containing organic matter, ignition must follow digestion. The problem of rendering the refractory materials soluble still remains, and certain minerals are very resistant, e.g., tourmaline, sillimanite, zircon. The use of sodium peroxide as a sinter and fusion mixture is frequently recommended, (15, 20, 21) but this reagent is frequently impure and not applicable to certain investigations. Recently, interest has centered on the use of lithium tetraborate or metaborate either with or without fluoride additions (18, 22). Fluoride addition allows removal of silicon by evaporation, and boron may be removed by a subsequent acid digestion (18). This fusion technique is very rapid and effective with most minerals. For the small number, e.g., chromite, which take a considerable time to react, the substitution of lithium fluoride by sodium fluoride gives a higher melting fusion mixture and reaction is extremely rapid. Table I11 gives the details of the preliminary treatment of a sample of Irvinebank tailings and allows a comparison of results. Results using sodium peroxide-sodium carbonate, lithium fluoride-boric acid, and sodium fluoride-boric acid fusions are equivalent, despite the fact that differing amounts and types of undissolved residue were left by each method. (20) R. J. W. McLaughlin and V. S. Biskupsky, Anal. Chim. Acta, 32, 165 (1965). (21) T. A. Rafter, Analyst, 75, 485 (1950). (22) N. H. Suhr and C. 0. Ingamells, ANAL.CHEM., 38,730 (1966).
This suggests that any of these three methods can be used satisfactorily. The lithium or sodium fluoride-boric acid is to be preferred because generally there was considerably less residue than with peroxide methods. The wet digestion methods, either on ignited or nonignited samples, invariably gave considerably lower tin values compared with the fusion methods and were not used subsequently for quantitative studies. Presumably this method does not provide complete dissolution of tin. Geochemical Survey for Tin. Conventional ac polarography has been used without the necessity to control temperature or to degas the solution to provide a cheap, simple, and reliable method of determination of tin in rock samples from a geochemical survey conducted over a wide area of Australia. Samples with tin contents down to 10 ppm were analyzed. Those in which tin could not be detected were simply classified as containing less than 10 ppm. For surveys of this type, currently being made in several laboratories, the attainable sensitivity of 5 ppm, or lower using stripping techniques, was not required, RECEIVED for review September 8, 1969. Accepted May 22, 1970. Acknowledgment is made of a grant from the Australian Research Grants Committee for the instruments used in this and the preceding paper (7).
Computer Approach to Ion-Selective Electrode Potentiometry by Standard Addition Methods M. J. D. Brandl and G . A. Rechnitz Department of Chemistry, State University of New York, Bufalo, N. Y. 14214 Methods are described for determining an unknown concentration using an ion-selective electrode without prior calibration of the electrode. The methods are based on standard addition procedures. In the simplest case, only two standard additions are required and a simple calculation i s described which can be performed by a computing calculator. To obtain high accuracy in the determination of unknown concentrations, multiple additions are made and a least squares curve fitting method is used to evaluate the unknown concentration, electrode slope, and standard potential. A computer program to accomplish this calculation, ADDFIT, is given in Fortran IV. The effectiveness of these methods is demonstrated by experiments on lead and chloride samples. ANALYTICAL METHODS using ion selective membrane electrodes may be largely classified as direct determinations of ionic activity or indirect titrimetric procedures (1). Direct potentiometry is based on the relation of the measured electrode potential to the logarithm of ionic activity through the equation
E
=
E,
+ S log a
(1 )
I Present address, Research Department, Imperial Chemical Industries Ltd., Agricultural Division, Billingham, Teesside, U. K.
