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Errors in the gran addition method
Anulytlru
Chlmico
Acrd
447
Elwvicr Publishing Company. ~msterdnm Printed in 171~Nclherlnnds
ERRORS
IN
THE
GRAN
PART III. EXPERIMENTAL FLUORIDE-SELECTIVE
N.
PARTHASARATHY.
Dbpartemene (Received
de Chimie 18th October
ADDITION
DETERMINATION ELECTRODE
J. BUFFLE Min~rule
and
et Ancdytiyue
D.
METHOD OF
ERRORS
BY
MEANS
OF A
MONNIER de I’UniversitZ,
Geneva
(Switzerland)
1971)
The importance of the possible errors in the Gran plot method which are reported here, has not been studied before, although several authors have used this method l- 3. In Part I of this series4 the errors introduced by various parameters in the addition technique were discussed and a theoretical evaluation of the titration conditions to achieve maximal precision in the determination of the unknown concentration was described. In order to verify the theory, cxpcriments were performed with a fluoride-selective electrode. EXPERIMENTAL
Apparatus A Beckman fluoride-selective electrode and a reference saturated calomel electrode were used with a Metrohm digital pH-meter. All measurements were made at 23 +O.l” in a polyethylene beaker. The solutions were stirred magnetically during the measurements.
Reayen ts Total Ionic Strength Buffer (TISAB) was prepared by dissolving 170 g of sodium nitrate and 57 ml of anhydrous acetic acid in 500 ml of distilled water. The pH was adjusted to 5.4 with 5 M sodium hydroxide and the solution was diluted to 1 I with water. Standard fluoride solutions were obtained by diluting 0.1 M sodium fluoride solution (Beckman solution no. 566342) and stored in polyethylene bottles.
Procedure The “sample” of known concentration (20 ml) was placed in a polyethylene beaker and 20 ml of TISAB was added to it. The fluoride-selective electrode and the reference electrode were immersed in the solution and the e.m.f. was measured. Then 0.5-m] aliquots of the standard solution obtained by (1 + 1) dilution of pure fluoride solution with TISAB were added up to 10 ml and the corresponding potentials were measured. The above procedure was repeated for several concentrations of the standard and sample solutions. V, and AV,/l V,( were evaluated by carrying out statistical calAnal. Chim. Acta,
59 (1972)
448
N.
PARTHASARATHY.
J. BUFFLE.
D. MONNIER
culations, using the values s,,/V, =O. 1 ‘;/,, sv/V-, =0.025 ‘pi, and sit = 0.05 mV. Independent measurements were made to determine these errors. EXPERIMENTAL
DETERMINATION
OF
STATISTICAL
ERROR
ON
l’,
oj’AV.JKl =-j(R) curve The proccdurc described above was carried out for C,,= 10.. ’ M, V, =40ml, V,n,,x= 10 ml, N =20, and several concentrations of standard solutions. The value of k used for the determination of V, was calculated by the least squares method5 for each titration. The rcgrcssion line Y =f( V) was calculated by using the weighting factor described previously4, and AV,/I V,l was evaluated from the variability of the slope of
Determination
TABLE
I
EXPERIMENTAL EVALUATION OF v, AND AI’,/1 V,I FOR DIFFERENT TITRATION TI ONS (C0=10-4M. V0=40ml. l&.=lOml. N=21) - _.._----..---..-.- ._.._-__- __._ _. .._.. _ --__I-..---_...-_..-.--.-...C,Jburd (M) v, (fd) bf(R) AK/l KI (‘22 C(M) 10-2_~---_-1;4----_-_.o,41-----_--_--~~ 1.12*10-4 1.02.10-J 4.8 5*10-J I.1 0.82 1.02. 10-J 3.45 4-10-3 1.0 1.02 1.01 ’ 10-4 1.33 2.5. lo-” 0.8 1.61 l.oo*lo-4 2.26 1.5. 10-3 0.6 2.70 1.01~10-~ 1.26 5.0. 10-J 0.097 8.05 l.OO* 10-4 1.03 2.5. 10-4 -0.21 16.06 l.oo~Io-~ 1.38 l.S* lo-” - 0.43 26.7 9.98, lo-” 2.4 8.0, lo-’ -0.7 49.9 9.96. 10 -’’ 3.7 6.0. 10-s - 0.82 66.4 9.94*lo-5 4.76 5.0. 10-?# -0.91 79.5 --___ --.._ -_--_---.--.
