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Graphic method for the determination of titration end-points
Talanta.1966.Vol. 13.pp. 1431to 1442. PergamonPressLtd. Printedin NorthernIreland
GRAPHIC
Department
METHOD FOR THE DETERMINATION OF TITRATION END-POINTS *
FOLKE INGMAN and EBBE STILL of Analytical and Inorganic Chemistry, Abe Akademi, Abo, Finland
(Received 21 April 1965. Revised 9 April 1966. Accepted 10 May 1966) Summar-Aa extension of Gran’s method for the determination of equivalence points in potentiometric acid-base titrations is presented. Equations are derived assuming that the stability constants of the reactions are known. The equations make it possible to locate the equivalence point with fair accuracy when acids with stability constants ranging from KEa N 10’ to Kz* - lOlo are titrated. The limit of KL above which Gran’s equations do not yield satisfactory results is about 10’. Conditional constants and a-coefficients are used to extend the method to apply to more complex systems, where more than one
speciesreacts with the titrant. AMONG
the graphical methods for the determination of titration end-points published by various authors, those of Gran 1*2have attracted considerable attention. Gran’s first method1 involves the plotting of either AV/ApH or AV/AE as a function of V, the volume of added titrant, after values of E or pH have been experimentally determined for several values of V. In the ideal case the resulting plot consists of two straight lines, which intersect each other and the V-axis at the equivalence point. One of the drawbacks of this method is that the accuracy of the TABLE I.-GRAN’S
Substance titrated Strong acid Weak acid
SECOND METHOD
Functions plotted on the acid side of the on the alkaline side of the equivalence point equivalence point (V, + V)lO(k-pa) =f(V) V.lO”-*=’ =f(V)
(V, + V)lO’*H-“‘) = f(V) (V, + V)lO’p=-k” =f( V)
The titrant is in each case assumed to be a strong base. V, is the volume of the solution before the titration is started, Vthe volume of added titrant, and k, k’ are arbitrarily chosen constants.
plot is influenced by the accuracy of the experimental data, and another that the lines for titrations of weak electrolytes are more or less curved and do not yield reliable results, because the change in pH or E at the equivalence point is too small. The second method of Gran2 is based on an original idea of Ssrensen,8 who plotted {H}, the antilogarithm of -pH, as a function of V for titrations of strong protolytes, disregarding the effect of the changing volume. Sorensen employed an analogous method when titrating halides with silver nitrate. Gran introduced a correction for the volume change during the course of the titration. He also extended his method to titrations of weak protolytes, and to other types of titrations than those involving neutralisations. The functions plotted when Gran’s second method2 is applied to acid-base titrations are presented in Table I. + Presentedin part by E.S. in December 1962 at Finska Kemistsamfundet, Helsingfors. 1431
F.
1431
INGMAN
and E. STILL
Gran’s second method yields excellent results when a moderately weak acid such as acetic acid is titrated with a strong base. Two straight lines are obtained and the point where these lines intersect each other and the V-axis is the equivalence point. For weak acids with stability constants of 10’ to 10s the lines obtained by this method are no longer straight but slightly curved. Moreover, the asymptotes do not intersect at the V-axis, and the point where they intersect only occasionally coincides with the true equivalence point. These deficiencies originate in the simplifying assumptions made by Gran when deriving his equations. The aim of the present work is to show how it is possible, when the stability constant for the reaction is known, to derive an expression which, when plotted against the titrant volume V, yields a straight line that intersects the V-axis at the equivalence point. The expression for the titration of a weak acid with a strong base will first be derived. It will further be shown how it is possible to generalise this expression by the introduction of conditional constants and a-coefficients.4 The determination of the equivalence points of titrations of complex systems is thereby facilitated. As an example, the titration of a weak acid in the presence of another weak acid or a metal ion will be considered. TITRATION
OF A WEAK
ACID
WITH
A STRONG
BASE
A weak acid HA is titrated with a strong base (sodium hydroxide). The initial concentration of the acid is denoted by C,, and the initial volume by V,. The concentration of the strong base is denoted by CNaOHand the volume added by V. The consumption of the base at the equivalence point is denoted by V,,. The derivation is based upon the following three conditions: (a) The law of mass action must hold: HA+
KH _ tHA] HA- (H)[A] *
H 4-A;
(1)
(For convenience, all signs of charge are omitted in this paper). The constant Kg* is a “mixed constant”, valid at a certain ionic strength. Mixed constants, where the activities of the hydrogen and hydroxide ions (denoted by {H} and {OH)) are used in conjunction with concentrations of all other species, are very convenient for equilibrium calculations, since pH-values are nowadays almost solely determined potentiometrically. The activity of the hydrogen ion is related to the concentration by the expression {H}
=_fiJHl.
(2)
where fHis the activity coefficient of the ion. (b) The solution must be electrically neutral, meaning that
[Al + [OHI =
Wal
+ [HI.
(3)
(c) The equation ve&NsOH
must be valid at the equivalence point.
=
v0cHA
(4)
Determination
The concentration
1433
of titration end-points
of the sodium ions is
M
=
2vo+ v cKaOH
(5)
which, when combined with equation (3), gives
[Al = ?-
+ WI -
v, + v C&H
[HA] = +
C,,
- [A] = +$
0
WI
CNaoH-
[HI + [OHI.
