1112
ANALYTICAL CHEMISTRY
calcium carbonate and resin. The volat’ilefiber finish is included in the values for resin content and could only be estimated. The results preqented (a) indicate the magnitude of the error int,roduced by calcium carbonate decomposition on results obtained using normal analytical ignition procedures for t,he determination of resin content, and ( b ) provide a simple, but preciw, tcchniqrie for correcting the resultant errors. The method described is wefill in t h o x cases n-here resin content is limited by specifiration, and where failure to correct for the loss of carbon dioxide n-orild result in the rejertion of the material for its specified I I Y , :mc1 n-here fairly accurate resin contents are needed.
LITERATURE CITED
(1) Hamilton, L. F., Sinnpson, S.G., “Talbot’s Quantitative Chemical Analysis,” p . 320, Xacmillan, Yew York, 1948. (2) Hodgman. C. D., “Handbook of Chemistry and Physics,” 34th ed., p. 462, Cherniral Rubber Co., Cleveland, Ohio, 1952. (3) XIilitary Specification IIIL-P-8013 (L7.SA.F.), “Plastic AIate-
rials, Glass Fabric Base, Low Pressure Lamimted, Aircraft Structural,” Test lIet,hod 4.2.2.1.2. RECEIVED for review May
4, 1935. Accepted April 19, 1956. T h e x v r k reported here \vas sponsored by t h e Materials Laboratory, Directorate of Research, Wright Air Dewlopment Center, V.S.A.F., under Contract No. 33(038)51-4060.
End Point Calculation in Conductometric and Photometric Titrations ERNEST GRUNWALD’ The Weizmann lnstitote o f Science, Rehovoth, lsrael
i11 objeetibe inchhod of end point calculation is described for conducton~r.tricor photometric titration curves which have no sharp end-point indicating change of slope. These include all titrations which Mould be difficnlt or impractical by the potentiometric method. The method makes use of the cur\ed part of the titration cline near the equikalence point and niinimises end point errors due to kariations in the equibalent conductivities, dekiations from Beer’s law, and sj stematic experimental errors. 4n end point accuracy of better than 0 . 5 7 ~is obtained eben in cases where the end point error may be as large as lO7c by other methods of calculation. Yo additional data are required.
procedure has a sound basis i n theory. There is a linear r:ingc\ before the equivalence point where the added titrant reacts quantitatively x i t h the substrate; there is a linear range past the equivalence point \There the added titrant no longer reacts with the substrate, the substrate-titrant reaction being complete; and the intersection of the tn-o straight lines defined in this way occurs at the equivalence point.
C
O X D U C T O l I E T R I C and photometric methods of titi,ation can give accurate results in many cases where thp potentiometric method fails because the inflection point on the plot of electroinotive force us. volume of titrant is not clearly dpfinrd. For esaniple, the photometric titration of p-bromophenol i p I i ~ 9.2) with aqueous sodium hydroxide can be accurate to hrtter than lye a t concentrations as low as 0.001M ( 3 , 4). Similarly, 0.001.1f p-nitrophenol (pEi.4 7.0) can be titrated photomrtrically in the presence of 0.001JI m-nit,rophenol (p& 8.3) with an accuracy better than 2Ce ( 3 ) . However, in all cases where the potentiometric method fails, the conductometric and photometric titration curves are also difficult to interpret because there is no sharp change of slope at the equivalence point. The difficulties are illustrated by the titration of 0.01.Y aqueous sodium acetate with 1 S hydrochloric acid, a titration which would be difficult by the potentiometric method, The conductometric titration curve in Fater a t 25” C. is shown in Figure 1. The ordinate, K , is the conductivity, and the abscissa, E, the ratio of the equivalents of titrant to those of substrate. The curve was computed using the h o l m values of the dissociation const,ants and equivalent conductivities a t the ionic strengths existing during the titration ( 7 ) . -4s the figure shows, there is no sharp end-point indicating change of dope at the equivalence point. However, there are trro nearly linear segments, one before and the other past the equivalence point. The end point is usually taken as the intersection of the two straight lines drawn through the points in these linear ranges. Under certain idealized conditions (no changes in the equivalent conductivities and no volume changes during the titration), this 1 Present address, Chernktry D e p a r t m e n t , Florida Tallahassee, F l a .