(1) R. A. Durst, in “Ion Selective Electrodes,” R. A. Durst, Ed., National Bureau of Standards Special Publication 314, Washington, D. C., 1969, p 375. 1172
where E, is the standard potential and S is the slope, usually near to the Nernstian (RT/nF). It may readily be shown (2) that an error of A 0.1 mV in measurement of the electrode potential leads to an error of 0.39 Z in the value of a, under Nerstian conditions with n = 1. However this accuracy is rarely achieved for a single determination except under ideal laboratory conditions with very stable electrodes. If the error is random, in principle it can be made as small as is desired by making a sufficient number of replicate determinations. Thus an accuracy of +0,2% has been obtained for the determination of silver at the 10-’M level (3). The differential technique of null point potentiometry (4, 5) has been applied to direct potentiometry with ion selective electrodes. This method can in theory achieve high accuracy but it has been shown ( 5 ) that with nonideal electrodes having different values of S, the accuracy is limited by the measurement of the cell null potential. Standard addition methods provide an alternative approach to obtaining high accuracy by direct potentiometry. Methods
*
( 2 ) T. S. Light, ibid., p 356. (3) D. C. Muller, P. W. West, and R. H. Muller, ANAL.CHEM., 41, 2038 (1969). (4) R. A. Durst, ibid., 40,931 (1968). (5) M. J. D. Brand and G . A. Rechnitz, ibid., 42, 616 (1970).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
based on known additions of the ion to be determined (6, 7) and on known additions of a reagent which complexes with the ion (7,s) (standard subtraction) have been described. Durst (9) has described a method in which known volumes of the unknown solution are added to a known volume of a standard solution of the ion. In general, for any number, ( j - l), of additions of Vi ml of solution of known concentration Ccto Vml of solution of unknown concentration C
Ei
=
E,
+ S log
cv +
j
civ,
i=l
V + k
vi
(2)
i=l
where V I = 0. The use of concentration rather than activity implies that the ionic strength of the unknown is not seriously altered by the standard additon. If the slope S is known, in principle only a single addition is necessary to determine C; Equation 2 gives E, and E2 which can be solved simultaneously for C (6). For the more general case of multiple additions the graphical method of Gran (10) has been applied (11) to linearize Equation 2. This procedure is far superior to that of a single addition because the combined weights of several data points are used to calculate the unknown concentration and random errors will tend to be eliminated. The possibility also exists of solving Equation 2 without a prior knowledge of S; a minimum of two additions are required to solve Equation 2 for i = 1 to 3. For the case of multiple additions, the system of equations is over-defined and it is necessary to select the best values of E,, S, and C to fit the data to Equation 2. At the present time, computer methods have not been widely applied in analytical potentiometry. An analog computer method has been described (12) for calculating cation activities from glass electrode potentials and Dyrssen et al. (13) have given a number of computer methods for locating the end point in potentiometric titrations. This paper describes a simple approximation for the solution of Equation 2 for two additions and a nonlinear least squares method for multiple additions. THEORY
Two Addition Method. The method of making two additions and solving Equation 2 for C without prior knowledge of E, or S is not highly accurate but represents the simplest possible way of obtaining analytical information from an ion selective electrode. The solution of Equation 2 for i = 1 to 3 cannot be given in an explicit form and it is necessary to use a successive approximation method. The most obvious approximation is to put S = RT/nF, then any pair of Equation 2 can be solved simultaneously for E, and C. These values could then be used to obtain a better estimate for S and the process repeated. Unfortunately, this method does not lead to convergence of C and an alternative approach is required. (6) Newsletter, Orion Research Inc., Cambridge, Mass., 1, 9 (1969). (7) Zbid.,2, 5 (1970). (8) Zbid., 1, 25 (1969). (9) R. A. Durst, Mikrochim. Acta, 3, 611 (1969). (10) G. Gran, Analyst, 77, 661 (1952). (11) A. Liberti and M. Mascini, ANAL.CHEM., 41,676 (1969). (12) S. M. Friedman and F. K. Bowers, Anal. Biochem., 5 , 471 (1 963). ( I 3) D. Dyrssen. D. Jagner, and F. Wengelin, “Computer Calculation of Ionic Equilibria and Titration Procedures,” Wiley, New York, 1968.
