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, 3 5 , 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 :
,
P'(z) A&o,
-
Ai(20, 21)
d
d (2
21, $2)
A3(xO, 21, (z
2 2 , $3).
- zl)
(5
(2
- 20)
- ZO)(2 - $1)
-k
& d (5 - 20) x
-
+ ....
22)
(6)
A simple expression can be obtained by taking the first two terms in Equation 7 : f"(2)
= P"(x) = ~ & ( - C O , xi, $2) 2A3(xO, $1, 2 2 , [ 3 . ~- (20
Equal volume increments (1 drop of solution equals 0.04 ml.) : NaOH used (ml.) 26.50 PH 7.15
23)
ZZ)]
61
(8)
Y1
7 . 5 0)0.72 > ' ~ ~ > 0 . 3 7> - 0 . 5 5 8.22 )-0.18 >O .54 8.76
Using Equation 10, with 62 the end point is found to be
52
=
x
=
26.54
-
(s)
(0.04)
3%
=
=
and from Equation 3,
Unequal volume increments:
&(ZO,21, -CZ) =
NaOH used (ml.) PH
6 2f
A3(20, 51, 2 2 , 2 3 )
x
= 2,
- -.
6 3f
26.58 8.20
26.68 9.20
67
= -
6(8x)3
0.04,
26.567
26.48 7.05
26.55 7.65
The divided differences are tabulated as :
Equation 9 then reduces to : 6'f
26.62 8.76
63
7.15
Y3
As a special case, when equally-spaced intervals of 2 are used,
62
Yo
Y2
21
26.58 8.22
X
$1
On setting Equation 8 equal to zero, as required by the process of finding the inflection point, and solving for x:
+ +
26.54 7.50
The differences are tabulated as:
+ +
20
of equivalence point determination outlined above, as well as Kolthoff's method, is a cubic function approach. Small volume increments favor the validity of these methods. The unequal volume method, as compared with Kolthoff's method, uses a slightly more elaborate calculation but avoids the experimental difficulties in adjusting small volume increments such as exactly one drop from the tip of a buret. It is well known that a cubic curve possesses a real inflection point and is symmetric with respect to that point. Thus accuracy can also be improved by choosing four points located as symmetrically to the inflection point as possible.
However, accuracy can be improved by taking small volume increments. As an example for the use of Equation 9, a sample of acetic acid solution was titrated with standard sodium hydroxide solution. Using equal and unequal volume increments for the same t i t m tion, the following data were obtained:
AI
A3
(6%)
which is an interpolation expression of Kolthoff's method. It has been attempted to derive Equation 10 directly from the interpolation theory of equally spaced "difference," but this attempt has never been successful. Thus the above formulation provides also a satisfactory mathematical basis for Kolthoff's method. The validity of Equation 9 and 10 depends on the relative magnitude of the third and higher terms in Equation 7, as compared with the first two terms. Unfortunately these terms do not always converge rapidly; a deviation from this cubic function approach is to be expected in actual titration curves.
21
26.55, y1
22
26.58, yz
23
26.68, y3
/ox
8.20\ 1 .o
The end point is calculated from Equation 9 :
ACKNOWLEDGMENT
Thanks are due to Stanley Bruckenstein for his beneficial suggestions and encouragement, and to G. H. Brown and J. E. Taylor for help in preparing the final form of the manuscript.
% =
1 [26.48 3
+ 26.55 + 26.58 -
z] (- 809)
=26.577
The best inflection point in an enlarged plot of the titration curve is 26.58. It can be concluded that the method
LITERATURE CITED
(1) Cavanagh, Bernard, J. Chem. Sot. (London) 1930, 1425. (2) Fenwick, ~ lA ~ ~CHEM. ~ 4, ~~ 144 (1932). VOL. 37, NO. 12, NOVEMBER 1965
1589
.
~
(3) Fortuin, J. M. H., Anal. Chim. Acta 24, 175 (1961). (4) Hahn, F. L., 2. Anal. Chem. 163,
169 (1958). (5) Hartree, D. R., “Numerical Analysis,” pp. 86-9. Oxford University Press, London, 1958. (6) Hostetter, J. C., Roberts, H. J., J. Am. Chem. SOC.41, 1337 (1919). (7) Kolthoff, I. M., Laitinen, H. A., “pH
and Electrotitrations,” 2nd ed., p. 110, Wiley, New York, 1944. (8) Kolthoff, I. M., Sandell, E. B., “Textbook of Quantitative Inorganic Analysis,” p. 488, Macmillan, New York, 1952. (9) Korn, G. A., Korn, T. M., “Mathematical Handbook for Scientists and Engineers,” pp. 651-2, McGraw-Hill, New York, 1961.