S t a t e University,
.5
c
0I
03
0.5
0.7
0.9
E
1.1
1.3
1.5
1.7
1.9
Figure 1. Conductometric titration of 0.01.V sodium acetate with 11%’ hydrochloric acid in water at 25’ C.
I-nfortunately, it is difficult in practice to recognize these linear ranges. For example, in Figure 1, only the portion of the titration curve between E = 0.9 and E = 1.1has pronounced curvature. Because of experimental error, different portions of the nearly linear remainder will appear to be linear in different titrations, and the end point is sensitive to the €-range selection which is made. -4s shown in Table I, errors of the order of 2y0 may arise from this cause. (This error is, of course, in addition to that causrd by the uncertainty of the experimental data.) Moreover, the actual choice of the linear ranges is subjective. T o make the selection of the linear ranges objective, one poesi-
V O L U M E 28, NO. 7, J U L Y 1 9 5 6
1113
Iile approach is to precalculate their positions t1,oni tliwry. This approach \vas sucwssfiil in the recent n-ork on photometric titration (S), hiit in general is subject to serious limitations. First, t h(>relevant ecluilibriiim constants and substrate concentrations must be known in advance, a t least approximately. Second, bec:ruse the linear ranges lie well before and past the equivalence point, progwssive changes in the equivalent conductivity or pi,ogressive devi:itions from Beer's 1:in. dwing the titration c:an largP ( ~ r t ~ o i ~This s . second limitation is illustrated b y thr coiitliic~toiiirti~i(~ titration of 0.01S :mimoni:i ivith 1.V : r [ v t i c * :u:id i n :tnh>.tlmri!: inethanol a t 18" C.! Reaction 1. SH1
+ CIIJCOOH
SH:
+ CHICOO-
.I)
Therrlevmit data are shonn in Tal)le 11. arid the tiiuitioii ~ ~ u is w e plotted in Figure 2. [The s y i n i ~ o K l is used i n this papru to denote thr,rniotl!.riiiiiii(, ec1dibi~iuniconstants, and k t o tietiotc. z(1iiilil,t~iiinic'oiistmts bawd on molar concentrations (6).] The d;it:i i n c~oliiniri4 of Table I1 shorn that more than 98', of thc 0.5, and that the protoit trmsfet~ iiddrd t i t i w i t isprotol>,zeda t E to thr :immonia is better than 9iyc complete \\-hen E >, 1.5. (There is some dilution diiririg the titration.) =\ccorclitigly, t h e liii~nrr:ingcls might IIP sr~lwtcdas 0 < E < 0.5, anti E >, 1.5.
Irased on the fact that, for any two points on the titration curve = 1, there exist two conjugate points above E = 1 (actub t ~ l Eo ~ :illy, an infinite set of conjugate points) such that the straight lines d r a m through the two points below E = 1 and through the two conjugate points zbove intersect exactly a t E = 1. The €-values of these conjugate points can be calculated when an :inalytical expression for the titration curve has been ohtiiintd. In the geneid case, the E-values depend in a complicated miy on the substrate conrrnti~stionand on the equilibrium constant for the siilostr:itr-titi~aiit~,rttcfion. However, in the importmt sperial case where s i i b s t ~ t aiid c ~ titrant react in 1 to 1 molar amoiint? - - ~that is, in :dl xrid-liaae titrations and in many precipitation and i,.:idatioii-r~~ductionp r o c ~ -the e-values turn out to be indeum aiid theii~ p~wclent of concentratioii iind ~ c ~ u i l i h i ~ i constant! c.:ili.iilation is a simple matter.
1 such that the straight lines represented by Equation.. 3 and 4 intersect exactly at E = 1.