A satisfactory solution is obtained by initially setting C 0. Equation 2 is then solved for i = 2 and 3 to give I$ and S. These values are then substituted in Equation 2, i = 1, to allow a new estimate for C to be made, and the process is repeated until estimates for C converge. Usually a large number of cycles are required and the process is tedious by hand calculation. However, the method was originally developed for use with a computing calculator, of which a number are now commercially available. The rather limited memory provided in this type of calculator necessitates the use of relatively simple programs. A program to perform the above calculation has been written for the 9100 A calculator (Hewlett-Packard, Palo Alto, Calif.) and is available from the authors. Multiple Addition Method. Although the method of two additions has the advantage of speed and simplicity, if accurate values of concentration are required, several further standard additions are necessary. It is required to find values of E,, S, and C which give the best fit of the experimental data to Equation 2. This problem may be solved by use of a nonlinear least squares curve fitting procedure for an equation of the form =
Y = f ( x , a, b, c )
(3)
where y = E,, x = V , and a, 6, and c are the undefined constants E,, s, and C. The principles of this type of curve fitting method have been given elsewhere (14-17). In general these methods require that initial estimates for a, b, c, Le., a,, bl, cl, are known. By successive approximations the estimates for a, b, and c are refined until there is little difbn+l,cn+l,when the process ference between a,, b,, cn and anfl, is stopped, One method (17) of making these successive approximations is applicable when the partial derivatives of f ( x , a, b, C) with respect to a, b, c are easily obtained. Taylor’s Formula then gives
The coefficients&a, 6b, 6c are now the unknowns in this linear equation and their values can be obtained by a linear least squares fit. Then
+ 6a bn+l = bn + 66 = Cn + 6c a,+1
=
an
Cn+1
This method is applied here to Equation 2. Initial estimates for E,, S, and C are obtained by assuming that the electrode slope is Nernstian, which is usually a close approximation, S = 59in
(5)
where n is the ionic charge (negative for anions). Values for E, and C are obtained by simultaneous solution of Equation 2 for El and E?, C=
C?v2
(V?
+ V)exp[2.303(E~- EI)/SI - V
(6)
W. E. J. Wenworth, J . Chem. Educ., 42,96, 162 (1965). P. N. Murgatroyd, Elecfrotz. Lerr., 5 (26), 701 (1969). H. Kim, J . Chem. Educ., 47, 120 (1970). R. H. Pennington, “Introductory Computer Methods and Numerical Analysis,” Macmillan, New York, 1965, p 377.
(14) (15) (16) (17)
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
1173
J , K ,
N ,VOL
I
CALCULATE
INITIAL
AC = AS = A L = 0
Figure 1. Flow chart for computer program ADDFIT
s = s
+E
WRITE
CALCULATE A E & , A C
Z ( I I = A L t A S XlI)
E,
=
El
- Slog C
(7)
The partial derivatives of Equation 3 are bEi - = bEo
(*)
r
1
2 civi
cv+ - -- log bS
i=l
bEi
2 vi
v+ L
S Vi log e
dEi - =
j
bC
-
i=l
CV+
J
=
xi
Yi
civi
(10)
i=l
An error function is calculated, (11)
Values of AEo, AS, and AC are then calculated by a least squares fit to the equation.
Zi = AE, 1174
+ AS xi + AC yi
ANALYTICAL CHEMISTRY, VOL. 42,
(12)
+&W)
The estimates E,, S,C are corrected by addition of AEo, AS, and A C , respectively, and the calculations of Equations 8 to 12 repeated until the values of E,, S, C converge. This program was originally written for use with the 9100A calculator and is not too tedious for a single experiment because, as will be seen later, convergence is obtained after three or four cycles; this is undoubtedly so because the initial estimate for S is quite accurate. Five separate programs are required, the last three of which perform the least squares fit to Equation 2 by means of the library program 09100-70814 (Hewlett-Packard, Palo Alto, Calif.). The complete program is available from the authors. However, for use with large numbers of data points or with many experiments, the program is excessively time-consuming. The program has therefore been rewritten in Fortran IV to allow its use on a large computer (CDC 6400). A flow chart for this program is shown in Figure 1 and the program list is given in the Appendix. EXPERIMENTAL
The chloride electrode used was a prototype silver chloride membrane electrode (Corning Glass Works, Medfield, Mass.).