CONDI-
I -1.0
-0.5
0
0.5
1.0
1.5
Fig. 1. Relationship bctwccn statistical error in V,, AV,.l V,l. and the titration conditions. C, = 10m4 M, V, = 40 ml, V,,,,, = 10 ml, N=21, ~v,,/V~=0.1‘~~ sv/V,=O.O25”/;;, s,=O.O5 mV. (1) Experimental curve obtained by statistical extrapolation by using a weighting factor; (2) thcorctical curve cvaluatcd for the same titration conditions4. Anal. Chim. Acta, 59 (1972)
THE
GRAN
METHOD
FOR
FLUORIDE
449
ELECTRODES
this line. The results of these titrations are tabulated in Table I and the corresponding plot is shown in Fig. 1. For the definition of symbols, see Part 14. Compurison of the tatisticul method and the graphicul method oj’extrapolation The same experimental procedure as above was used for three different concentrations of standard solutions; 10 sets of titration data were recorded in each case. In each case V, was determined by statistical calculations. In Table II are tabulated the values of V, and AVX/JVX/,I evaluated by using the weighting factor (columns A), and without this weighting factor (columns B). In columns C the values of V, obtained graphically are reported. TABLE
II
COMPARISON OF STATISTICAL DIFFERENT C/C, RATIOS
AND
GRAPHICAL
METHODS
OF EXTRAPOLATION
FOR
(C,= 10-4M,V,=40ml, V,,,= lOml, N =20;A=Statistical resultswith theweightingfactor,B=statisticaI results without the weighting factor, C=graphical determination) --. .-.. Titration no.
B
A
K (4 c/c,= 1 2 3 4 5 6 7 8
c _...
100
c/c, 1 2 3 4 5 6 7 8 9
= 5
c/c, 1 2 3 4 5 6 7 8 9’ 10
= 1
0.41 0.42 0.42 0.41 0.4 1 0.41 0.40 0.41
AK/l
Kl(%)
12.6 9.8 4.3 11.9 7.3 12.3 10.9 5.6
K W) 0.36 0.34 0.42 0.38 0.36 0.32 0.33 0.36
- .._.
AK/I K.1(“/;‘,I
--K (4
7.3 16.3 6.7 16.1 11.5 13.6 11.6 6.4
0.37 0.31 0.40 0.33 0.35 0.33 0.40 0.35
8.01 8.03 8.03 8.30 8.05 8.27 7.96 7.96 8.0
0.98 1.03 1.1 1.2 1.5 1.3 1.01 0.79 0.96
7.95 7.98 7.96 8.23 8.00 8.32 7.92 7.93 7.98
1.23 1.09 1.12 1.2 1.5 1.4 1.01 0.84 1.29
8.04 8.00 8.07 8.24 8.11 8.37 7.93 7.9 8.04
39.94 40.00 40.00 40.00 40.95 40.00 40.00 40.00 40.43 40.00
1.4 < 0.01 CO.01 co.01 0.9 co.01 c 0.01 CO.01 0.7 co.01
39.96 40.00 40.00 40.00 40.03 40.00 40.00 40.00 40.36 40.00
c 0.01 co.01 (0.01 CO.01 0.97 c 0.01 < 0.01 CO.01 0.7 co.01
40.3 39.6 39.96 39.6 41.22 39.96 39.8 39.9 40.5 39.96 ._.~
_Anal.
Claim. Acta,
59 (1972)
450
N.
PARTHASARATHY.
