Anal. Chem. 1994,66,830-835
Models for Conductance Measurements in Quality Assurance of Water Analysis Stephen 0. Hughes, Elizabeth L. Taylor, and Peter D. Wentzell' Trace Analysis Research Centre, Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, B3H 4 3 , Canada
Ross F. McCurdy and Robert K. Boss Fenwick Laboratories Ltd., Suite 200, 5595 Fenwick Street, Halifax, Nova Scotia, B3H 4M2, Canada Models for the prediction of conductance in nonbrine water samples through the measurement of ionic concentrations and other parameters are compared. Such predictions are often used for quality assurance purposes by comparing them with actual measurementsto determinewhether gross analysis errors have been made. A currently recommended method for making such predictions is a semiempirical relation that adapts the Debye-Hiickel-Onsager equationto mixed electrolytesystems by incorporating modified definitions of ionic charge and concentration. The limitations of this model are examined, and extensions to it are considered. Other predictive methods, including multiple linear regression (MLR), principal components regression (PCR), partial least-squares (PLS) regression, continuum regression (CR), and neural networks (NN), are also considered. Models employ the concentrations of 10 ions as well as, in some cases, additional water quality measurements. Best results were obtained with an extended form of the Debye-Hiickel-Onsager equation and an optimized MLR model. PCR, PLS, CR, and NN did not offer significant advantages. Perhaps no resource is as important to a region as its water, and this is increasingly the case. The analysis of nonbrine waters is critical in a number of areas including public health (e.g., the assessment of potability), industrial monitoring (e.g., boiler or cooling water), and pollution control (e.g., wastewater management). Procedures for making chemical and physical measurements on such samples are ~ell-established,l-~ and an enormous number of routine analyses are conducted worldwide each day. Because of the importance of the results of such analyses and the large number that are carried out, quality assurance becomes an especially important concern. Various methods are commonly used for this purpose, including the use of control samples. While these can test the reliability of the analytical methods employed, it is still possible to obtain erroneous results for individual samples due to errors in measurement, data entry, or other steps in the procedure. (1) Grcenbcrg, A. E., Clesffiri, L. S., Eaton, A. D., Eds. Standard Methods for rhe Examinafionof Water and Wastewafer,18th ed.;American Public Health Association: Washington, DC, 1992. (2) United Statu EnvironmentalProtection Agency. Methods for rhe Chemical Analysisof Water and Wastes;U.S. GovernmentPrintingOffice: Washington, DC, 1979. (3) American Society for Testing and Materials. Manual on Wafer;The Society: Philadelphia, PA, 1978. (4) Hunt, D. T. E. The Chemical Analysis of Wafer: General Principles and Techniques, 2nd ed.; Royal Society of Chemistry: London, 1986, (5) Fresenius, W. WaferAnalysis; Springer-Verlag: New York, 1988.
030 Anal)rticel Chemk3t?y, Vol. 66,No. 6, Much 15, 1994
One way to circumvent this problem is to run replicates for each sample, but this can be time-consuming and costly. In the analysis of inorganic ions, another method that can be used is measurement of the conductivity of the sample and comparison of this result with that predicted based on the concentrations of major ions. Additionally, this approach can indicate the presence of major ions not included in the model. While this technique has limitations in identifying erroneous results for all ions, it can serve as a part of the overall quality control program. One of the recommended approaches to calculating the conductivity in nonbrine water samples is a semiempirical method that will be referred to as the Rossum model (ref 1, pp 1-12). Details of this model were reported in 1975,6and despite its acceptance as a standard procedure in 1989 and its obvious practical utility, very little has been written concerning its validity or general applicability, not to mention possible alternative models. Furthermore, the initial report was limited to samples with conductivities not exceeding about 1400 rS cm-l. The goal of the work presented here was to conduct a more rigorous evaluation of the Rossum model through a comparison with a range of other models. Included in this comparison were empirical models based on multiple linear regression (MLR), principal components regression (PCR), partial least squares (PLS), continuum regression (CR), and neural networks (NN), as well as semiempirical models based on extensions to the original Rossum model. The data employed were obtained for a wide range of samples processed by a commercial laboratory.