(7)
0
Substitution of (6) and (7) into (1) and subsequent rearrangement of the terms yields Veq -
V = V{H}K&
+
~([Hl-
tOHI)(1+ (H}K,H,I
The values of most of the constants have been determined at an ionic strength of ,U = 0.1. It is thus practical to adjust the ionic strength to this value, and in our experiments sodium perchlorate was used for this purpose. It may be noted that in equation (8), the concentrations [H] and [OH], caused by the electroneutrality rule, occur in addition to the hydrogen ion activity {H} determined potentiometrically. For accurate analyses the [H] and [OH] values in equation (8) should be calculated from the activities. This means that at an ionic strength of 0.1, [H] = {H}/10-“~08 and [OH] = {OH}/10-0.12. A comparison of equation (8) with the Gran equations for the titration of a weak acid reveals that Gran plotted only the first term V{H}K&, written in the form VIO(k-*H), on the acid side of the equivalence point, k being a constant arbitrarily chosen to yield numerical values that could conveniently be handled. On the alkaline side of the equivalence point Gran plotted the term (V. + v>[OH]/C,,,, against V. Gran wrote the term in the form (V, + V)lO(pH-K), k’ being a constant similar to k. When plotted against V, equation (8) yields a straight line with a slope of 45”. The titration curves of p-alanine (log K& = 10.2) and ammonium nitrate (log K& = 9.4) with sodium hydroxide are shown in Fig. 1. The titrations were performed at an ionic strength of ,u = 0.1. Kielland’s values6 were used for the activity coefficients, and the values of the stability constants were taken from the literature,6 The use of equation (8) is illustrated in Tables II and III and Fig. 2. For comparison, Fig. 3 shows the curves that result when Gran’s formulae are used for the titration of ammonium nitrate. The method described can be used only if the value of the stability constant K& is known. This could naturally be considered a serious drawback. The method is, however, intended for use in cases where no other means are available for the determination of the equivalence point with satisfactory accuracy. Such cases are encountered in practice when very weak acids (7 < log K& < 10) are titrated. The limit beyond which the simple Gran equations do not yield satisfactory results can be considered as log K& N 7. For most analytical purposes the value of the stability constant can be estimated with satisfactory accuracy from the pH value at the point where VN V,,/2, i.e., 3
F. INGMAN and E. STILL
1434 12 77 10 9 PH
18 7 6 5 4 0
-1
2
3
4 5vs
7
a
g
10
FIG. l.-Titration of 50 ml of 1.07 x lO-SM p-alanine (log KL = 10.2) and titration of 77 ml of 1.02 x lo-*M ammonium nitrate (log K& = 9.36) with O*lOOM sodium hydroxide. At the equivalence point, the consumption is theoretically 535 ml for &alanine and 7.85 ml for ammonium nitrate. TABLEII.-/I-ALANNE log v, + V
V ::; 2.0 2.5 3.0 3.5 40 4.5 5.0 5.5 6.0
pU3
CNaOE
955 9.77 9.95 10.12 10.28 10.43 10.59 10.75 10.91 11.07 11.21
1%
2.71 2.72 2.72 2.73 2.73 2.73 2.74 2.74 2.75 2.75 2.75
-4 -4.1133 -3.93 -3.76 -360 -3.43 -3.29 -3.13 -2.97 -2.81 -2.67
(HI&
1+
(W-II - [HD
1 + l~“6 100.” = 1 + 100.86 = 1 + 100.08 = 1 + 10-0.08 = 1 + lO-0.as = 1 + lO-0.*0 = 1 + 10-0.65 = 1 + 10-0.1’ = 1 + 10-0.81 = 1 + 10-1.0’ =
K, 100*‘4 loo.67 100.U 1OO.U 100.N lOWI 100.16 100.” 100.08 100.06 100.04
4.47 4.03 3.56 300 2.50 2.06 164 1.27 098 0.59 0.06
-
V, = 50 ml, log K, = 1400, log K& = 10.20 (from Fig. l), log foH = -0.12, VW -
V = V(H)K:A + E(D-U
-
-
v
0.13 0.15 0.17 0.20 0.25 0.30 040 0.53 0.73 0.98 1.32
= = = = = = = = = =
logf,
4.34 3.88 3.39 2.80 2.25 1.76 1.24 0.74 0.25 -0.39 -1.26 = -0.08
[OHD(1+ U-W&I
TABLE III.-AMMONIUM NITRATE
log V
3.0 4.0 5.0 6.0 7.0 8.0 9.0 10-o
PUS
8.50 8.89 9.15 9.38 960 9.85 10.18 10.69 11.14 11.37
v, + v
cN,O,
290 2.90 2.91 2.92 2.92 2.93 2.93 2.94 294 294
logWH1
[HI)
-
-538 -4.99 -4.73 -4.50 -4.28 -4.03 -3.70 -3.19 -2.74 -2.51
1 +
100.86
5
1 + 100’” = 1 + 100.8’ = 1 + 10-0.0’ = 1 + lO-0.S’ = 1 + 10-O.&@= 1 + lO-o.s* = 1 + lO-l.88 = 1 + 10-1.18 = 1 + 10-5.0’ =
I’, = 77 ml, log Km = 1400, log Kg* = 9.36, logfoX
V,, - V = V{H}KL
+
ZWl
v,, - v
1 -t U-XV&
= -0.12,
1(-y.=
l@*EO 100.“’ lOo.P@ lOWSO 100.” 100.0 100.0’ 100.0’ lOc!.OO
7.24 5.91 4.87 3.82 2.88 1.95 1.06 0.37 0.15 0.10
-
logfH = -0.08
- PHDU + O-IV&)
0.03 0.03 0.04 0*05 0.07 0.11 0.20 0.59 1.62 2.69
= = = = = = = = = =
7.21 5.88 4.83 3.77 281 1.84 0.86 -@22 -1.47 -2.59
Determktion
2-
of titration end-points
I p-ALAN/NE
Y
1
o-
1435
\I
1 7
I
I
I\
I
3
4
51
6
I
I
--vu
I
,\I
f I
X
9mllO I\
I
-1
FIG. 2.-Straight lines that result when titration data in Fig. 1 are substituted in equation (8). The principles of calculation are illustrated in Tables II and III. For /k&nine the straight line intersects the V-axis at V = 5.22 ml, which corresponds to an error of -2-4 %. For ammonium nitrate the straight line intersects the V-axis at V = 7.78 ml, which corresponds to an error of -0.9 %.
2
-1
FIG. 3.-The straight line (a) that results when the titration data for ammonium nitrate shown in Fig. 1 are used to compute V, - V from equation (8), and the curves (b, c, c’, c”) that result when Gran’s second method is used. The abscissa of the point of intersection depends on the value of the arbitrarily chosen constants k and k’ (cJ Table I). The values of k’ used for computing c’ and c” differed by &@30 from the value of k’ used for computing c.
when approximately half of the acid has been titrated. If a very high degree of accuracy is sought, the method developed by Goldman and Meites’ can be used. If an incorrect value of the constant has been used, the plot will not have a slope of 45” and will not be linear. It may be mentioned that the results of the titrations given in Figs. 1 and 2 can be found without using any graphic procedure by calculating Vea for the various volumes of titrant (ix., in the last column of Table II, V is added to F’eq- v).