( G - G I ) / ( €( G - G ~ ) / (E
€1)
=
(G, - G I ) / ( € *-
€1)
(3)
(G, - G3)/(C4-
€3)
(4)
ANALYTICAL CHEMISTRY
1114
The result,ing expressions for G are:
This imposes the restraint,
[Gdl -
- Gi(1 - e z ) l / ( ~- € 1 )
€1)
E
=
- e4)1/(% - € 3 )
[G4(1 - €3) - G d l
(5)
Because Equation 5 is the sole restraint upon the four points, three of the points are independent. The fourth can be computed from Equation 5 when G is known as a function of e. Weak Base Plus Weak Acid. Equation 5 ail1 now be applied to a specific example, the titration of ammonia whose initial concentration is b, with acetic acid whose concentration is so large that volume changes during the titration can be neglected. At a given value of e before the equivalence point, the concentrations of ammonia, acetic acid, and ammonium acetate are, 21, 2, and ( b e - x). By use of Equarespectively, [b(l - e ) tion 2, G is given by
+
G
Go
+ gib + g2bE - 6gx
(6)
where
91
= QKHa
g? = gNH,Ac - 91 ag 92 - gHAo
Past the equivalence point, the concentrations of ammonia, acetic acid, and ammonium acetate are, y, [(E - 1) b+ y], and ( b - u ) , respectively. Therefore,
G = Go
+ g3b + gibe - 6gy
(7)
E
< 1: > 1:
G = Go
G = Go
+ gib + gib€ - 8gbEz/(l - E)kP + gab + gdbe - 6gb/(€ - l)kp
(9) (IO)
Equation 9 applies to the left-hand side of Equation 5 , and Equation 10 to the right-hand side. When the mathematical operations required by Equation 5 are carried out, all terms and factors involving the b, k p , or gi values can be eliminated, and this result is obtained:
+
€2 €1 - %e2 - E, f €3 - 2 (e, - 1)(E4 - 1) (1 - E2)(1 - E l )
(11)
Equation 11 contains no parameters characteristic of the specific example given; therefore, i t is a general result and will apply equally to any other titration of weak \lase with weak acid or of a Teak acid with weak base. According to Equation 11, for r t i i ~ .two points €1 and EP there is an infinite set of conjugate points E ) arid €4, because either € 3 or e4 is still independently variable. The variation of €3 with e4 for the specific case el = 0.5, e2 = 0.9 iu jhown in Figure 3. In order to minimize the end point error> it is advisable to choose ea and e4 not too far from the equivalence point, yet far enough apart to define a straight line with jatisfactory accuracy.
Table 111. End Points in Conductometric Titration of 0.01N-4mmonia with 1NAcetic Acid in Methanol at 18" C. End Point,
a here
€1
g3 =
- gr, and gr
gNH4do
= gHAa
€2
0.5
0.5
-
According to E q u a t i o n 11, increaaing distance from
I
I
0.5
0.9 1 5 1 125 0.7 2.d 1 57 0.5 3.3 2 0 Effect of deviations iron1 E q u a t i o n 1 1
0.5
0.9 0.8 0.9 0.9
0.3 0.3
I
I
I
1.4
-
I I I I I
0.5
0.6 0.5 0.5
I
0.9
1.5
1.5 1.3 1.4 1.5
E
€6
According t o E q u a t i o n 11, pointa near E = 1 1.3 1,125 0.9 0.95 1, 3 1.055 0.8 1.5 1.33
0.7
1.6
6:
1.11 1.11
1.11 1.11 1 21
1.004
0.993 1.000 1 1.004 0.967 0.947
E =
1.000 0.977 1.006 0,998 1,019
1
I I I
P W
I
I
1.2
-
I I
I
_ _ _ _LI _ _ _ - _ _ _ _ - - - - - - - - I
I I
1
I O I O
I
I
I
1.2
14
1.6
€3
Figure 3.
Plot of
El = 0.5;
€2
€ 3 US. € 4
= 0.9
Kow x and y must be expressed as functions of E. This is done by means of the equilibrium constant for Reaction 1namely,
Equation 8 may be simplified on the following basis. The terms 6 g x in Equation 6 and 6gy in Equation 7 are the deviations of the (G,e) plots from a straight line. Except very close to e = 1, these deviations are small, and x can be approximated with sufficient accuracy by be2/(1 - e ) k p , and y by b / ( e - 1)kp.