J. BUFFLE.
D. MONNIER
Discussion The experimental curve of Fig. 1 passes through a minimum as predicted by the theory4, but this curve is slightly displaced from the theoretical one. This may be due to the fact that in the theoretical calculations, SE, the error in E, was assumed to be constant. In practice, this assumption is probably not true. The change in E produced when C is close to Cc is small, whereas when C differs considerably from Co the change in E is large and sE probably is also larger than in the first case. In any case, the optimal precision is obtained when the concentration C is close to Co, a condition which was also observed by Manahan3. Table II gives the results of titrations for three different concentrations of standard solutions. As can be seen, when C is not considerably different from Co (i.e. for C/C ,=5 or l), the graphical and the least squares tit methods are in good agreement. However, when C is very different from Co (i.e. for C/C, = lOO),either the graphical method or the least squares method without the weighting factor can lead to important errors in the determination of V,. Hence, in that case, the graphical method is not applicable, but it is possible to use the statistical one provided that a correct weighting factor can be determined. EFFECT
OF SYSTEMATIC
ERROR
ON k, Ak
Three titrations were selected from Table I (log(R) =0.6, 0.097, -0.21), and k,, the most probable value of k, was determined by a least squares fitsn6; V, was then determined by plotting Y against V and extrapolating to zero. In each case values of Ak/k were chosen arbitrarily and cortedpmding values of k were calculated from the relationship : Ak/k = (k,- k)/k,. For each set of experimental data and the corresponding - 3 .alculated values of k, plots of Y us. V were constructed. In Fig. 2, some of these curves are reported for the condition log(R) =0.6 (C/C, = 15). From this Fig., it is seen that the non-linearity in the curve occurs at 4 Y (ml) 6 ,’
.:,- *-I
0
1
2
3
4
5
6
7
a
Vtml) 9
,
Fig. 2. Plots of Y = f(V) for various values of Ak/k. Co= low4 M, C = 1.5 * 10-j M. Vu =40 (1) A/+=-15%; (2) Ak/k=-7%; (3) Ak/k=-1%; (4) Ak/k=S%; (5) Ak/k=lO$:,. Anal. Chim. Acta,
59 (1972)
ml, V,,, =
10 ml.
THE
GRAN
METHOD
FOR
FLUORIDE
451
ELECTRODES
fLk/kP/o)
-20
l30 40 Fig. 3. The variation of error A V,/JV,[ made in V, obtained by graphical cxttapolution M (0); (2) C=5.0*10^ in k. Co= lo- 4 M, V, = 40 ml. V,,, = 10 ml. (1) C=1.S’10-3 10’4M(x).
as a function oferror ’ M (A); (3) C-2.5.
very negative values of Ak, conditions which are hardly encountered in practice. An approximate value of Ak/k above which the curve is linear can be estimated and The theoretical calculations similar to those done in Part II was found to be -7%. (ref. 7) but for the conditions used here also indicated the departure from linearity for Ak/k d - 8%. Hence the result obtained from the experiment is in good agreement with the theory. It can also be seen from Fig. 2 that as Ak increases algebraically, V, also increases. The relative systematic errors in V, were calculated for the three conditions mentioned above and plotted against Ak/k {Fig. 3). As can be seen, a small error in k causes a significant error in V,, and, as was also shown theoretically, a systematic error in iv, can be produced even when the relation Yoersus V is linear. It should be noted that curve 1 in Fig. 3 does not pass through the origin; this is presumably because Anal. Chim. Acta, 59 (1972)
N. PARTHASARATHY.
452
J. BUFFLE,
D. MONNIER
C/C, is high (C/C,= 15) for this curve. As was seen above, for high C/C0 ratios, the graphical method produces the same type of error as the statistical method without the weighting factor. This error has to be added to the systematic error caused by Ak. We wish to thank electrode.
Beckman
Instrument
Inc. for providing
us with the fluoride
SUMMARY
The errors incurred in the Gran addition tcchniquc have been cxpcrimentally determined with a fluoride-sclcctive clcctrodc. The results obtained agreed with the theory proposed in the previous Parts. To obtain maximal precision. the following conditions must be satisfied : (u) the concentration of the standard solution should bc as close as possible to that of the unknown solution; (b) the electrode slope, k, must bc detcrmincd with utmost precision; and (c) 1 c R = { V,,;,, *(C+ C,)! / I V,- (Co+ C,)j < 3. Rf?SUMJ?