F. INGMAN and E. ST~L
1436
However, this procedure is not always appropriate Table V). TITRATION
OF A WEAK
BASE
(cJ the titration presented in
WITH
A STRONG
ACID
An analogous expression can be derived for the determination of the equivalence point when a weak base is titrated with a strong acid (hydrochloric). The equation is : Veq -
V = V{OH}K,O& +
owl
- [HI)0 + WW%d
(9)
where
POW K% = [Bl(OH)’ This case will not be treated in this paper. TITRATION
OF TWO
WEAK
ACIDS
WITH
A STRONG
BASE
The determination of the total amount of acid in a mixture containing two weak acids HAr and HA,, is usually quite easily accomplished, provided both acids are strong enough to be titrated in the conventional manner, i.e., provided that both K&, and K&11 < 10’. When, however, the amount of the stronger acid (here taken to be HA,) in the mixture is to be determined, complications often arise owing to the fact that the difference between the stability constants of the two acids is too small to permit a conventional stepwise titration. When conditional constants and a-coefficients4 are used, it is quite easy to derive an expression analogous to the one already derived for the titration of one weak acid, and this will permit the determination of the first equivalence point. In this case the equation HA, + OH + A, + H,O
(10)
HA,, + OH =S A,, + H,O
(11)
represents the main reaction. The reaction
is considered a side-reaction, the effect of which is taken into account by introducing the appropriate cr-coefficient.8 The electroneutrality rule states that in this case
Dal + WI = (PHI + LW + [AI1= PH’I + LW
(12)
Comparison of this equation with equation (3) shows that [OH’] should be substituted for [OH] in equations (6) and (7), and this in turn means that the only change that need be made in equation (8) is the substitution of [OH’] for [OH], which yields
ve,-
J-,’= V(H}K& +
E
(WI - tOH’I)(l + (H}K&J.
(13)
As usual, [OH’] denotes the concentration of not only the free hydroxide ion but also of all the hydroxide ion added to the solution that has not reacted with HA, according to the main reaction. There will be no noticeable error if the activity
Determination of titration end-points
1437
coefficient foaP is considered equal to foH. The concentration that has reacted with HAzr is given by [A,,]. Hence WI’1 = [OH] +
IM =
of hydroxide ion
%H[OHI
(14)
where (15) To test the equation derived several titrations of chloroacetic acid (log K& = 2GQs in the presence of acetic acid (log K& = 4.7)” were performed using sodium hydroxide. As can be seen from Table IV and Figs. 4 and 5 the results were in close agreement with the theoretical. Figure 4 shows one of the titrations curves and Fig. 5 shows the straight line that results when Yeq - Y computed from equation (13) is plotted against Y to determine the location of the equivalence point.
‘3 PH
I
7 6
3
0
1
2
3 4 ---+v
5
6
7
FIG. 4.-Titration of 50 mi of a solution 0.72 x 10-W in cbloroacetic acid and 0.50 X lO-#M in acetic acid (A log K& = log Ii’& - logKm,, = 1.9) with O*lOOM sodium hydroxide. At the equivalence point, the consumption is theoretically 360 ml for chloroacetic acid. TABLEIV.-MONO~HLOROACETZCACID(HA*) AND ACETICACID@A~~) Y 0.0 0.5 1.0 1~5 2.0 2.5 3.0 3.5 4.0 4.5
~0.0 2.59 2.68 2.78 2.91 3.06 3.24 348 3.80 4.14 4.47
log v, + v ClWOli 2.70 2-71 2.71 2.72 2.72 2.73 2.73 2.73 2.74 2.74
[HI - [OH’1
1O-%8’ _ ] O-9.80_ IO-P.70_ 10-1.8’ _ IO-P.08_ IO-B.16_ IO-“‘0 _ ~0-9.72_ IO-“.00_ IO-“WJ_
v, = 50 ml, log K&, = 2~35, bg K& = 9.24, [HA,,1 log& = -0.08,
V,, -
VW-- v
I+ W&
~O-l.85=E*(ye.sa IO-4.8’ = ,0-W’ *o-*.1, =E*O-8.1’ IO-4.0P=i IO-#.86 10-m = 10-8.03 10-a.,* 3 IO-a.ao IO-Em =E 10-u* ,0-w, 5 _ 10-W’ ,O-“‘8” 1 _,O-8.87 10-8’61= - ,0-2.6#
V-
V(H}K&
0% + 0.74 + 1.18 + 1.30 + I.23 + 1.02 + 0.70 + 0.39 0.20 0.11 -
427 = 4.27 3.09 = 3.83 219 = 3.37 1.35 =I 2.65 0.79 = 2.02 0.38 = I.40 0.05 = 0.75 0.30 =i 0.09 0.78 = -0.58 1.70 = --I*59
10-a’so, log aoa = 6-94 log fox = --0*12~
+ ~GHI-
WU)(1
+ WK&)
F. INGMANand E. Snu
1438
FIG. 5.--Straight line obtained when titration data plotted in Fig. 4 are used to compute V,, - V from equation (13). The different steps of the calculations are shown in Table IV. The straight line intersects the V-axis at V = 3.62 ml, which corresponds to an error of +0.6x. TITRATION
OF A WEAK
ACID
IN
THE
PRESENCE
OF A METAL
ION
A very weak acid can be made apparently stronger if its conjugate base A produced in the reaction HA+OH+A+H,O
(16)
can be forced to take part in a side-reaction. The equilibrium of the main reaction will then shift to the right and the numerical value of the conditional constant4 of reaction (16), Kiu = CC&~, will increase. Many metal ions react readily with the conjugate base of an acid to form a complex according to the scheme A+M+MA.
(17)
As has been pointed out by Ringborn, 4 this side-reaction permits the titrating of many acids that are too weak to be titrated directly. However, the metal ions are themselves acids, and will consequently react to some extent with hydroxide ions. This side-reaction is undesirable, but cannot be avoided. It may be written M+OH+MOH.
(18)
Frequently matters are further complicated by the fact that the metal hydroxide starts to precipitate at a pH that is very close to the pH at the equivalence point of the titration. As a result, the titration curve will be deformed as in Fig. 6. In order to determine the location of the equivalence point on a titration curve of this shape, it is necessary to employ a method that utilises points on the titration curve that lie on the acid side of the equivalence point only. The value of the conditional constant of the acid may be low enough to permit the use of one of Gran’s simple equations. In other cases one may be forced to use an expression which can be derived similarly to the expressions already presented in this paper. The effect of these two side-reactions is taken into account by introducing the coefficients : aA
=
QOH(M) =
1 -I MKMA
(19)
1 +
(20)
WKHOH
Determination of titration end-points
1439
where Knaa and &on denote the stability constants of the complexes MA and MOH respectively. Assuming that a bivalent metal is added to the solution in the form of MB, the condition of electroneutrality now is: 2[B] + [A] + [OH] = [H] + [Nal+
2[Ml+
[MAI + [MOW.
(21)
When [MA] + [MOH] is added to both sides and ([Ml + [MA] + [MOH]) is substituted for [B], it is found that ([Al + [MA]) + (PHI
+ NOHI)
= [A’] + [OH’] = WI.