The application of Equation 11 will now be illustrated by the data, which are listed in Table 11, for the titration of 0 . 0 1 N ammonia with 1 N acetic acid in methanol. Because three of the four points may be chosen independently, let us choose el = 0.5, €2 = 0.9, and ea = 1.5. The value of e4 is then obtained from Equation 11 as 1.125. The corresponding values of 1 0 3 ~ are 0.384, 0.603, 0.685, and 0.666, and the equations of the two conjugate straight lines are
1 0 3 ~= 0.110 1 0 3 ~= 0.609
+ 0.5-18e + 005ie
The intersection lies a t e = 1.004. Other end points computed by the same method are shown in Table 111. The end points based on Equation 11 are remarkably accurate when the points are fairly close to E = 1. As the distance from e = 1 increases, the end point error becomes larger because of increasing deviations from Equation 2. Hen-ever, by proper choice of the four points, an accuracy of better than 0.5% is readily attainable. This is the more remarkable when it is remembered that the titration is difficult not only because A varies, but also because the equilibrium constant for Reaction 1 is relatively small. i l t the equivalence point, k p = 65. In order for a titration of this type to be successful by potentiometric methods, k p should be a t least 104. The present method, therefore, extends the range of titratability to much lowx k p values without substantial loss in accuracy.
V O L U M E 28, NO. 7, J U L Y 1 9 5 6 Table I11 also shows the sensitivity of the end point to errors in the €-values. I n general, the sensitivity increases with the curvature a t el, €2, c3, or E ~ . However, it appears that errors of the order of 0.01 in E can usually be tolerated. Because of the approximations made in expressing 5 and y as functions of E in Equations 6 t o 10, none of the four points should be chosen very close to 1.000. I n practice, the accuracy is satisfactory if the range 0.95 < E < 1.05 is excluded. Weak Base plus Strong Acid. Consider the titration of sodium acetate whose initial concentration is a, with hydrochloric acid whose concentration is so large that volume changes during the titration can be neglected. If 2 equals the concentration of hydrogen ion before the equivalence point, and is the concentration of acetate ion past the equivalence point, these equations are obtained. E t
< 1: > 1:
G = Go G = Go
+ gba $. g6aE
- 692:
+ g7a + gaae - 6gy
(12)
1115
roo 650
-
600
-
550
-
9
LT 4.50
I
I
98
+-
gHCl
I
I
I
I
1
4.00
1
/I
9
-
I
I
-
--
-
I
Figure 4.
(13)
I
- y5
f g8
- gHfAc-
According to the same approximation shown previously z = k ~ ~ / (-1E), and y = k ~ / ( e - 1). Using these values in Equations 12 and 13 and substituting in Equation 5, Equation 14 is obtained as the restraint upon el, E*, E ~ and , c4.
I
I
I
I
I
I
I
I
I
I
Conductometric titration of 0.008N sodium azide with 0 . 1 s hydrochloric acid in water [Zdl
gXaC1
I
503-
where 96 = gNaAo gB = gHAo gNaCl g7 g8 gHAo
I
- €1)
- r , ( l - e z ) l / ( E o - €1) = - € 3 ) - Y3(1 -
%)]/(E4
-
€3)
(15)
When the substrate and titrant react in 1 to 1 molar amounts, the equation among €1, € 2 , €3, and e4 does not involve any parameters characteristic of the specific titration, as shown by the following general result. Let the titration be represented by the chemical Equation 16.
A
+B
=
mC
+ nD
(16)
The relationship among the four e-values is then found to be Again all parameters characteristic of the specific example have disappeared, and Equation 14 is therefore applicable t o any titration of a weak base B+th strong acid, or of a weak acid with strong base. When Equation 14 is applied to the end point determination for the titration of 0.01A' sodium acetate n i t h 111' hydrochloric acid (for which the titration curve is shoir-n in Figure l), the results shown in Table IV are obtained, which demonstrate that an accuracy of 0.37, is pos&le. This titration viould have been very difficult by the potentiometric method.