On a dCterminC: cxp&rimcntalement Its diff&-entcs crrcurs qui peuvcnt apparaitre lors de I’utilisation de la mCthodc d’addition, et ccci au moyen dc I’tilectrode selective dc F-, Ccs rCsultats scmblent confirmer les calculs thkoriques dbcrits dans les p&&dents articles. Dcux conditions importantcs doivcnt Ctrc rcspcctees: la conccntration dc la solution standard doit Ctrc aussi proche que possible de cellc de la solution inconnue, ct la pcnte de l’Clectrode, k, doit &re dktcrminie avcc la plus grandc pr& cision. D’autre part, la condition
1< R = i Ku,, .(C+C,)~.l~~,.(C,+C,)S<3 doit Cgalcmcnt
Ctrc respcct@e.
ZUSAMMENFASSUNG
Die beim Zumischverfahrcn von Gran auftretenden Fehlcr wurden mit einer fluoridsclektivcn Elcktrodcn cxperimcntcll bcstimmt. Die erhaltencn Ergebnisse stimmten mit dcr in den vorausgcgangenen Tcilen vorgeschlagenen Theorie iiberein. Fiir maximale Genauigkeit miissen folgende Bedingungen erfiilit sein : (lz) die Konzentration der Standardl6sung sol1 so dicht wie mijglich bei dcrjenigen der unbekannten Liisung liegen; (b) die Elektrodensteilheit, k, muss mit gr6sstmCiglicher Genauigkeit bestimmt werden ; und (c) 1 < R = { I&,, . (C + C,) )./ .(V, *(C, + C,) 1 < 3. REFERENCES 1 2 3 4 5 6 7
T. Anfalt, D. Dyrssen and D. Jsgner, Attar. Chh. Actor. 43 (1968) 487. D. Dyrssen and D. Jogncr, And. Clrh. Actu, 35 (1966) 407. S. Munahun, And. C/rem.. 42 (1970) 128. J. ButTIe, N. Pnrthwarathy ilnd D. Monnier. Anal. Cltira Acttr. 59 (1972) 427. M. J. D. Brand and G. A. Rcchnitz, &IO!. Chert~.. 42 (1970) 1172. E. W. Bauman, Anal. Chim. Acta, 42 (1968) 127. J. Buflle, And. Chim. Actor, 59 (1972) 439.
Anal. Chim. Acta,
59 (1972)
Chlmico
Acrd
447
Elwvicr Publishing Company. ~msterdnm Printed in 171~Nclherlnnds
ERRORS
IN
THE
GRAN
PART III. EXPERIMENTAL FLUORIDE-SELECTIVE
N.
PARTHASARATHY.
Dbpartemene (Received
de Chimie 18th October
ADDITION
DETERMINATION ELECTRODE
J. BUFFLE Min~rule
and
et Ancdytiyue
D.
METHOD OF
ERRORS
BY
MEANS
OF A
MONNIER de I’UniversitZ,
Geneva
(Switzerland)
1971)
The importance of the possible errors in the Gran plot method which are reported here, has not been studied before, although several authors have used this method l- 3. In Part I of this series4 the errors introduced by various parameters in the addition technique were discussed and a theoretical evaluation of the titration conditions to achieve maximal precision in the determination of the unknown concentration was described. In order to verify the theory, cxpcriments were performed with a fluoride-selective electrode. EXPERIMENTAL
Apparatus A Beckman fluoride-selective electrode and a reference saturated calomel electrode were used with a Metrohm digital pH-meter. All measurements were made at 23 +O.l” in a polyethylene beaker. The solutions were stirred magnetically during the measurements.
Reayen ts Total Ionic Strength Buffer (TISAB) was prepared by dissolving 170 g of sodium nitrate and 57 ml of anhydrous acetic acid in 500 ml of distilled water. The pH was adjusted to 5.4 with 5 M sodium hydroxide and the solution was diluted to 1 I with water. Standard fluoride solutions were obtained by diluting 0.1 M sodium fluoride solution (Beckman solution no. 566342) and stored in polyethylene bottles.
Procedure The “sample” of known concentration (20 ml) was placed in a polyethylene beaker and 20 ml of TISAB was added to it. The fluoride-selective electrode and the reference electrode were immersed in the solution and the e.m.f. was measured. Then 0.5-m] aliquots of the standard solution obtained by (1 + 1) dilution of pure fluoride solution with TISAB were added up to 10 ml and the corresponding potentials were measured. The above procedure was repeated for several concentrations of the standard and sample solutions. V, and AV,/l V,( were evaluated by carrying out statistical calAnal. Chim. Acta,
59 (1972)
448
N.