(22)
A comparison of this equation with equation (3) shows that the following changes should now be made in equation (8): K,Hk = K&/ccAcMI should be substituted for K&, and [OH’] should be substituted for [OH]. The final result is then V., - V = V{H}Kgf
+ F([H]
- [OH’])(l + {H}K,f)
(23)
NaOH Note. If a univalent metal is added as MrBr, the solution will be electrically neutral when WI1 + [Al + COHI = [HI + Ml + Ml. (24)
As tB,l = [WI + MA1
+ MOHI, substitution will again lead to equation (22). However, most of the metals that can be used for the purpose discussed are bivalent.
To determine whether equation (23) is valid in practice, an aqueous solution of acetylacetone (about lO_aM) containing an approximately three-fold excess of copper(II) salt was titrated with sodium hydroxide. In this titration the reaction between copper ions and the acid proceeds towards completion during the titration; therefore the conditional “constant” Kgf will not be constant but will attain the calculated value at the end of the titration. Nevertheless, an approximately straight line will be obtained, and the intersection with the V axis will coincide with the equivalence point. The stability constant is KHfIA= IV.@, corresponding to KiH = 105.1. When the appropriate values = 1 + 10-1.51(j3.2 = 106.7 (25)
uA(Cu)
and aOB~Cu~ f
1 + 10-1.5105.7= 104.?
(26)
are inserted, the conditional constant of the base formed will be Ki,u’ =
ctA(CujKzH_ 10e*7105’1 lo,_6
-_Z
a0H(Cu)
104.2
(27)
This constant corresponds to an acid constant of 10s.4 which determines the shape of the titration curve.0 The titration is thus a borderline case and could possibly be accomplished with visual end point indication, were it not for the fact that copper hydroxide begins to precipitate at pH very close to that at the equivalence point. Furthermore, the colour of the copper compounds may affect the colour change of the indicator.
F. INGMAN and E. STILL
1440
The titration results given in Table V and Figs. 6 and 7 are seen to be in good agreement with the theoretically calculated values. The ionic strength in this titration is not 0.1 but O-13. This, however, means a difference in the log fa value of only O-01, which is neglected. Figure 6 also illustrates the difficulties encountered when an attempt is made to locate the equivalence point on a titration curve which is deformed by the precipitation of a metal hydroxide. The pH of the equivalence point is 5.0 andcopper hydroxide begins to precipitate at pH N 5.2 (if the solubility product of copper hydroxide is 10-18.2,the point where precipitation should occur can be calculated to be pH N 5.6). 72 11
10
9
8 t ri7 6
0
1
2
3
4
-v
5
FIG. 6.-Titration of 50 ml of 0.91 x 10-eM acetylacetone
6
7
(log K& = 8.9) containing
about a three-fold excess of copper ([C&t,, = 0.04M) with O.lOOM SOCCU~ hydroxide. At the equivalence point the consumption is theoretically 4.55 ml. TABLE V.-ACXTYLACETONECOMPLEXEDWITHCOPPER
log v, + V V
p{W
CNsOH
o-0
2.63 2.76 2.93 3.19 3.38 3.66 4.30
2.70 2.71 2.72 2,73 2.73 274 2.74
1.0 2.0 3.0 3.5 4.0 4.5
log
1 -t- {J$KHA H,Aj
([HI - W-U) 1 1 1 1 1 1 1
-2.55 - 2.68 -2.85 -3.11 -3.30 -3.58 -422
+ + + + + + +
lO-0.aa = 10-M’ = 10-0.88 = 10-0.84 = 10-L” = lO-X.01= 10-8.06 =
V, = 50 ml, log K& = 8.95, log Kcti = 8.2, log KcuOH = 5.7, tLA(cu)= 6.7, log aOHICu,= 2.25, and log KziA’ = 2.25 logfn = -0.08, logfen = -0.12 H,A' V,,
-
V=
VIWKHA
+
E
K, -
100.X loo.‘* 100.08 100.06 100.08 1OwJa 100.00
000 0.31 0.42 0.35 0.26 0.16 0.04
+ + + + + + +
200 1.41 089 047 029 0.15 0.03
[Cu’] = 3 x lo-*
WI - [OH’l)(l + {H)@iA’)
v = = = = = = =
200 1.72 1.31 0.82 0.55 0.31 0.07
gives log
Determination
Oo
1
of titration end-points
2
3 -v
4
1441
5 ml
FIG. 7.-Straight line obtained when titration data plotted in Fig. 6 for a titration of acetylacetone complexed with copper are used to calculate V,, - Vfrom e uation (231. The intermediate and final results of the calculations are collected in Taa le V. The straight line intersects the V axis at F = 4.63 ml, which corresponds to an error of +1-s %.
EXPERIMENTAL The experimental work was performed using a Metrohm Potentiograph E 336 recording potentiometer equipped with an EA 107 glass electrode and an EA 404 saturated calomel reference electrode. All solutions were prepared from reagent-grade chemicals. Special care was taken to ensure that the O~lOOM sodium hydroxide solution was free from carbonate. The following procedure was followed in the titrations: A lO-ml sample of an approximately 5 x lo-*Macid solution was transferred to a W-ml beaker. Any additions of other acids or metal salts were made and the solution was diluted to 50 ml after the addition of sodium perchlorate to adjust the ionic strength to 0.1, The solution was titrated with @lOOM sodium hydroxide solution with magnetic stirring at the slowest practical speed of titration. Esumb-On propose une extension de la mdthode de Gran pour determiner les points d%quivalence dam les titrages potentiometriques acide-base. On dtduit des equations en supposant que les constantes de stabilite des reactions sont connues. Les equations rendent possible la localisation du point d’&privalence avec une assez grande precision lorsqu’on titre des acides dont les constantes de stabilite se situent entre K$ N 10’ et KS N lOlo. La limite de KL au-dessus de laquelle les equations de Gran ne donnent pas de resultats satisfaisants est d’environ 10’. On utilise. des constantes conditionnelles et des coefficients a atin d’etendre la mtthode pour l’appliquer a des systemes plus complexes, oh plus dune esp&ce reagit avec l’agent de titrage. Zusammeafassnng-Es wird eine Erweiterung der Gran’schen Methode zur Bestimmung von Aquivalenzpunkten bei potentiometrischen +%ure-Basen-Titrationen gezeigt. Unter der Voraussetzung, daB die Stabilitltskonstanten der Rcaktionen bekannt sind, werden Gleichungep abgeleitet. Die Gleichungen ermbglichen die Lokalisierung des Aquivalenzpunktes mit ziemlicher Genauigkeit, wenn Stiuren mit Stabilitiitskonstanten von K~A - 10’ bis K& - lOlo titriert werden. Die Grenze von K&, tiber welcher die Gran’schen Gleichungen keine zufriedenstellenden Ergebnisse liefem, ist ungefahr 10’. Mediumabhangige Stabilittbskonstanten und a-KoefIizienten zur Erweiterung der Methode gebraucht, urn sie auf kompliziertere Systeme anzuwenden, wo mehr als eine Spezies mit dem Titranten reagiert.