BACKGROUND Rossum Model. The theoretical basis for predicting conductance in dilute solutions of simple electrolytes has been established for many years and is well-treated in several It was Kohlrausch who first correctly observed that, in dilute aqueous solutions, the equivalent conductance, A, varies with the square root of the concentration of a strong (6) Rossum, J. R., J.-Am. Wafer Works Assoc. 1975,67, 204-205. (7) Harned, H. S.;Owen, B. B. The Physical Chemisrry ofElecrrolyricSolufioonr, 3rd ed.;Reinhold: New York, 1958. ( 8 ) Fuoss, R. M.; Accascina, F. Elecrrolyfic Conducfance;Interscienffi Publishers: New York, 1959. (9) Robinson, R. A., Stokcs, R. H. Electrolyte Soluflons,2nd ed.;Buttenvorth: London, 1959. (10) MacInncs, D. A. The Principles of Electrochemistry; Reinhold New York, 1961. (1 1) Atkins, P. W. Physical Chemisrry, 4th ed.; Freeman: New York, 1990; p 760. 0003-2700/94/0368083Q$~.SQ~Q
Q 1994 Amorloan Chemkal Socbty
Tabla 1. Lkt ol Symbok a d Their Untts
equivalent concentration of an ion (mM) equivalent concentrationof a salt = (molar concentration)/z z+ or 1 2- (M)
c, c+, c-
C
C'
D e
; G G+', G-' ll
A, h+, h xo, h + O , L O
A+', h '
A A'
R z, z+, z-
5,z-
composite equivalent concentration for a mixed electrolytesolution (Rossummodel) (mM) dielectric constant (relative permittivity) of solution (no units) proton charge (C) vacuum permittivity (J-1C2 m-1) Faraday's constant (Cmol-') solution conductivity (rS cm-l) composite ionic conductivities in a mixed electrolyte solution (Rossum model) (&3 cm-1). solution viscosity (P) equivalent conductanceof an ion (Scm2 mol-') limiting equivalent conductance of an ion
(Scma mol-') composite limiting equivalent conductance
of ions in a mixed electrolyte solution (Rossum model) (Scma mol-') equivalentconductanceof a solution (Scm2 mol-') limiting equivalent conductance of a solution (Scmz mol-') gas constant (J mol-' K-l) absolutevalue of ionic charge (no units) composite ionic charges for a mixed electrolyte solution (Rossum model) (no units)
electrolyte. He also demonstrated that the limiting equivalent conductance of a salt, Ao, could be expressed as the sum of the limiting equivalent conductances of the individual ions, X + O and Lo. The theoretical basis for the concentration dependence was derived later and resulted in the DebyeHiickel-Onsager equation for a single electrolyte in dilute solution:
A = Ao - (A
+ BX0)C1/'
(1) where Cis the equivalent concentration of the electrolyte and A and Bare constants which depend on solution temperature, dielectric constant, and viscosity, as well as other physical constants. The constant A accounts for relaxation of the counterion cloud around the moving ion, and B compensates for the electrophoretic effect as the ion moves through solution. For 1:l electrolytes in aqueous solution at 25 OC, A = 60.20 S cm2 mol-' M-1/2 and B = 0.229 M-1/2.11 A more general form of the equation, adopted by Shedlovsky,12 gives the equivalent conductance of a single ion formed from a binary salt in dilute solution as
where
28.95 S cm2 mol-l M-1/2 K1/2P (note the latter value will be 2.895 when SI units of kg m-l s-l are used for viscosity). To calculate the conductivity, G, of a solution consisting of a single salt, the equivalent conductance of the anion and cation are added and multiplied by the equivalent concentration. This equation is reasonably accurate for the prediction of conductance in relatively dilute samples, especially for 1: 1 electrolytes. For example, on the basis of data from Robinson and Stokes? 5 mM solutions of NaCl, BaC12, and Lac13 give errors of -0.4%,-1.5% and -5.3%, respectively. Errors arise from the truncation of higher-order terms in the derivation of the Debye-Hiickel-Onsager equation and, at high concentrations, from ion association. The prediction of conductance measurements in water samples is complicated by the fact that they are multielectrolyte systems. The problem of nonbinary electrolytes has been addressed,7J3 but the solution is algebraically complex. Rossum6 treated the problem of multielectrolyte systems semiempirically by considering them as binary salts in which parameters used in the Debye-Hiickel-Onsager equation are essentially *average" or composite values. His equation takes the same form as eq 2 but is presented in terms of conductivity (in pS cm-l): G = G+O
+
G-O
+ Z-)c13/2
[(Z+
-
(3) where Z+Z-A0
w = z z 2Q +
-1
+ Ql/2
= (Z,
+ Z-)(Z+A/ + Z-X,)'
K,= 247~(e,DRT)~/~ K2 = e@
1O'eP 6?r(e,,DRT )'I2q
The constants K1 and K2 are derived from those in eq 2 with adjustments for dimensions. Rossum gave values of K1 = 115.2 mM1l2and K2 = 0.668 S cm2mol-' mM-V2 (or p S cm-1 mM-3/2) at 25 O C . The composite equivalent concentration, C', is in millimolar, and G will be in microsiemens per centimeter. The composite parameters are given as
and the other symbols are defined in Table 1 . Note that conversion factors have been included with a and @thatare valid if the units used are as given in Table 1 . Shedlovsky gives the constants as a = 9.838 X lo5 M-l/* K3l2and B =
The summations areover all positiveor negative ions. Rossum did not provide justification for these definitions other than that they appear logical and seem to work for real samples. The absence of a rigorous theoretical foundation, however, makes the recommendation of this approach as a standard method somewhat questionable. Furthermore, the DebyeHiickel-Onsager equation is known to deviate as the ionic concentration increases.*-10912 Rossum limited his tests to samples below ca. 1400 p S cm-l. Additional terms have been used by other workers to extend the range of the original Debye-Hiickel-Onsager model and the use of these for the
(12)Shcdlovsky, T.J. Am. Chem. Soc. 1932, 54, 1405-1411.