NO. 11, SEPTEMBER 1970
Table I. Standard Additions to a Hypothetical Monovalent Cation Electrode Where CV > C,Vl Data Initial concentration = 1.OOO X lO+M; V = 50.00 ml; n = 1 i Ci,M Vi,ml Ei,mV 0 0 -65.0 1 2 0.1 0.1 -60.7 3 0.1 0.1 -57.1 4 0.1 0.1 -53.9 5 0.1 0.1 -51.2 6 0.1 0.1 -48.7
Table 11. Standard Additions to a Hypothetical Monovalent Cation Electrode Where CV cv CIV, Data Initial concentration i
= 1.OOO X 10-3M; V C,,M Vi,ml
1 2
0 0.1
3
0.1
4 5
0
0.5 0.5 0.5 0.5 0.5
0.1
0.1 0.1
6
Results C, M
Linear least squares fit ... 2 Addition fit (i = 1 to 3) 0.920 X low3 Multiple addition fit (i = 1 to 6) 1.006 X
S,mvl decade
E,, mV
No. of cycles
54.90
99.68
..,
51.02
89.93
100
55.15
100.32
4
Its potential was measured with respect to a saturated mercurous sulfate electrode Model 40455 (Beckman Instruments Inc., Fullerton, Calif.) using a Digital Electrometer Model 101 (Corning Glass Works, Medfield, Mass.). The lead electrode was a solid state membrane electrode Model 94-84 (Orion Research Inc., Cambridge, Mass.) and its potential was measured against a double junction reference electrode Model 90-02 (Orion Research Inc.) with a Model 801 digital pH/mV meter (Orion Research Inc.).
50.00 ml; n E,, rnV -65.0 -48.7 -39.2 - 32.6 -27.5 -23.4
1
=
Results S, mvl C, M
Linear least squares fit ... 2 Addition fit (i = 1 to 3) 1.019 X.10-3 Multiple addition fit (i = 1 to 6) 0.996 x 10-3
decade
E,, mV
No. of cycles
55.00
100.00
..*
55.73
101.72
25
54.86
99.69
4
Table 111. Determination of Chloride with a Chloride Electrode by Standard Additions Data Initial concentration = 1 X 10-2M; V = 20.00 ml; n = -1 Ionic strength of solution adjusted to 0.1 with NaN03
Ci,M
i
1 2 3
RESULTS AND DISCUSSION To test the performance of the two point addition calculation and the program ADDFIT for multiple additions, a hypothetical monovalent cation electrode with a defined slope of 55.00 mV/decade and a standard potential of 100.0 mV was used. The potentials during imaginary standard addition experiments were calculated for an unknown concentration of 1.00 X 10-3Mboth for the case where CV ‘v CIVl and CV > CIVl. Calculated potentials were rounded to the nearest 0.1 mV, and the calculated total concentration after each addition was rounded to four significant figures. To test the programs under actual conditions, standard addition experiments were performed with a divalent cation (lead) electrode and a monovalent anion (chloride) electrode using “unknowns” of known Concentration. For each set of data the total concentration after each addition was known, and a linear least squares fit was used to calculate the electrode slope and standard potential. Subsequently, the unknown concentrations, slope, and standard potential for each set of data were calculated by the two addition method and by ADDFIT. The results obtained for each set of data are shown in Tables I to IV. For calculations of the two addition method using the computer calculator, each new cycle corresponding to a better estimate for C is initiated manually (by pressing the CONTINUE key). The process was discontinued when the change in C was less than 0.01 Z or when the number of cycles reached 100. ADDFIT invariably converged to within 0.01 of C in less than 100 cycles, and it was thought unnecessary to test the number of cycles in this program. Tables I to IV show that ADDFIT converges in 3 or 4 cycles for the data used. In Tables I1 to IV, the two addition calculation converges in about 25 cycles. In Table I the two point method converges only very slowly because CV > CIVl. In several experimental tests of this program it has been found that the
=
4
5 6
Vi,ml
0
0
0.1 0.1 0.1 0.1 0.1
2 2 2 2 2
Ei,mV 107.9 92.9 84.9 79.8 75.9 73.1
Results decade
E,,, mV
No. of cycles
-57.77
-7.63
...