1442
F. INGMANand E. SELL REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
G. Gran, Acta Chem. &and., 1950,4,559. G. Gran, Analyst, 1952,77,661. P. Sprrensen, Kern. Maanedsblad, 1951,32,13. A. Ringbom, Complexation in Analytical Chemistry, Wiley/Interscience, New York/London, 1963. I. M. Kolthoff and P. J. Elving, Treatise on Analytical Chemistry, Part I, Vol. 1, p. 242. Interscience, New York, 1959. A. E. Martell and L. G. Sillkn, Stability Constants of Metal-ion Complexes, Special Publication No. 17. The Chemical Society, London, 1964. J. A. Goldman and E. Meites, Anal. Chim. Acta, 1964,30, 28. E. W&n&en, Tukzntu, 1963, 10,221. B. Eklund and A. Ringbom, Finska Kemists. Medd., 1962,71, 53.
GRAPHIC
Department
METHOD FOR THE DETERMINATION OF TITRATION END-POINTS *
FOLKE INGMAN and EBBE STILL of Analytical and Inorganic Chemistry, Abe Akademi, Abo, Finland
(Received 21 April 1965. Revised 9 April 1966. Accepted 10 May 1966) Summar-Aa extension of Gran’s method for the determination of equivalence points in potentiometric acid-base titrations is presented. Equations are derived assuming that the stability constants of the reactions are known. The equations make it possible to locate the equivalence point with fair accuracy when acids with stability constants ranging from KEa N 10’ to Kz* - lOlo are titrated. The limit of KL above which Gran’s equations do not yield satisfactory results is about 10’. Conditional constants and a-coefficients are used to extend the method to apply to more complex systems, where more than one
speciesreacts with the titrant. AMONG
the graphical methods for the determination of titration end-points published by various authors, those of Gran 1*2have attracted considerable attention. Gran’s first method1 involves the plotting of either AV/ApH or AV/AE as a function of V, the volume of added titrant, after values of E or pH have been experimentally determined for several values of V. In the ideal case the resulting plot consists of two straight lines, which intersect each other and the V-axis at the equivalence point. One of the drawbacks of this method is that the accuracy of the TABLE I.-GRAN’S
Substance titrated Strong acid Weak acid
SECOND METHOD
Functions plotted on the acid side of the on the alkaline side of the equivalence point equivalence point (V, + V)lO(k-pa) =f(V) V.lO”-*=’ =f(V)
(V, + V)lO’*H-“‘) = f(V) (V, + V)lO’p=-k” =f( V)
The titrant is in each case assumed to be a strong base. V, is the volume of the solution before the titration is started, Vthe volume of added titrant, and k, k’ are arbitrarily chosen constants.
plot is influenced by the accuracy of the experimental data, and another that the lines for titrations of weak electrolytes are more or less curved and do not yield reliable results, because the change in pH or E at the equivalence point is too small. The second method of Gran2 is based on an original idea of Ssrensen,8 who plotted {H}, the antilogarithm of -pH, as a function of V for titrations of strong protolytes, disregarding the effect of the changing volume. Sorensen employed an analogous method when titrating halides with silver nitrate. Gran introduced a correction for the volume change during the course of the titration. He also extended his method to titrations of weak protolytes, and to other types of titrations than those involving neutralisations. The functions plotted when Gran’s second method2 is applied to acid-base titrations are presented in Table I. + Presentedin part by E.S. in December 1962 at Finska Kemistsamfundet, Helsingfors. 1431
F.
1431
INGMAN
and E. STILL
Gran’s second method yields excellent results when a moderately weak acid such as acetic acid is titrated with a strong base. Two straight lines are obtained and the point where these lines intersect each other and the V-axis is the equivalence point. For weak acids with stability constants of 10’ to 10s the lines obtained by this method are no longer straight but slightly curved. Moreover, the asymptotes do not intersect at the V-axis, and the point where they intersect only occasionally coincides with the true equivalence point. These deficiencies originate in the simplifying assumptions made by Gran when deriving his equations. The aim of the present work is to show how it is possible, when the stability constant for the reaction is known, to derive an expression which, when plotted against the titrant volume V, yields a straight line that intersects the V-axis at the equivalence point. The expression for the titration of a weak acid with a strong base will first be derived. It will further be shown how it is possible to generalise this expression by the introduction of conditional constants and a-coefficients.4 The determination of the equivalence points of titrations of complex systems is thereby facilitated. As an example, the titration of a weak acid in the presence of another weak acid or a metal ion will be considered. TITRATION
OF A WEAK
ACID
WITH
A STRONG
BASE
A weak acid HA is titrated with a strong base (sodium hydroxide). The initial concentration of the acid is denoted by C,, and the initial volume by V,. The concentration of the strong base is denoted by CNaOHand the volume added by V. The consumption of the base at the equivalence point is denoted by V,,. The derivation is based upon the following three conditions: (a) The law of mass action must hold: HA+
KH _ tHA] HA- (H)[A] *
H 4-A;
(1)
(For convenience, all signs of charge are omitted in this paper). The constant Kg* is a “mixed constant”, valid at a certain ionic strength. Mixed constants, where the activities of the hydrogen and hydroxide ions (denoted by {H} and {OH)) are used in conjunction with concentrations of all other species, are very convenient for equilibrium calculations, since pH-values are nowadays almost solely determined potentiometrically. The activity of the hydrogen ion is related to the concentration by the expression {H}
=_fiJHl.
(2)
where fHis the activity coefficient of the ion. (b) The solution must be electrically neutral, meaning that
[Al + [OHI =
Wal
+ [HI.
(3)
(c) The equation ve&NsOH
must be valid at the equivalence point.
=
v0cHA
(4)
Determination
The concentration
1433
of titration end-points
of the sodium ions is
M
=
2vo+ v cKaOH
(5)
which, when combined with equation (3), gives
[Al = ?-
+ WI -
v, + v C&H
[HA] = +
C,,
- [A] = +$
0
WI
CNaoH-
[HI + [OHI.