PARTHASARATHY.
J. BUFFLE.
D. MONNIER
culations, using the values s,,/V, =O. 1 ‘;/,, sv/V-, =0.025 ‘pi, and sit = 0.05 mV. Independent measurements were made to determine these errors. EXPERIMENTAL
DETERMINATION
OF
STATISTICAL
ERROR
ON
l’,
oj’AV.JKl =-j(R) curve The proccdurc described above was carried out for C,,= 10.. ’ M, V, =40ml, V,n,,x= 10 ml, N =20, and several concentrations of standard solutions. The value of k used for the determination of V, was calculated by the least squares method5 for each titration. The rcgrcssion line Y =f( V) was calculated by using the weighting factor described previously4, and AV,/I V,l was evaluated from the variability of the slope of
Determination
TABLE
I
EXPERIMENTAL EVALUATION OF v, AND AI’,/1 V,I FOR DIFFERENT TITRATION TI ONS (C0=10-4M. V0=40ml. l&.=lOml. N=21) - _.._----..---..-.- ._.._-__- __._ _. .._.. _ --__I-..---_...-_..-.--.-...C,Jburd (M) v, (fd) bf(R) AK/l KI (‘22 C(M) 10-2_~---_-1;4----_-_.o,41-----_--_--~~ 1.12*10-4 1.02.10-J 4.8 5*10-J I.1 0.82 1.02. 10-J 3.45 4-10-3 1.0 1.02 1.01 ’ 10-4 1.33 2.5. lo-” 0.8 1.61 l.oo*lo-4 2.26 1.5. 10-3 0.6 2.70 1.01~10-~ 1.26 5.0. 10-J 0.097 8.05 l.OO* 10-4 1.03 2.5. 10-4 -0.21 16.06 l.oo~Io-~ 1.38 l.S* lo-” - 0.43 26.7 9.98, lo-” 2.4 8.0, lo-’ -0.7 49.9 9.96. 10 -’’ 3.7 6.0. 10-s - 0.82 66.4 9.94*lo-5 4.76 5.0. 10-?# -0.91 79.5 --___ --.._ -_--_---.--.
CONDI-
I -1.0
-0.5
0
0.5
1.0
1.5
Fig. 1. Relationship bctwccn statistical error in V,, AV,.l V,l. and the titration conditions. C, = 10m4 M, V, = 40 ml, V,,,,, = 10 ml, N=21, ~v,,/V~=0.1‘~~ sv/V,=O.O25”/;;, s,=O.O5 mV. (1) Experimental curve obtained by statistical extrapolation by using a weighting factor; (2) thcorctical curve cvaluatcd for the same titration conditions4. Anal. Chim. Acta, 59 (1972)
THE
GRAN
METHOD
FOR
FLUORIDE
449
ELECTRODES
this line. The results of these titrations are tabulated in Table I and the corresponding plot is shown in Fig. 1. For the definition of symbols, see Part 14. Compurison of the tatisticul method and the graphicul method oj’extrapolation The same experimental procedure as above was used for three different concentrations of standard solutions; 10 sets of titration data were recorded in each case. In each case V, was determined by statistical calculations. In Table II are tabulated the values of V, and AVX/JVX/,I evaluated by using the weighting factor (columns A), and without this weighting factor (columns B). In columns C the values of V, obtained graphically are reported. TABLE
II
COMPARISON OF STATISTICAL DIFFERENT C/C, RATIOS
AND
GRAPHICAL
METHODS
OF EXTRAPOLATION
FOR
(C,= 10-4M,V,=40ml, V,,,= lOml, N =20;A=Statistical resultswith theweightingfactor,B=statisticaI results without the weighting factor, C=graphical determination) --. .-.. Titration no.
B
A
K (4 c/c,= 1 2 3 4 5 6 7 8
c _...