-57.88
-7.81
27
-58.06
-8.02
3
S,mvl C,M
Linear least squares fit ... 2 Addition fit (i = 1 to 3) 1.001 X Multiple addition fit (i = 1 to 6) 1.008 X
Table IV. Determination of Lead with a Lead Electrode by Standard Additions Data Initial concentration 1
2 3 4
5 6
=
Cis M
i
2 X 10-eM; V Vi, ml 0
0
2x 2x 1x 1x 1x
=
1 .OO 1 .oo
10-4 10-4 10-3 10-3
1.00
1.00 1.00
10-3
100.00 ml; n Ei,mV -224.6 -214.8 -209.5 -197.1 -191.6 -187.9
=
2
Results Linear least squares fit 2 Addition fit (i = 1 to 3) Multiple addition fit(i = 1 to6)
C,M
S,mvl decade
E,, mV
...
30.01
-53.02
.
1.613 X loe6
28.33
-60.5
23
1.486 X 10-8
27.16
-66.36
4
No. of
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
cycles , .
1175
value of C converges only very slowly or sometimes not at all when CV > CIVl. The two addition method cannot be recommended unless it is known that CV < CIVL. The results obtained for the concentrations by ADDFIT shown in Tables I to I11 are within 1 of the actual value. The concentration value obtained in Table IV is significantly different from the initial concentration taken. This probably reflects the difficulties of accurately preparing very dilute solutions. It may be noted that the concentrations of standard solutions added are at least two orders of magnitude greater and can be prepared accurately. The method of making two additions provides a quick convenient means of obtaining an unknown concentration accurate to a few per cent. The program used to calculate the result is simple but it could be made more sophisticated if a calculator with a slightly larger memory were available (the 9100 B model, for example). This would enable decisions to end the program by counting the number of cycles and by checking convergence of C to be handled by the machine. The program ADDFIT is very much more reliable in practice. The program is not exceptionally large and could certainly be handled by a small dedicated computer, allowing the possibility of completely automating a standard addition method. In this respect, the only disadvantage of the standard addition method is that it requires a discrete sample and is not applicable to continuous monitoring. In the examples given here only five additions have been made, but for greater accuracy a larger number of additions may usefully be made. ADDFIT has been written for the case of standard additions but the same method of calculation may be applied to related methods (7-9). Application of this method allows direct potentiometry to approach the accuracy which has previously been attainable by potentiometric titration.
ACKNOWLEDGMENT Lead determinations were kindly performed by N. C. Kenny. We gratefully acknowledge the assistance of the Center for Scientific Measurement and Instrumentation and thank the Corning Glass Works for the loan of equipment.
RECEIVED for review March 20,1970. Accepted June 15,1970. This work supported by grant from The National Science Foundation and the Office of Saline Water.
APPENDIX Program ADDFIT