Table IV. End Points in Conductometric Titration of 0.01N Sodium Acetate with 1NHydrochloric Acid in Water at 23" C. End Point, €1
€2
€3
€4
e
K h e n the stoichiometry is 1 t o 1 but two or more substrates are present, the relationship among €1, €2, et, and involves also the ratios of their concentrations. For example, if the weak acid HA (initial concentration a ) is titrated with strong base in the presence of a weaker acid H B (initial concentration b ) the relationship among the four +values ( E now being defined as equivalents of titrant per equivalent of HA) can be shon-n t o be
When the stoichiometry is not 1 to 1, Equation 15 is still applicable, but the final expressions involve the substrate concentration and the equilibrium constant for the substrate-titrant reaction. The method is still useful, but approximate values of these quantities must be known. APPLICATION OF METHOD
Other Cases. The method just described is readily extended to other types of titration. The fundamental equation is 5, and G must be known as a function of e . The relationship of G to E can be expressed simply and with sufficient accuracy as follow. At E < 1, the roilcentration, 5, of unreacted titrant, and a t E > 1, the concentration, y, of unreacted substrate, are computed from the appropriate equilibrium expressions by neglecting the effect of incomplete re:iction on the other concentrations. If, moreover, Equation 2 is valid (this is always the case near the equivalence point), Equation 5 reduces to the simpler form of Equation 15.
Convergence of End Point Calculations. When the present method is applied, a graph of the titration curve is constructed and a tentative estimate of the end point is obtained by conventional procedures. By use of this estimate and a convenient set of el, €2, €3, and €4 values, four points are located on the titration curve, the two conjugate straight lines are constructed, and a second estimate of the end point is obtained. This process is repeated until successive estimates converge. The data in Table I11 show clearly t h a t the convergence will be best if the four points are chosen so that they lie outside the region where there is great curvature. Even a rather large error in the e-values will then produce only a small error in the end point. This effect is illustrated by the following data for the
1116
ANALYTICAL CHEMISTRY
titration of 0.01N ammonia with 13- acetic acid in methanol a t 18” C. If the four +values are taken as 0.5, 0.9, 1.5, and 1.123 and if the first end point lies a t E = 0.900, successive end points nil1 be 0.932, 0.953, 0.967, 0.975. . . On the other hand, if the four E-values are taken somewhat further from the equivalence point as 0.5, 0.8, 1.5, and 1.33, successive end points will br 0.900, 0.962, 0.985, 0.992, 0.998. For the titration of 0.01.V sodium acetate p i t h 1 Y hydrochloric acid in water a t 23‘ C., if the E-values are taken as 0.5, 0.8, 1.4, and 1.29, successive end points xi11 be 0.900, 0.986, 1.004, I n this case the convergence is very satisfactorv. These calculations point out the practical limits of the present method. As the change in slope a t the equivalence point be(xinips less sharp and the curved part of the titration curve \\.idens, the accuracy of the first end point estimate arid the convc’i’gence of successive estimates both become poor. The two effects work in concert, and in cases of very gradual change of slope the calculat,ion becomes prohibitively long. However, eveii i n these cases the convergence limit of successive end points is very close to the equivalence point. I n the titration of sodium acetate the first, estimate of the end [mint is not likely to be in error by more than 3y0 (Table I). A siiiyle calculation suffices t o find the final end point. Tliis ea, and €4 values. may then be verified with a second set of el, 111 the titration of ammonia with acetic acid in methanol, the first rstimate may be in error by IO$%; with practice this error c:in be made smaller. Even so, two or three successive calcula:we likely to be required. hen the convergence of successive end point estimates is poor, an alternative niethod of calculation may be used. In&:id of making a single first estima,te of the end point, several estimates are made in such a way as to bracket the equivalence point. For each of these estimates a single four-point calculation is made, and the calculated end point is plotted vs. the estimnted point. The correct end point is found on the plot as the point where the calculated and the estimated end points are equal. This method has the advant,age that the convergence limit’ is :ipproached from both directions. Practical Example. Figure 4 shows the conductometric titration curvc for 0.008N sodium azide with 0.lA‘ hydrochloric arid in water. The titration data were obtained on a Shedlovsky bridge (8) with earphone detector. The temperature of the tit,ration eel1 was constant t o within 0.2’ c. The cell constant was too large to obtain maximum accuracy in the resistance measurements ( 5 ) , but the experiniental points fell on a smooth curve with a mean deviation of 0.570. I n Figure 4, the ordinate Vo)/VoR,where V is the volume of titrant, Vo the initial is (1’ volume of sodium azide, and R the measured resistance. T h e factor (1’ VO)/VOcorrects for the dilution during the titration.