(13) Onsager, L.;Fuoss, R. M.J. Phys. Chem. 1932.36, 2689-2118.
oi3)1/2eP
(1
ff=
24~(e,,R)~/~
@=
67~(e,,R)~/~
Ana&tical Chemistry, Vol. 66,No. 6,M r c h 15, 1994
031
~
Table 2. Statktlcal Summary ol Water Ouallty Parameters
parameter sodium potaesium calcium magnesium sulfate chloride nitrate t nitrite ammonia orthophosphate reactive silica iron manganese copper zinc PH alkalinity TOC conductivity carbonate bicarbonate hardness Langelier index saturated pH
units mM mM mM mM mM mM mM mM mM mM mM mM mM mM mg/L CaC08 mg/L C p S cm-1 mM mM mg/L CaC03
mean value 2.09 0.36 2.17 0.70 0.82 1.94 0.39 0.20 0.01 0.37 0.05 0.01 0.00 0.01 7.54 214 11.6 801 0.02 4.24 287 -0.65 8.18
min value
mas value
Measured Quantities 0.01 71.7 0.00 22.5 0.00 17.5 0.00 8.52 0.01 16.2 0.00 90.3 0.00 7.43 0.00 14.6 0.00 1.19 0.00 1.80 0.00 1.42 0.00 0.29 0.00 0.02 0.00 2.49 4.3 12.0 0 2500 0.5 850 4.1 10400 Calculated Quantities 0.00 0.76 0.00 50 0.0 2100 -9.15 4.47 6.04 15.8
Rossum model is investigated here, along with other empirical models. Other Models. Given the need to predict conductance based on ionic concentration of water samples and the recommendation of the Rossum model as a standard method, it was felt that a more rigorous examination of alternative models should be investigated. These models included extensions and modifications to the original Rossum model, as well as several empirical calibration methods that have appeared in the chemometrics literature. The database used for this investigation included 23 parameters, which are listed in Table 2. Some of these parameters are measured directly, while others are derived. Not all of the parameters measured were found to be useful. For methods based on the Rossum model, only the ionic concentrations were used. For other models, additional parameters were also investigated. The following is a list of the models tested in this work. For ease of reference, each of the models has been assigned an acronym. RM: This is the original Rossum model. MRM: The modified Rossum model uses an alternative method for the calculation of the composite charge. Specifically,
z+= c c + z + / c c +
(4) and likewise for 2-.Rossum claimed that his definition of the composite charge provided better results. This conclusion was examined. ERM: The extended Rossum model includes an additional term in the composite equivalent ionic concentration, C'. The parameter associated with this term is determined empirically. EMRM: The extended modified Rossum model incorporates the ERM and MRM. MLR: This represents a class of models generated empirically using multiple linear regression. MLRa uses the ionic concentrations directly, MLRb uses an optimized set of 032 Analytical Chemhtry, Voi. 66, No. 6, March 15, 1004
SD
median value
percentile 25th-75th
6.17 1.55 2.42 0.86 1.97 7.42 1.18 1.03 0.08 0.25 0.21 0.02 0.00 0.12 0.7 228 64.6 1070
0.66 0.06 1.66 0.55 0.23 0.62 0.01 0.00 0.00 0.36 0.00 0.00 0.00 0.00 7.6 200 1.7 528
0.24-1.71 0.034.18 0.61-2.74 0.15-0.93 0.09-0.52 0.20-1.43 0.00.05 0.00.01 0.00.00 0.18-0.49 0.00.01 0.00.00 0.00.00 0.00.00 7.2-8.0 70-280 0.8-4.9 288-956
0.05 4.56 301 1.73 1.26
0.01 3.94 236 0.03 7.75
0.00-0.02 0.67-2.73 79.4-360 -1.17-0.38 7.39-8.61
ionic concentrations, and MLRc employs variables beyond the simple ionic concentrations. PCR: This represents the best model obtained by principal components regression. PLS: This is the best model obtained by partial least squares. CR: This is the best model determined by continuum regression. NN: This is the best model obtained using back-propagation neural networks. The principles of principal components regression, partial least squares, continuum regression, and neural networks as they apply to calibration problems in chemistry have been described elsewhere and will not be discussed here.