(10) Lingane:, J. J., “Electroanalytical Chemistry, p. 70, Interscience, New York, 1958. JOHNSON F. YAN Department of Chemistry Kent State University Ohio WORKsponsored by the National Institutes of Health, Project No. GM 08961-02.
Detection and Recovery of Biological Oxidation Products of Hydrocarbons by Gas Chromatography
0
4
0
12
20 24 28 32 36 RETENTION TIME (MINUTES)
16
40
44
48
52
Figure 1. Chromatogram of a synthetic mixture of alcohols and fatty acids 0.25% Carbowax 2 0 M and 0.4% ;so-phthalic acid on 200-micron microbeads (acid-washed); 6-foot X ‘/cinch 0.d. column; He, 20 ml./min.; injector and detector at 2 4 0 ’ C.; programmed from 60’ to 180’ C. a t 4’/min. Instrument: F & M Scientific; Model 700, dual Hz flame ionization detector
SIR: Growth of certain microorganisms on long-chain alkanes and alkenes (GO to CM) leads to an accumulation of oxidation products in the culture medium. Thin-layer chromatographic (TLC) analyses of ether extracts of culture media invariably indicated the presence of long-chain monocarboxylic acids, usually longchain alcohols, and in some instances hydroxy and dicarboxylic acids. Monocarboxylic acids and alcohols were separated as classes by preparative TLC or column chromatography using silica gel. The two classes of compounds were further analyzed by gas liquid chromatography (GLC). In order to reduce the time required for analytical procedures and to minimize sample losses, fatty acids were determined as free acids with the column packing suggested by Nikelly who used acid-washed glass microbeads coated with a polar liquid phase and {so-phthalic acid as an additive to eliminate adsorption and dimerization of the fatty acids (3). This method proved expedient because glass microbeads are easy to coat and once coated the packing of the column is-readily achieved. Moreover, little if any column conditioning is needed before use. This column 1590
ANALYTICAL CHEMISTRY
I
6
IE
24 30 36 TIME (MINUTES)
42
48
54
Figure 2. Preparative chromatogram of free fatty acids produced by Pseudomonas oeruginoso 3/a-inch a.d. X &foot aluminum column packed with 200-micron, acidwashed, glass microbeads coated with 0.35% Carbowax 2 0 M and 0.4% ;so-phthalic acid. Temperature 175’ C. Flow rate 150 cc. per minute. Wilkens Autoprep, Model A - 7 0 0 (1 ) Tetradecanoic acid (2) 13-Tetradecenoic acid
is also suitable for the analyses of longchain alcohols. Figure 1 shows a chromatogram of a programmed analysis of a known mixture of alcohols and monocarboxylic acids. Tetradecanoic acid and an unsaturated C14 acid, as well as several other saturated and unsaturated acid pairs, were identified from the culture fluid of Pseudomonas aeruginosa grown on tetradecene-1. Unsaturated acids were determined by peak disappearance after bromination ( 2 ) . Good separation of the saturated-unsaturated pairs prompted attempts to separate the pairs by preparative GLC using the same packing. The best results were obtained with a column 4 feet long packed with 200-micron, acidwashed, glass microbeads to which the amounts of Carbowax 20M and isophthalic acid added were 0.35% and 0.4% by weight, respectively. At best there was still some peak overlapping between saturated and unsaturated acid pairs (Figure 2). Therefore it was
more feasible to collect each pair by preparative GLC and then separate the pairs by column chromatography on Agl\’Os-impregnated silica gel (1). By these methods we have recovered a sufficient amount of the CI4pair to identify the components as tetradecanoic acid and 13-tetradecenoic acid. Analysis of the alcohol fraction is, a t the moment, preliminary. Thus far, long-chain saturated and some unsaturated alcohols have been tentatively identified by GLC on the free-fatty-acid column. LITERATURE CITED
(1) . . DeVries, B., J. Am. Oil Chemists’ SOC.40, 184 (1963). (2) . , James, A. T., Martin, A. J. P., Biochek. J. 63, i44 (1956). (3) Nikelly, J. G., ANAL. CHEM. 36, 2244 (1964). A. J. MARKOVETZ M. J. KLUG
Department of Microbiology University of Iowa Iowa City, Iowa