.
+
+
Table 1.. End Point Calculation for Titration of 0.0085 Sodium Azide with 0.LV Hydrochloric h i d in Water t
0 7 0 85 0 li 0 (i
€2
e3
€4
0.9 0.89 0.93 0.915
1.3 1.2 1.2 1.3
1.11 1.1 1.1 1.1
End Point, 111. 19.90 19.99 19.75 19.97 .I\-. 1 9 . 9 0 f 0 08
-4first estimate of 19.44 ml. for the end point was obtained by extrapolation of the nearly linear segments observed for 0 < T. < 10 and T 7 > 22. .A single calculation, using the four points 0.7, 0.9; 1.3, :tnd 1.11 multiplied by 19.44 ml., led to an end point voliinie of 19.89 ml. -4second calculation, using 19.89 ml. as the end point estimate, led to 19.90 ml. This end point vas chwked with three other sets of values of €1, E ? , €3, and €4. The results are shown in Table T’. T h e mean deviation of the four values \vas only 0.4%: all five calculations took less than 30 minutes. This titration is comparable to that of sodium acetate because the K A values of acetic and hydrazoic acids are nearly equal. The relatively large discrepancy between the first estimate and the final end point is not surprising. For similar +range selections, the tentative end point in the titration of sodium acetate occurred a t E = 0.974, as shown in Table I. This is in good agreement with the observed ratio, 19.44/19.90 = 0.977. By the potentiometric method this titration would have been very difficiilt. ACKNOWLEDGMENT
The author wishes to thank the Yad Chaiin Weizmann for t h e award of a Chaim Weizmann Fellowship, and Dan Golomb for supplying the titration data for Figure 4. LITERATURE CITED
(1) Bacarella, A. L., Grunwald, E., hlarshall, H. P., Purlee, E. L., J . Org. Chem. 20, 747 (1955). (2) Bjerrum, S . , Unmack, A., Zechmeister, L., KgZ, Dambe T’idenskab. Selskab., .Vat.-fus. M e d d . 5 , 11 (1921). (3) Goddu, R. F., Hume, D. S . , i l x a ~CHEM. . 26, 1679 (1954). (4) Ibid., p. 1740. (5) Golomb, D., Weizmann Institute of Science, Rehovoth, Israel, private communication. (6) Grunwald, E., AXAL.CHEM.26, 1696 (1954). (7) Harned, H. S., Owen, B. B., “Physical Chemistry of Electrolytic Solutions,” Reinhold, New York, 1943. (8) Shedlovsky, T., J . Am. Chem. SOC.52, 1793 (1930). R E C F I V Efor D review September 26, 1955.
Accepted March 24, 1956
Automatic Photometric Titrations THOMAS L. MARPLE and DAVID N. HUME Department o f Chemistry and Laboratory for Nuclear Science, Massachusetts Institute o f Technology, Cambridge 39, Mass.
A simple logarithmic attenuator circuit which makes possible the direct recording of absorbance from the output of a Beckman RIodel spectrophoton,eter is described. When used in conjunction with a stripchart recording potentiometer and a constant-deliverY reagent the apparatus is to automatic Photometric titrations. The application to automatic iodometric and acidimetric procedures is described.
A
\TOKG several physicochemical methods available for lo-
cating titration end points automatically, measurement of light absorption has received surprisingly little attention. Several workers have suggested arrangements in which a large change in transmittance at the equivalence point could operate a trigger mechanism or be recorded as an indicator-instrument deflection ( 2 , 5, 7 , 14, 15). For many reactions, however, the equivalence point must be located by examination of an absorbance us. volume curve ( 4 ) . Heretofore, the only reported application of this technique to autornatic titration has been that of