(7)
0
Substitution of (6) and (7) into (1) and subsequent rearrangement of the terms yields Veq -
V = V{H}K&
+
~([Hl-
tOHI)(1+ (H}K,H,I
The values of most of the constants have been determined at an ionic strength of ,U = 0.1. It is thus practical to adjust the ionic strength to this value, and in our experiments sodium perchlorate was used for this purpose. It may be noted that in equation (8), the concentrations [H] and [OH], caused by the electroneutrality rule, occur in addition to the hydrogen ion activity {H} determined potentiometrically. For accurate analyses the [H] and [OH] values in equation (8) should be calculated from the activities. This means that at an ionic strength of 0.1, [H] = {H}/10-“~08 and [OH] = {OH}/10-0.12. A comparison of equation (8) with the Gran equations for the titration of a weak acid reveals that Gran plotted only the first term V{H}K&, written in the form VIO(k-*H), on the acid side of the equivalence point, k being a constant arbitrarily chosen to yield numerical values that could conveniently be handled. On the alkaline side of the equivalence point Gran plotted the term (V. + v>[OH]/C,,,, against V. Gran wrote the term in the form (V, + V)lO(pH-K), k’ being a constant similar to k. When plotted against V, equation (8) yields a straight line with a slope of 45”. The titration curves of p-alanine (log K& = 10.2) and ammonium nitrate (log K& = 9.4) with sodium hydroxide are shown in Fig. 1. The titrations were performed at an ionic strength of ,u = 0.1. Kielland’s values6 were used for the activity coefficients, and the values of the stability constants were taken from the literature,6 The use of equation (8) is illustrated in Tables II and III and Fig. 2. For comparison, Fig. 3 shows the curves that result when Gran’s formulae are used for the titration of ammonium nitrate. The method described can be used only if the value of the stability constant K& is known. This could naturally be considered a serious drawback. The method is, however, intended for use in cases where no other means are available for the determination of the equivalence point with satisfactory accuracy. Such cases are encountered in practice when very weak acids (7 < log K& < 10) are titrated. The limit beyond which the simple Gran equations do not yield satisfactory results can be considered as log K& N 7. For most analytical purposes the value of the stability constant can be estimated with satisfactory accuracy from the pH value at the point where VN V,,/2, i.e., 3
F. INGMAN and E. STILL
1434 12 77 10 9 PH
18 7 6 5 4 0
-1
2
3
4 5vs
7
a
g
10
FIG. l.-Titration of 50 ml of 1.07 x lO-SM p-alanine (log KL = 10.2) and titration of 77 ml of 1.02 x lo-*M ammonium nitrate (log K& = 9.36) with O*lOOM sodium hydroxide. At the equivalence point, the consumption is theoretically 535 ml for &alanine and 7.85 ml for ammonium nitrate. TABLEII.-/I-ALANNE log v, + V
V ::; 2.0 2.5 3.0 3.5 40 4.5 5.0 5.5 6.0
pU3
CNaOE
955 9.77 9.95 10.12 10.28 10.43 10.59 10.75 10.91 11.07 11.21
1%
2.71 2.72 2.72 2.73 2.73 2.73 2.74 2.74 2.75 2.75 2.75
-4 -4.1133 -3.93 -3.76 -360 -3.43 -3.29 -3.13 -2.97 -2.81 -2.67
(HI&
1+
(W-II - [HD
1 + l~“6 100.” = 1 + 100.86 = 1 + 100.08 = 1 + 10-0.08 = 1 + lO-0.as = 1 + lO-0.*0 = 1 + 10-0.65 = 1 + 10-0.1’ = 1 + 10-0.81 = 1 + 10-1.0’ =
K, 100*‘4 loo.67 100.U 1OO.U 100.N lOWI 100.16 100.” 100.08 100.06 100.04
4.47 4.03 3.56 300 2.50 2.06 164 1.27 098 0.59 0.06
-
V, = 50 ml, log K, = 1400, log K& = 10.20 (from Fig. l), log foH = -0.12, VW -
V = V(H)K:A + E(D-U
-
-
v
0.13 0.15 0.17 0.20 0.25 0.30 040 0.53 0.73 0.98 1.32
= = = = = = = = = =
logf,
4.34 3.88 3.39 2.80 2.25 1.76 1.24 0.74 0.25 -0.39 -1.26 = -0.08
[OHD(1+ U-W&I
TABLE III.-AMMONIUM NITRATE
log V
3.0 4.0 5.0 6.0 7.0 8.0 9.0 10-o
PUS
8.50 8.89 9.15 9.38 960 9.85 10.18 10.69 11.14 11.37
v, + v
cN,O,
290 2.90 2.91 2.92 2.92 2.93 2.93 2.94 294 294
logWH1
[HI)
-
-538 -4.99 -4.73 -4.50 -4.28 -4.03 -3.70 -3.19 -2.74 -2.51
1 +
100.86
5
1 + 100’” = 1 + 100.8’ = 1 + 10-0.0’ = 1 + lO-0.S’ = 1 + 10-O.&@= 1 + lO-o.s* = 1 + lO-l.88 = 1 + 10-1.18 = 1 + 10-5.0’ =
I’, = 77 ml, log Km = 1400, log Kg* = 9.36, logfoX
V,, - V = V{H}KL
+
ZWl
v,, - v
1 -t U-XV&
= -0.12,
1(-y.=
l@*EO 100.“’ lOo.P@ lOWSO 100.” 100.0 100.0’ 100.0’ lOc!.OO
7.24 5.91 4.87 3.82 2.88 1.95 1.06 0.37 0.15 0.10
-
logfH = -0.08
- PHDU + O-IV&)
0.03 0.03 0.04 0*05 0.07 0.11 0.20 0.59 1.62 2.69
= = = = = = = = = =
7.21 5.88 4.83 3.77 281 1.84 0.86 -@22 -1.47 -2.59
Determktion
2-
of titration end-points
I p-ALAN/NE
Y
1
o-
1435
\I
1 7
I
I
I\
I
3
4
51
6
I
I
--vu
I
,\I
f I
X
9mllO I\
I
-1
FIG. 2.-Straight lines that result when titration data in Fig. 1 are substituted in equation (8). The principles of calculation are illustrated in Tables II and III. For /k&nine the straight line intersects the V-axis at V = 5.22 ml, which corresponds to an error of -2-4 %. For ammonium nitrate the straight line intersects the V-axis at V = 7.78 ml, which corresponds to an error of -0.9 %.
