Indirect Polarographic Determination of Boron
50 ml. of distilled water. Heat until the melt is dissolved. Xeutralize to Table 1. Boric Acid Diffusion Currents the methyl red end point with dilute Hh'O3, then add a slight excess. Boil gently for 10 minutes to remove CO1. Cool, transfer to a 100-ml. volumetric flask, and make up to volume with 1.38 4.17 0.33 distilled water. 2.64 4.00 0.66 0.99 4.10 4.14 PREPARATION OF BORON- FREE 5.39 4.08 1.32 BLANK. Select an aliquot of the sam6.68 4.05 1.65 ple solution and evaporate to dryness. 2.31 9.20 3.98 Add equal volumes of methyl alcohol 13.60 4.12 3.30 and 12M hydrochloric acid. Evaporate 15.90 4.02 3.96 to dryness. Add concentrated nitric Av. 4.07 acid and evaporate to hard dryness. Std. dev. 0.07 Cool and take up residue with distilled water, and make up to original volume a Solutions were prepared from 0.066M with distilled water. H3B03 in 0.5M KN03. Eight grams of PREPARATION OF POLAROGRAPHIC mannitol were added to each solution. SOLUTION.To a 100-ml. volumetric Total volume was 100 ml. flask, add a sample aliquot containing from 0.2 to 4 mg. of boron. Add 5 ml. of the saturated quinone solution and 2 drops of methyl red which serves as Table II. Analysis of Colemanite" a suppressor. Fill to volume with 0.5M Polarographic KNO3. Transfer to a 150-ml. beaker method, Kramer's method, and adjust pH to 5.5 with dilute "03 % B203 % B203 and dilute KOH. Add 8 grams of mannitol and stir until dissolved. 32.47 32.8 32.66 32.9 Record the final pH. Transfer the 32.6 solution to the polarographic cell and deaerate with high purity nitrogen for a All determinations were made at room a minimum of 10 minutes. Scan from temperature. +0.5 to -0.2 volt us. SCE at a suitable sensitivity setting of the polarograph. Prepare a comparison standard by selecting an aliquot of equal size from the blank and following the same proEXPERIMENTAL For comparison purposes, a colemancedure through the addition of mannitol. Should the addition of mannitol ite sample was first analyzed for boron Apparatus. d Sargent Model XV cause the pH to drop more than 0.2 by Kramer's method (4) which employs polarograph was used in conjunction unit, discard the solution and prepare a with an H cell containing a saturated ion exchange to separate boron from new boron-free blank introducing an calomel reference electrode (SCE). the interfering metal ions in the sample. additional evaporation with methyl alNo damping was used. The prepared solution is then titrated cohol and 12N hydrochloric acid. ReThe dropping mercury electrode had with standard sodium hydroxide in the adjust pH to 5.5 if a smaller drift is a drop time of 5.96 seconds a t a height presence of mannitol. The results, noted. Add standard boric acid slowly of 59.8 cm. of mercury in 0.5M KNO3 which appear in Table 11, are compared with a calibrated syringe until the pH with no applied potential. The merwith those obtained by the polaroapproximates that of the boron sample cury outflow was 1.45 mg. per second. solution. Transfer to the polarographic graphic method. The solutions were thermostated a t cell and scan over the same voltage To confirm the reliability of the pro25" + 0.2' C., except where noted in range. Superimpose the wave produced the text. posed method, the boron content of on that of the sample. The two waves All pH measurements were made National Bureau of Standards Glass Yo. should be parallel and in near coinciwith a Beckman Zeromatic pH meter. 93 (B2O3, 12.76%) was determined. dence. Measure the diffusion currents Reagents. Quinone, Eastman pracThe results were = 12.7% with a a t corresponding points on the two tical grade was purified by a single standard deviation of 0.1 6. plateaus. Correct the wave height of sublimation. A fresh saturated soluThe direct comparison method (3) standard for the volume of boric acid tion of quinone in 0.5M KNOI was was used to evaluate the wave heights in used. prepared each day. the analysis of all samples. Inasmuch Standard boric acid was prepared as the half-wave potentials shift proby weighing B203directly. B203was RESULTS AND DISCUSSION gressively to more negative values with obtained by fusing boric acid, Fisher A-74, in platinum. decreasing pH it is important that the -411 other chemicals used were of To establish the limits of the linear final pH values of the sample solution reagent grade. relationship between the concentration and comparison standard be in close Procedure. PREPARATION OF of boric acid complexed with mannitol agreement; otherwise, a large error may GLASS A N D MINERAL SAMPLES. and the diffusion current (id), the data be introduced by failing t o measure the Weigh 100 mg. of the sample into a in Table I were obtained. The contwo waves a t corresponding points. 30-ml. platinum crucible, add 600 mg. stancy of i d / C was maintained until the Most of the ions usually associated anhydrous potassium carbonate, and concentration of boric acid exceeded with boron in minerals were tested for mix. Fuse over a Fisher burner. four millimoles per liter of polarographic adverse effects. The halides interfere After cooIing slightly, place the crucible in a 250-ml. beaker containing solution. unless their concentrations are very low SIR: I n a recent publication, Abbott and Collat (1) have shown that the polarographic reduction of 1,4-benzoquinone (quinone) can be used for the indirect determination of certain acids. In this study, the procedure of Abbott and Collat was adapted, with minor modifications, to the analysis of boron in glass and colemanite. The boron content of these materials is usually determined by methods which utilize the titration of the mannitol-boric acid complex with standard base; however, the preliminary separation of boron from the hydrolyzable sample ions is always required in order to obtain sharp and accurate end points (8). The prior separation of boron is omitted in the procedure outlined in this paper. Support for by-passing this step is provided by the results of numerous preliminary tests and the analysis of both synthetic and standard samples. These results indicate that under the conditions and techniques of the proposed method, low concentrations of hydrolyzable ions in the polarographic solution do not adversely affect the boron evaluation. By circumventing this step much time is saved, and a possible source of error is avoided. About 20 minutes is required for a single determination.