EXPERIMENTAL SECTION Sample Parameters. All of the water samples used in this work were analyzed at Fenwick Laboratories Ltd. (Halifax, NS) using their Rapid Chemical Analysis package (RCAp). Methods used were based on standard procedures for water analysis1p2and will only be summarized here. Metallic ions (Na, K, Ca, Mg, Fe, Mn, Cu, Zn) were determined by flame atomic absorption on a Varian SpectrAA 40 spectrometer (Varian Canada, Georgetown, ON). Colorimetric methods were used for chloride, ammonia, nitrate + nitrite, orthophosphate, reactive silica, and alkalinity. These were carried out on an automated COBAS FARA centrifugal analyzer (Roche Diagnostics, Mississauga, ON). Chloride was determined using mercuric thiocyanate in the presence of Fe2+. The method for ammonia utilized the formation of the blue indophenol by reaction with hypochlorite and alkaline phenol (phenate method). Nitrate was reduced to nitrite with hydrazine sulfate and total nitrate plus nitrite was then determined by reaction with sulfanilamide and N-( l-naphthy1)ethylenediamine. Orthophosphate and reactive silica
were determined by reaction with ammonium molybdate and ascorbic acid (heteropoly blue method). Selectivity for phosphate was achieved kinetically, while, for silica, the phosphate interference was removed through the addition of oxalic acid. Alkalinity, which is "the sum of all titratable bases",' was measured colorimetrically using an automated methyl orange procedure. In a typical water sample, alkalinity depends predominantly on the concentration of carbonate, bicarbonate, and hydroxide ions, so the first two can be estimated indirectly from the pH and alkalinity,' as was done here. Sulfate was determined by a turbidimetric method on a nephelometer (Turner Designs Model 40, Mountain View, CA) using BaC12 to precipitate BaS04. Total organic carbon (TOC) was determined using the persulfate ultraviolet oxidation method implemented on a Technicon autoanalyzer (Technicon,Tarrytown, NY) with a phenolphthaleinindicator. A Radiometer GK2401C glass electrode (Radiometer America, Westlake, OH) was used to measure pH and a flowthrough conductance electrode (Radiometer Model CDC3 14) was used to measure conductivity, which was corrected to 25 OC. The data used in this study were obtained from 483 typical samples and represented a variety of sources. The bulk of samples analyzed by Fenwick Laboratories are groundwater samples (ca. 70%), with surface water (ca. 25%) and wastewater (ca. 5%) making up the majority of the rest. Conductivities ranged from about 4 to 11 000 p S cm-l. For the purposes of this study, a subset of 422 samples with conductivities below 1400 p S cm-' was also considered. The samples were analyzed during the period August 1991 to March 1992. Software. Most calculations were performed on a 486compatible computer under DOS 6.0 using the Excel 4.0 spreadsheet program (Microsoft, Redmond, WA) and programs written in MATLAB 4.0 (The Math Works Inc., Natick, MA) for DOS and Microsoft Windows. Some of the MATLAB programs were written by the authors and others were provided by Dr. Barry Wise.14 Programs for the artificial neural networks were written in MATLAB (486 PC) and C (Sun SparcStation 10/4 1).
RESULTS AND DISCUSSION Rossum Models. A statistical summary of most of the RCAp parameters is given in Table 2. Because the distributions of measurements are generally not Gaussian, the median and upper and lower quartile ranges have been given in addition to the mean and standard deviation. For the Rossum model and its variations, the first seven measured ionic concentrations were used, along with ammonium, carbonate, and bicarbonate concentrations. It was assumed that there was no nitrite present (as is normally the case), only nitrate. The concentration of NH4+ was calculated from the total ammonia concentration and the pH. For thesamples analyzed, the bulk of the total ammonia was generally present as the ammonium ion. The carbonate and bicarbonate were calculated from the alkalinity and pH, with the amount of carbonate generally being negligible. Unfortunately, the pH was reported to only one decimal place for these calculations, (14) Wise, B. M. PLS ToolBox, v 1.2, Wise Software, 1991.
n
E
0
\
3 x
.-> 4
0 a,
+J
.I? -0
1400
8oool 6000
Measured Conductivity @S/cm) Figure 1. Comparison of measured and predlcted conductivities for test data using the Rossum model over two ranges: (a)
