations, (k) the liquid-junction potential is invariant (eq d), (1) because the electrode is perfect or no interferents are present, the Nernst equation applies instead of the Nicolsky-Eisenman equation (eq e). For various types of analyses, one or more of the above simplifications may be justified but not all at once, because optimization of one aspect, e.g., 6) by means of ionic strength buffering (41, usually entails disadvantages, such as (l), i.e., detection-limit and linearity problems. Thus, if only the Nernst equation (eq e) is invoked and (i)-(1) are used
El = Eo + s In c,
(m)
E2 = Eo + s In (co + Ac)
(n)
with El and E , being the true emfs of the ISE for the concentration of the sample before addition (cot to be estimated) and after addition (c, + Ac). The slope of the calibration plot emf vs. In (activity) is s. c, + Ac is calculated as (couo + clul)/(uo + ul), where c1 is the concentration of the added
solution, u, is the volume of the sample before addition, and + u1 is the volume after addition. Equation g is found as
uo
Assuming the measurements entering into eq m, n, and o are free of emf and concentration errors (within the limits imposed by calculator round-off artifacts), any values for Eo,uo, CO, c1, ul, and s entered into eq m and n will yield Eo (from eq 0) identical with co. Quantization effects require numerical examples for demonstration; if one lets eq f operate on El and E2 in the following specific case one finds the conditions co = 1.0 mM Na+, Eo = 177.4800 mV, s = 59.1600 mV (25 “C, logarithm to base 10) and the results ;o
Ac, mM
emftheor,
0.1 1.0 3.0
2.4488 17.8089 35.6179
mV
(mM) in emf
quantization step q (mV) q=1 q = 0.1 q = 0.01 1.2353 1.0601 1.0327
1.0213 1.0007 1.0009
1.0038 1.0007 1.0004
Eo was chosen so that El = 0.00 mV, which simplifies the numerical work. For a correct analysis, any value 0 to qmax = 1 mV could be added to Eo (Monte Carlo technique); thus for E , = 177.4800 + 0.3725 = 177.8525 mV, the middle line of the above table would change to Eo = 0.9853,1.0007,0.9999 mM, respectively. Obviously values near 1.0 mM result, but the effects are systematic, not random. The problem was posed to chemistry students taking an introduction to BASIC programming as part of an analytical chemistry lab course, the idea being to simulate a Ca2+ISE, a DVM, and an automatic buret and to find the most cost-effective combination of commercially available instruments that would satisfy the requirement cou (C,) I 2%. To this end, calcium was “added” to a 20-mL sample of 1 mM Ca2+in steps of 1, 25, and 100 pL at different concentrations, such as 5,10, 50, and 100 mM. In the present work only results pertaining to 25-pL steps and the addition of 10 mM Ca2+are shown. The quantization of the DVM was taken to be either 0.01, 0.1, or 1mV (see Figure 1). Equation a was taken into account. Instead of a straight line one gets a modified saw-tooth function. An analyst aware of the problem could measure El and by proper choice of the added volume produce any value within f0.62% (q = 0.1 mV) or f3% ( q = 1mV) of the true co (these values were read off the full-sized figures for u1 = 3-6 mL). Figure 1 shows the effect of the DVM quantization. If higher concentrations of titrant are used (see Figure 2), the quantization error of 2, decreases somewhat at the expense of higher uncertainty in eq a. Random noise in the concentrations is directly superimposed; random noise in the added volume u1 is transformed via the saw-tooth function from an abscissa to an ordinate value (dilation or compression of this function in the volume direction); random noise in the true potentials El and E2 by way of eq f has an effect especially pronounced near the vertical transitions of the saw-tooth function. So, unless random noise in the emf readings is much greater than q, quantization effects must be expected. In the case of the double-known addition method, the implicit eq p must be
0003-2700/85/0357-0373$0 1.50/0 0 1984 American Chemical Society
374
ANALYTICAL CHEMISTRY, VOL. 57, NO. 1, JANUARY 1985
h
1 m 0 2x 3
m l 0.01H Ca
m l
0.01M C a
m l 0.01M C a
Figure 1. Singleknown addition method: 20 mL of 1.0mM Ca2+are titrated with x mL of 10.0 mM Ca2+solution. A perfect Ca2+ISE is assumed. Three differentDVMs are simulated: ordinate scale, expected concentration, E o = 1.0 mM f 5 % ; abscissa scale, volume added, buret with 25-pL step size. The peak-to-peak "noise" in the region 3-6 mL (b) is equal to about 1.2%,Le., *OB%. The seemingly smooth arcs that appear in several figures are artifacts due to the sampling process, here an emf measurement after every addition of 25 pL, that would evolve into a saw-tooth pattern If the sampling rate were chosen sufficiently high.