VOL. 37, NO. 12, NOVEMBER 1965
1587
in the polarographic solution. Low concentrat.ions of iron and aluminum do not interfere, but adverse effects were noted when relatively large amounts of these ions mere introduced
not show adverse effects a t concentrations within the limits imposed by the recommended size of the sample aliquots.
(3) Kolthoff, 1. M., Lingane, J. J., 3,5, “Polarography,” Interscience, 2nd New ed., York, Vol. 1952, I, p. (4) K ~ H., A~ ~ cHEM. ~ ~ ~~27, 1M . ~
LITERATURE CITED
in tests’ As little 10 p.p.m. of arsenate and chromate interfere. Silicate and phosphate do
(1) Abbott, J. C., Collat, J. W., ANAL. cHEM, 35, 859 (1963). (2) Frank, A. J., Ibid., 35, 830 (1963).
CLIFFORD C. BOYD Department of Chemistry East Tennessee State University Johnson City, Tenn.
(1955). .~.~
, .
A Method for the Determination of Equivalence Point in Potentiometric Titrations Using Unequal Volume Increments SIR: ,Methods for determination of the equivalence point in potentiometric titrations can be classified into three groups: first, Kolthoff (7, 8 ) and other authors (2, 6, 10) used first and second differences of dependent variable to locate the end point; second, Hahn (4) offered a method of calculation through difference “quotient” parameters; third, Cavanagh (1) and Fortuin (5) derived a “general” expression for the potential-volume curve which is based on the Nernst equation. In all of these methods, there is one point in commonthat is, the choice of equal volume increments. The method outlined below, however, permits the use of unequal volume increments and is based on fitting a cubic function approximation to experimental data. Starting from the assumption that the inflection point in a potentialvolume curve exactly coincides with its equivalence point, then the second derivative of the relation E = E ( v ) , if any, must be zero. For those cases where no potential-volume relations are available, as are normally encountered in actual titrations, the method of “divided difference” in the interpolation theory serves as a useful tool. If a function y = f(z) can be approximated by a polynomial P ( z ) in a given interval of the independent variable z, then for unequally spaced increments of x, the first, second, . . . . . . and r-th order “divided differences” are defined as follows:
and Newton’s Interpolation Formula (5, 9) provides an expression to relate x and y:
= P(2) = ( 5 - ~ o ) A i ( z o ,5 1 ) (5 - 20)(5 - z J A z ( 2 0 , 21, z2)
y = f(z)
Yo
+
+
.
.
.
I
.
.
1588
$2,
. . ., 2,) ES
.
+
.
+
The definition and evaluation of the divided difference may be tabulated for more convenient operation, thus 2 50
Ai
Y yo
(r = 0 , 1, 2, . . , .) ( 5 )
A2
- Vo
\Vl
For equally spaced intervals of
5,
say
where
ax, the r-‘border divided difference is sidply
“difference”
- ryn+r-l
...
11
(2
- 2k)
(3)
r!o’
where 6‘f is the +-order of y = f (5) :
- 50
=
k-0
6.f
Ar =
Z t
I-,
J - L
Fd.1
A ~ - ~ ( x I52, , . . . ,G)- A ~ - I ( ~ 21, o,
ANALYTICAL CHEMISTRY
.
(2)
yn-,
Ar(2oJ 21,
.
One of the important properties of divided difference is that +-order divided differences of a polynomial of the r-th degree are constant, hence the (r+ l)-th-order divided differences of the polynomial vanishes. This property holds true also for the equally spaced “difference.” Differentiating Equation 2 r times, as given in reference (Q),results in
j
%-I)
( j = 0,1, 2,
. . *).
The first and second derivatives of the function y = f (2)are obtained by setting
- 1) + r(r yn+r-z + . . . + (-1PY” 2!
=
r = 1 and r = 2, respectively, in Equation 5 :
,