2
-1
FIG. 3.-The straight line (a) that results when the titration data for ammonium nitrate shown in Fig. 1 are used to compute V, - V from equation (8), and the curves (b, c, c’, c”) that result when Gran’s second method is used. The abscissa of the point of intersection depends on the value of the arbitrarily chosen constants k and k’ (cJ Table I). The values of k’ used for computing c’ and c” differed by &@30 from the value of k’ used for computing c.
when approximately half of the acid has been titrated. If a very high degree of accuracy is sought, the method developed by Goldman and Meites’ can be used. If an incorrect value of the constant has been used, the plot will not have a slope of 45” and will not be linear. It may be mentioned that the results of the titrations given in Figs. 1 and 2 can be found without using any graphic procedure by calculating Vea for the various volumes of titrant (ix., in the last column of Table II, V is added to F’eq- v).
F. INGMAN and E. ST~L
1436
However, this procedure is not always appropriate Table V). TITRATION
OF A WEAK
BASE
(cJ the titration presented in
WITH
A STRONG
ACID
An analogous expression can be derived for the determination of the equivalence point when a weak base is titrated with a strong acid (hydrochloric). The equation is : Veq -
V = V{OH}K,O& +
owl
- [HI)0 + WW%d
(9)
where
POW K% = [Bl(OH)’ This case will not be treated in this paper. TITRATION
OF TWO
WEAK
ACIDS
WITH
A STRONG
BASE
The determination of the total amount of acid in a mixture containing two weak acids HAr and HA,, is usually quite easily accomplished, provided both acids are strong enough to be titrated in the conventional manner, i.e., provided that both K&, and K&11 < 10’. When, however, the amount of the stronger acid (here taken to be HA,) in the mixture is to be determined, complications often arise owing to the fact that the difference between the stability constants of the two acids is too small to permit a conventional stepwise titration. When conditional constants and a-coefficients4 are used, it is quite easy to derive an expression analogous to the one already derived for the titration of one weak acid, and this will permit the determination of the first equivalence point. In this case the equation HA, + OH + A, + H,O
(10)
HA,, + OH =S A,, + H,O
(11)
represents the main reaction. The reaction
is considered a side-reaction, the effect of which is taken into account by introducing the appropriate cr-coefficient.8 The electroneutrality rule states that in this case
Dal + WI = (PHI + LW + [AI1= PH’I + LW
(12)
Comparison of this equation with equation (3) shows that [OH’] should be substituted for [OH] in equations (6) and (7), and this in turn means that the only change that need be made in equation (8) is the substitution of [OH’] for [OH], which yields
ve,-
J-,’= V(H}K& +
E
(WI - tOH’I)(l + (H}K&J.
(13)
As usual, [OH’] denotes the concentration of not only the free hydroxide ion but also of all the hydroxide ion added to the solution that has not reacted with HA, according to the main reaction. There will be no noticeable error if the activity
Determination of titration end-points
1437
coefficient foaP is considered equal to foH. The concentration that has reacted with HAzr is given by [A,,]. Hence WI’1 = [OH] +
IM =
of hydroxide ion
%H[OHI
(14)
where (15) To test the equation derived several titrations of chloroacetic acid (log K& = 2GQs in the presence of acetic acid (log K& = 4.7)” were performed using sodium hydroxide. As can be seen from Table IV and Figs. 4 and 5 the results were in close agreement with the theoretical. Figure 4 shows one of the titrations curves and Fig. 5 shows the straight line that results when Yeq - Y computed from equation (13) is plotted against Y to determine the location of the equivalence point.
‘3 PH
I
7 6
3
0
1
2
3 4 ---+v
5
6
7
FIG. 4.-Titration of 50 mi of a solution 0.72 x 10-W in cbloroacetic acid and 0.50 X lO-#M in acetic acid (A log K& = log Ii’& - logKm,, = 1.9) with O*lOOM sodium hydroxide. At the equivalence point, the consumption is theoretically 360 ml for chloroacetic acid. TABLEIV.-MONO~HLOROACETZCACID(HA*) AND ACETICACID@A~~) Y 0.0 0.5 1.0 1~5 2.0 2.5 3.0 3.5 4.0 4.5
~0.0 2.59 2.68 2.78 2.91 3.06 3.24 348 3.80 4.14 4.47
log v, + v ClWOli 2.70 2-71 2.71 2.72 2.72 2.73 2.73 2.73 2.74 2.74
[HI - [OH’1
1O-%8’ _ ] O-9.80_ IO-P.70_ 10-1.8’ _ IO-P.08_ IO-B.16_ IO-“‘0 _ ~0-9.72_ IO-“.00_ IO-“WJ_
v, = 50 ml, log K&, = 2~35, bg K& = 9.24, [HA,,1 log& = -0.08,
V,, -
VW-- v
I+ W&
~O-l.85=E*(ye.sa IO-4.8’ = ,0-W’ *o-*.1, =E*O-8.1’ IO-4.0P=i IO-#.86 10-m = 10-8.03 10-a.,* 3 IO-a.ao IO-Em =E 10-u* ,0-w, 5 _ 10-W’ ,O-“‘8” 1 _,O-8.87 10-8’61= - ,0-2.6#
V-
V(H}K&
0% + 0.74 + 1.18 + 1.30 + I.23 + 1.02 + 0.70 + 0.39 0.20 0.11 -
427 = 4.27 3.09 = 3.83 219 = 3.37 1.35 =I 2.65 0.79 = 2.02 0.38 = I.40 0.05 = 0.75 0.30 =i 0.09 0.78 = -0.58 1.70 = --I*59
10-a’so, log aoa = 6-94 log fox = --0*12~
+ ~GHI-
WU)(1
+ WK&)
F. INGMANand E. Snu
1438
FIG. 5.--Straight line obtained when titration data plotted in Fig. 4 are used to compute V,, - V from equation (13). The different steps of the calculations are shown in Table IV. The straight line intersects the V-axis at V = 3.62 ml, which corresponds to an error of +0.6x. TITRATION
OF A WEAK
ACID
IN
THE
PRESENCE
OF A METAL
ION
A very weak acid can be made apparently stronger if its conjugate base A produced in the reaction HA+OH+A+H,O
(16)
can be forced to take part in a side-reaction. The equilibrium of the main reaction will then shift to the right and the numerical value of the conditional constant4 of reaction (16), Kiu = CC&~, will increase. Many metal ions react readily with the conjugate base of an acid to form a complex according to the scheme A+M+MA.
(17)
As has been pointed out by Ringborn, 4 this side-reaction permits the titrating of many acids that are too weak to be titrated directly. However, the metal ions are themselves acids, and will consequently react to some extent with hydroxide ions. This side-reaction is undesirable, but cannot be avoided. It may be written M+OH+MOH.