a
DVM
reeolution
:
0. 1
mV
b
solved by iteration (1) z=
reeolution DVM : 1 m V
1'' 1 with El, E2, c1, ! .g5L 0
1
2
3
4
5
6
% 0
(P)
,
.950
1
m l 0. 1M Ca
2
3
4
5
Eo, uo,
6
ul, and s as above and
m l 0.1M Ca
Figure 2. Same as Figure 1 but with titrant concentration c 1 = 0.1 M.
a
1.05r
.109-r
1 p~
resolution DVM
1
1
2
3
4
5
1.05-
6
reeolution DVM : 0.1 m V
1
V2, m l 0.01M Ca
2
3
4
5
V2, m l 0.01M Ca
Figure 3. Double-known addition method: to 20 mL of 1.0 mM Ca2+ solution a fixed volume, v , = 2 mL, of 10.0 mM Ca2+is added and the emf is recorded as E,. The resultlng 22 mL is titrated with v2 mL of 10.0 mM Ca2+solution. A perfect Ca2+ ISE Is assumed. Two different DVMs are simulated. Note that the quantization nolse for q = 0.1 mV is =f1.4% or twice as large as the corresponding value for the SKAM!
V1
r
2 ml
c1 was chosen to be 10 mM and u1 was fixed at different values in the range 0.8-6 mL; u2 was then "scanned; Le., a titration was simulated. For every combination of El, E2, and E3 (appropriately clipped by eq f), the regula falsi with toas the independent variable was applied until IzI 5 In Figure 3 the DKAM simulations corresponding to Figure 1 (SKAM) are plotted. Note that the DKAM gives rise to larger quantization noise than the SKAM if the same DVM is used. This is in part due to the choice of ul; if a small change in u1 induces the truncated El to undergo a transition (e.g., El' = El f q , q = 0.01 mV), a bias is introduced. In Figure 4b,c the transition from u1 = 0.8 to 1.0 mL changes the resulting Eo by about -0.35% (u, = 6 mL). In figures not reproduced here, a change from u1 = 4.075 to 4.10 mL (u, = 6 mL, c1 = 0.01 M Ca2+,q = 0.01 mV) changes C0 by about +1%. The above is a simple demonstration of quantization noise. A thorough analysis would require the application of the Monte Carlo technique to the full set of eq a-g. Needless to say, quantization effects are also found if eq e is solved for the selectivity factor of an interfering ion, as with the separate-solution method.
V1
=
1
V1 = 0.8 m l
ml
0 3
1'
A
V2,
rnl 0.01M Ca
V2, m l 0.01M Ca
V2,
m l 0.01M Ca
Figure 4. Same as Figure 3 but with 9 = 0.01 mV and v , = 2, 1, and 0.8 mL. The uncertainty in B o amounts to about f0.2%. For v i 4 mL, similar figures apply but A t o increases to about f0.4%. T h e difference between v i = 4.075 and v,' = 4.10 mL (Le., the resolution of the automatic buret) suffices for a change in appearance as radical as that between Figure 4a and c. The effect is due to the fact that between v , and v,' the indicated potential E , changes by q = 0.01 mV; at the same time the bias introduced into E o changes sign.
375
Anal. Chem. 1985, 57, 375-376
ACKNOWLEDGMENT I thank W. Simon, ETH-Z, for letting me use his HP85/plotter system and G. Horvai for his critical comments.
LITERATURE CITED Horvai, G.; Pungor, E. Anal, Chem. 1983, 55, 1988-1990. (2) Meier, P. C.;et al. In "Medical and Biological Applications of Electrochemical Devices"; Koryta, J., Ed.; Wlley-Interscience: Chicester, 1980; Chapter 2.
(3)
Schwartz, Lowell, M. Anal. Chem. 1980, 52, 1141. C. E.; Hadjiloannou, T. P. Anal. Chem. 1982, 5 4 ,
(4) Efstathiou,
1525.
Peter C. Meier Ed. Geistlich Sohne AG CH-6110 Wolhusen, Switzerland
(1)
RECEIVEDfor review January 10, l984. Accepted August 31, 1984.