Input Format. The data for each experiment may be read from punched cards using the format given below. Each set of data may be identified by the variable K , which is negative for the last set of data in the deck. 1st Card
All other cards
Columns Field 1-3 I3 9-11
I3
K
17-18
I2
N
24-29
F 6.3
VOL
Comments No. of potential readings Indexes each experiment, K 1 Negative for last experiment Ionic charge. Negative for anions. Initial Volume
1-6
F 6.1
E(I)
Electrode potential
12-17 23-31
F 6.2 E 9.3
V(1) C(I)
Volume increment Concentration of added solution
PROGRAM ADDFIT(INPUT,OUTPUT,TAPES= INPUT,TAPE6 = OUTPUT) DIMENSION E(20), V(20), C(20), X(20), Y(20), Z(20), 1 SUMV(20), SUMCV(2O) C 1 READ(5,lOOl) J, K, N, VOL 1001 FORMAT(I3,5X,I3,5X,I2,5X,F6.2) READ(5,1002) (E(I), V(I), C(I), I = l,J) 1002 FORMAT(F6.1,5X,F6.2,5X,E9.3) KABS = IABS(K) WRITE(6,2004) KABS 2004 FORMAT(///19X,19H* **** EXPERIMENT ,I3,7H *****/I 1 5X,8HDATA. . . .) WRITE(6,2001) J, N, VOL 2001 FORMAT(lSX,4HJ = ,I3,5X,4HN = ,12,5X,6HVOL = ,F6.2//) WRITE(6,2002) (E(I),V(I),C(I),I= l,J) 2002 FORMAT(15X,1HE,11X,1HV,1OX,1HC//(12X,F6.1,5X,F6.2,5X,E9.3)) WRITE(6,2005) 2005 FORMAT(//SX,llHRESULTS. . . .//23X,3HCIN,12X,lHS,8X,SHESTAN//) C C CALCULATE SUMV(I), SUMCV(1) C SUMV(1) = V(l) SUMCV(1) = V(1) * C(1) DO 2 I = 2,J SUMV(1) = SUMV(1-1) V(1) SUMCV(1) = SUMCV(1-1) C(1) * V(1) 2 CONTINUE
+
+
C
INITIALISE S, CIN, ESTAN
C C
S = 59 /N
+
(C(2) * V(Z))/(((VOL V(2)) * (EXP(ALOG(10.0) * - E(l))/S ))))- VOL) ESTAN = E(l) - (S * ALOGlWCIN))
CIN 1
=
((E(2)
C C C 1176
INITIALISE VARIABLES
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
Variable J
>
L = l DELC = 0.0 DELS = 0.0 DELE = 0.0 C C C
CALCULATE NEW VALUES OF S, CIN, ESTAN
+
3 CIN = CIN DELC S=S DELS ESTAN = ESTAN DELE WRITE(6,2003) CIN, S, ESTAN 2003 FORMAT(20X,E9.3,5X,F7.3,5X,F8.3) IF(L - 1)5,5,4 4 IF(((lOO.O * ABS(DELC))/CIN) - 0.01)6,6,5 6 IF(K)9,9,1 C CALCULATE X(I), Y(I), Z(1) C C 5 DO 7 I = l,J X(1) = ALOGlO(((CIN*VOL) SUMCV(I))/(VOL SUMV(1))) Y(1) = (S * VOL * ALOGlO(EXP(l.O)))/(CIN * VOL SUMCV(1)) Z(I) = E(1) - ESTAN - (S * X(1)) 7 CONTINUE C THIS ROUTINE CALCULATES THE LEAST SQUARES COEFFICIENTS OF C Z = A B*X C*Y C C INITIALISE VARIABLES C C SUMX = 0.0 SUMY = 0.0 SUMZ = 0.0 SUMXY = 0.0 SUMXZ = 0.0 SUMYZ = 0.0 SUMX2 = 0.0 SUMY2 = 0.0 C DO 8 I = l,J SUMX = SUMX X(1) SUMY = SUMY Y(1) SUMZ = SUMZ Z(I) SUMXY = SUMXY (X(1) * Y(1)) SUMXZ = SUMXZ (X(1) * Z(1)) SUMYZ = SUMYZ (Y(1) * Z(1)) SUMX2 = SUMX2 (X(1) * X(1)) SUMY2 = SUMY2 (Y(1) * Y(1)) 8 CONTINUE C DNUM = J * ((SUMX2 * SUMYZ) - (SUMXZ * SUMXY)) 1 -SUMX * ((SUMX *.SUMYZ) - (SUMY * SUMXZ)) 2 SUMZ * ((SUMX * SUMXY) - (SUMY * SUMX2)) C DNOM = (J * SUMX2 * SUMY2) - (J * SUMXY * SUMXY) 1 -(SUMX *SUMX * SUMY2) (SUMX * SUMXY * SUMY) 2 +(SUMY * SUMX * SUMXY) - (SUMY *SUMX2 * SUMY) C DELC = DNUM/DNOM ERRM = SUMZ -- (DELC * SUMY) ERRN = SUMXZ - (DELC * SUMXY) DELS = ((J*ERRN) -(ERRM*SUMX))/((J*SUMX2)- (SUMX*SUMX)) DELE = (ERRM -- DELS * SUMX) / J L=L+l GO TO 3 9 CALL EXIT END
+
+
+
+
+
+
+
+ + +
+ + + + +
+
+
ANALYTICAL CHEMISTRY, VOL. 42, NO, 11, SEPTEMBER 1970
0
1177