100
c/c, 1 2 3 4 5 6 7 8 9
= 5
c/c, 1 2 3 4 5 6 7 8 9’ 10
= 1
0.41 0.42 0.42 0.41 0.4 1 0.41 0.40 0.41
AK/l
Kl(%)
12.6 9.8 4.3 11.9 7.3 12.3 10.9 5.6
K W) 0.36 0.34 0.42 0.38 0.36 0.32 0.33 0.36
- .._.
AK/I K.1(“/;‘,I
--K (4
7.3 16.3 6.7 16.1 11.5 13.6 11.6 6.4
0.37 0.31 0.40 0.33 0.35 0.33 0.40 0.35
8.01 8.03 8.03 8.30 8.05 8.27 7.96 7.96 8.0
0.98 1.03 1.1 1.2 1.5 1.3 1.01 0.79 0.96
7.95 7.98 7.96 8.23 8.00 8.32 7.92 7.93 7.98
1.23 1.09 1.12 1.2 1.5 1.4 1.01 0.84 1.29
8.04 8.00 8.07 8.24 8.11 8.37 7.93 7.9 8.04
39.94 40.00 40.00 40.00 40.95 40.00 40.00 40.00 40.43 40.00
1.4 < 0.01 CO.01 co.01 0.9 co.01 c 0.01 CO.01 0.7 co.01
39.96 40.00 40.00 40.00 40.03 40.00 40.00 40.00 40.36 40.00
c 0.01 co.01 (0.01 CO.01 0.97 c 0.01 < 0.01 CO.01 0.7 co.01
40.3 39.6 39.96 39.6 41.22 39.96 39.8 39.9 40.5 39.96 ._.~
_Anal.
Claim. Acta,
59 (1972)
450
N.
PARTHASARATHY.
J. BUFFLE.
D. MONNIER
Discussion The experimental curve of Fig. 1 passes through a minimum as predicted by the theory4, but this curve is slightly displaced from the theoretical one. This may be due to the fact that in the theoretical calculations, SE, the error in E, was assumed to be constant. In practice, this assumption is probably not true. The change in E produced when C is close to Cc is small, whereas when C differs considerably from Co the change in E is large and sE probably is also larger than in the first case. In any case, the optimal precision is obtained when the concentration C is close to Co, a condition which was also observed by Manahan3. Table II gives the results of titrations for three different concentrations of standard solutions. As can be seen, when C is not considerably different from Co (i.e. for C/C ,=5 or l), the graphical and the least squares tit methods are in good agreement. However, when C is very different from Co (i.e. for C/C, = lOO),either the graphical method or the least squares method without the weighting factor can lead to important errors in the determination of V,. Hence, in that case, the graphical method is not applicable, but it is possible to use the statistical one provided that a correct weighting factor can be determined. EFFECT
OF SYSTEMATIC
ERROR
ON k, Ak
Three titrations were selected from Table I (log(R) =0.6, 0.097, -0.21), and k,, the most probable value of k, was determined by a least squares fitsn6; V, was then determined by plotting Y against V and extrapolating to zero. In each case values of Ak/k were chosen arbitrarily and cortedpmding values of k were calculated from the relationship : Ak/k = (k,- k)/k,. For each set of experimental data and the corresponding - 3 .alculated values of k, plots of Y us. V were constructed. In Fig. 2, some of these curves are reported for the condition log(R) =0.6 (C/C, = 15). From this Fig., it is seen that the non-linearity in the curve occurs at 4 Y (ml) 6 ,’
.:,- *-I
0
1
2
3
4
5
6
7
a
Vtml) 9
,
Fig. 2. Plots of Y = f(V) for various values of Ak/k. Co= low4 M, C = 1.5 * 10-j M. Vu =40 (1) A/+=-15%; (2) Ak/k=-7%; (3) Ak/k=-1%; (4) Ak/k=S%; (5) Ak/k=lO$:,. Anal. Chim. Acta,
59 (1972)
ml, V,,, =
10 ml.