Silane Isomer Effect on the Capacity of Silica-Immobilized 8-Quinolinol S i c Immobilization of 8-quinolinol on silica surfaces is generally performed by diazo coupling of 8-quinolinol to the diazonium salt prepared from a silica-immobilized aromatic amine. The most recent synthetic procedure initially involves reaction of the silica surface with the mixed isomers of (aminopheny1)trimethoxysilane which is commercially available (1). The immobilized aromatic amine is then diazotized and coupled to 8-quinolinol. It has been indicated that the ortho isomer of the silane probably cannot become coupled to 8quinolinol since, once attached to the silica surface, the reactive amine functional group is in close proximity to the silica surface thereby prohibiting the coupling reaction to the rather bulky 8-quinolinol molecule (I). This prediction was also indicated by examination of molecular models of the bound species at each step of the immobilization reaction in order to study the effect of pore size and the related surface area on capacity (2). The above prediction has been corroborated in this work by separating the ortho isomer from the commercially available mixture and then performing the synthesis with the ortho isomer only. An expected increase in capacity when performing the synthesis with a silane fraction which lacks the ortho isomer, however, was not observed. EXPERIMENTAL SECTION Reagents. (Aminopheny1)trimethoxysilane (Petrarch Systems, Bristol, PA) was obtained as mixed isomers and stored under nitrogen in a refrigerator. Water for solution preparation was deionized by reverse osmosis followed by distillation in borosilicate glass still containing a quartz immersion heater. All other solvents and reagents were AR grade and used as received. Silica gel (Woelm TLC grade, ICN Pharmaceuticals, Cleveland, OH) was acid-washed with 0.1 M HC1 and dried in an oven at 120 "C for at least 12 h prior to use. Apparatus. A Hewlett-Packard Model 5880 Level Four gas chromatograph (Palo Alto, CA) equipped with an OV 101 methylsilicone oil column (Analabs, North Haven, CT, 39 m x 0.5 mm i.d.) and flame ionization detector was used f6r analytical separation of the isomers. Identification of isomers was performed via *H and 13C NMR spectroscopy on a Varian Model XL-300 FT NMR spectrometer (Palo Alto, CA). Preparative separation of the ortho isomer from a mixture of meta and para isomers was achieved by vacuum distillation with a 10 cm X 1.0 cm i.d. packed fractionating column at 0.2 mmHg. Procedure. Following fractionation of the commercially available silane by vacuum distillation, the various fractions were used for preparation of silica-immobilized 8-quinolinol by the previously described procedure ( I ) . The capacities of the resulting materials were then determined by using copper(I1) as the probe at pH 5.0 ( I ) .
RESULTS AND DISCUSSION The capillary gas chromatogram of commercially available (aminopheny1)trimethoxysilane is shown in Figure 1. The
B
i
i
~
,
0
~
5
~
,
,
r
IO
TIME (min) Figure 1. Capillary gas chromatogram of commercially available (aminophenyl)trimethoxysilane,mixed isomers: OV 101 column, 39 m X 0.5 mm Ld.; column temperature, 210 O C (isothermal);injection
port temperature, 200 O C ; detector temperature, 250 O C . three major peaks (A, B, and C) correspond to 29.2%, 45.6%, and 22.6% of the mixture, respectively, which indicates that appreciable quantities of all three isomers of the silane are present in the commercially available product. The stationary phase (OV 101, methylsilicone fluid) is typically of low selectivity, and separation according to boiling points is normally observed (3). Therefore, isomer A is thought to be the ortho isomer. The ortho isomer is the only one of the isomers that can undergo intramolecular hydrogen bonding in lieu of the intermolecular hydrogen bonding observed with meta and para isomers. The intramolecular hydrogen bonding thus lowers the boiling point. This effect is known for similar species such as anisidine (metboxyaniline), aminophenetole (ethoxyaniline), and aminophenol. However, compounds such as toluidine (methylaniline) show no such effect. Also, IH and 13C NMR spectroscopy of the first distillate indicate the ortho isomer. The lH NMR spectrum shows two doublets and two triplets which integrate for a total of four protons in the aromatic region of the spectrum. Two upfield singlets integrating for two protons and nine protons are indicative of the amine and methoxy protons, respectively. The proton decoupled 13C NMR spectrum gives seven peaks as expected; six in the aromatic region and the single upfield methoxy signal. The meta and p&a isomers have much closer boiling points and their separation from one another is more difficult. However, examination of molecular models indicated that no
0003-2700/85/0357-0375$01.50/00 1984 American Chemical Society
~
~
r