THE
GRAN
METHOD
FOR
FLUORIDE
451
ELECTRODES
fLk/kP/o)
-20
l30 40 Fig. 3. The variation of error A V,/JV,[ made in V, obtained by graphical cxttapolution M (0); (2) C=5.0*10^ in k. Co= lo- 4 M, V, = 40 ml. V,,, = 10 ml. (1) C=1.S’10-3 10’4M(x).
as a function oferror ’ M (A); (3) C-2.5.
very negative values of Ak, conditions which are hardly encountered in practice. An approximate value of Ak/k above which the curve is linear can be estimated and The theoretical calculations similar to those done in Part II was found to be -7%. (ref. 7) but for the conditions used here also indicated the departure from linearity for Ak/k d - 8%. Hence the result obtained from the experiment is in good agreement with the theory. It can also be seen from Fig. 2 that as Ak increases algebraically, V, also increases. The relative systematic errors in V, were calculated for the three conditions mentioned above and plotted against Ak/k {Fig. 3). As can be seen, a small error in k causes a significant error in V,, and, as was also shown theoretically, a systematic error in iv, can be produced even when the relation Yoersus V is linear. It should be noted that curve 1 in Fig. 3 does not pass through the origin; this is presumably because Anal. Chim. Acta, 59 (1972)
N. PARTHASARATHY.
452
J. BUFFLE,
D. MONNIER
C/C, is high (C/C,= 15) for this curve. As was seen above, for high C/C0 ratios, the graphical method produces the same type of error as the statistical method without the weighting factor. This error has to be added to the systematic error caused by Ak. We wish to thank electrode.
Beckman
Instrument
Inc. for providing
us with the fluoride
SUMMARY
The errors incurred in the Gran addition tcchniquc have been cxpcrimentally determined with a fluoride-sclcctive clcctrodc. The results obtained agreed with the theory proposed in the previous Parts. To obtain maximal precision. the following conditions must be satisfied : (u) the concentration of the standard solution should bc as close as possible to that of the unknown solution; (b) the electrode slope, k, must bc detcrmincd with utmost precision; and (c) 1 c R = { V,,;,, *(C+ C,)! / I V,- (Co+ C,)j < 3. Rf?SUMJ?
On a dCterminC: cxp&rimcntalement Its diff&-entcs crrcurs qui peuvcnt apparaitre lors de I’utilisation de la mCthodc d’addition, et ccci au moyen dc I’tilectrode selective dc F-, Ccs rCsultats scmblent confirmer les calculs thkoriques dbcrits dans les p&&dents articles. Dcux conditions importantcs doivcnt Ctrc rcspcctees: la conccntration dc la solution standard doit Ctrc aussi proche que possible de cellc de la solution inconnue, ct la pcnte de l’Clectrode, k, doit &re dktcrminie avcc la plus grandc pr& cision. D’autre part, la condition
1< R = i Ku,, .(C+C,)~.l~~,.(C,+C,)S<3 doit Cgalcmcnt
Ctrc respcct@e.
ZUSAMMENFASSUNG
Die beim Zumischverfahrcn von Gran auftretenden Fehlcr wurden mit einer fluoridsclektivcn Elcktrodcn cxperimcntcll bcstimmt. Die erhaltencn Ergebnisse stimmten mit dcr in den vorausgcgangenen Tcilen vorgeschlagenen Theorie iiberein. Fiir maximale Genauigkeit miissen folgende Bedingungen erfiilit sein : (lz) die Konzentration der Standardl6sung sol1 so dicht wie mijglich bei dcrjenigen der unbekannten Liisung liegen; (b) die Elektrodensteilheit, k, muss mit gr6sstmCiglicher Genauigkeit bestimmt werden ; und (c) 1 < R = { I&,, . (C + C,) )./ .(V, *(C, + C,) 1 < 3. REFERENCES 1 2 3 4 5 6 7
T. Anfalt, D. Dyrssen and D. Jsgner, Attar. Chh. Actor. 43 (1968) 487. D. Dyrssen and D. Jogncr, And. Clrh. Actu, 35 (1966) 407. S. Munahun, And. C/rem.. 42 (1970) 128. J. ButTIe, N. Pnrthwarathy ilnd D. Monnier. Anal. Cltira Acttr. 59 (1972) 427. M. J. D. Brand and G. A. Rcchnitz, &IO!. Chert~.. 42 (1970) 1172. E. W. Bauman, Anal. Chim. Acta, 42 (1968) 127. J. Buflle, And. Chim. Actor, 59 (1972) 439.
Anal. Chim. Acta,
59 (1972)