(18)
Frequently matters are further complicated by the fact that the metal hydroxide starts to precipitate at a pH that is very close to the pH at the equivalence point of the titration. As a result, the titration curve will be deformed as in Fig. 6. In order to determine the location of the equivalence point on a titration curve of this shape, it is necessary to employ a method that utilises points on the titration curve that lie on the acid side of the equivalence point only. The value of the conditional constant of the acid may be low enough to permit the use of one of Gran’s simple equations. In other cases one may be forced to use an expression which can be derived similarly to the expressions already presented in this paper. The effect of these two side-reactions is taken into account by introducing the coefficients : aA
=
QOH(M) =
1 -I MKMA
(19)
1 +
(20)
WKHOH
Determination of titration end-points
1439
where Knaa and &on denote the stability constants of the complexes MA and MOH respectively. Assuming that a bivalent metal is added to the solution in the form of MB, the condition of electroneutrality now is: 2[B] + [A] + [OH] = [H] + [Nal+
2[Ml+
[MAI + [MOW.
(21)
When [MA] + [MOH] is added to both sides and ([Ml + [MA] + [MOH]) is substituted for [B], it is found that ([Al + [MA]) + (PHI
+ NOHI)
= [A’] + [OH’] = WI.
(22)
A comparison of this equation with equation (3) shows that the following changes should now be made in equation (8): K,Hk = K&/ccAcMI should be substituted for K&, and [OH’] should be substituted for [OH]. The final result is then V., - V = V{H}Kgf
+ F([H]
- [OH’])(l + {H}K,f)
(23)
NaOH Note. If a univalent metal is added as MrBr, the solution will be electrically neutral when WI1 + [Al + COHI = [HI + Ml + Ml. (24)
As tB,l = [WI + MA1
+ MOHI, substitution will again lead to equation (22). However, most of the metals that can be used for the purpose discussed are bivalent.
To determine whether equation (23) is valid in practice, an aqueous solution of acetylacetone (about lO_aM) containing an approximately three-fold excess of copper(II) salt was titrated with sodium hydroxide. In this titration the reaction between copper ions and the acid proceeds towards completion during the titration; therefore the conditional “constant” Kgf will not be constant but will attain the calculated value at the end of the titration. Nevertheless, an approximately straight line will be obtained, and the intersection with the V axis will coincide with the equivalence point. The stability constant is KHfIA= IV.@, corresponding to KiH = 105.1. When the appropriate values = 1 + 10-1.51(j3.2 = 106.7 (25)
uA(Cu)
and aOB~Cu~ f
1 + 10-1.5105.7= 104.?
(26)
are inserted, the conditional constant of the base formed will be Ki,u’ =
ctA(CujKzH_ 10e*7105’1 lo,_6
-_Z
a0H(Cu)
104.2
(27)
This constant corresponds to an acid constant of 10s.4 which determines the shape of the titration curve.0 The titration is thus a borderline case and could possibly be accomplished with visual end point indication, were it not for the fact that copper hydroxide begins to precipitate at pH very close to that at the equivalence point. Furthermore, the colour of the copper compounds may affect the colour change of the indicator.
F. INGMAN and E. STILL
1440
The titration results given in Table V and Figs. 6 and 7 are seen to be in good agreement with the theoretically calculated values. The ionic strength in this titration is not 0.1 but O-13. This, however, means a difference in the log fa value of only O-01, which is neglected. Figure 6 also illustrates the difficulties encountered when an attempt is made to locate the equivalence point on a titration curve which is deformed by the precipitation of a metal hydroxide. The pH of the equivalence point is 5.0 andcopper hydroxide begins to precipitate at pH N 5.2 (if the solubility product of copper hydroxide is 10-18.2,the point where precipitation should occur can be calculated to be pH N 5.6). 72 11
10
9
8 t ri7 6
0
1
2
3
4
-v
5
FIG. 6.-Titration of 50 ml of 0.91 x 10-eM acetylacetone
6
7
(log K& = 8.9) containing
about a three-fold excess of copper ([C&t,, = 0.04M) with O.lOOM SOCCU~ hydroxide. At the equivalence point the consumption is theoretically 4.55 ml. TABLE V.-ACXTYLACETONECOMPLEXEDWITHCOPPER
log v, + V V
p{W
CNsOH
o-0
2.63 2.76 2.93 3.19 3.38 3.66 4.30
2.70 2.71 2.72 2,73 2.73 274 2.74
1.0 2.0 3.0 3.5 4.0 4.5
log
1 -t- {J$KHA H,Aj
([HI - W-U) 1 1 1 1 1 1 1
-2.55 - 2.68 -2.85 -3.11 -3.30 -3.58 -422
+ + + + + + +
lO-0.aa = 10-M’ = 10-0.88 = 10-0.84 = 10-L” = lO-X.01= 10-8.06 =
V, = 50 ml, log K& = 8.95, log Kcti = 8.2, log KcuOH = 5.7, tLA(cu)= 6.7, log aOHICu,= 2.25, and log KziA’ = 2.25 logfn = -0.08, logfen = -0.12 H,A' V,,
-
V=
VIWKHA
+
E
K, -
100.X loo.‘* 100.08 100.06 100.08 1OwJa 100.00
000 0.31 0.42 0.35 0.26 0.16 0.04
+ + + + + + +
200 1.41 089 047 029 0.15 0.03
[Cu’] = 3 x lo-*
WI - [OH’l)(l + {H)@iA’)
v = = = = = = =
200 1.72 1.31 0.82 0.55 0.31 0.07
gives log
Determination
Oo
1
of titration end-points
2
3 -v
4
1441
5 ml
FIG. 7.-Straight line obtained when titration data plotted in Fig. 6 for a titration of acetylacetone complexed with copper are used to calculate V,, - Vfrom e uation (231. The intermediate and final results of the calculations are collected in Taa le V. The straight line intersects the V axis at F = 4.63 ml, which corresponds to an error of +1-s %.
EXPERIMENTAL The experimental work was performed using a Metrohm Potentiograph E 336 recording potentiometer equipped with an EA 107 glass electrode and an EA 404 saturated calomel reference electrode. All solutions were prepared from reagent-grade chemicals. Special care was taken to ensure that the O~lOOM sodium hydroxide solution was free from carbonate. The following procedure was followed in the titrations: A lO-ml sample of an approximately 5 x lo-*Macid solution was transferred to a W-ml beaker. Any additions of other acids or metal salts were made and the solution was diluted to 50 ml after the addition of sodium perchlorate to adjust the ionic strength to 0.1, The solution was titrated with @lOOM sodium hydroxide solution with magnetic stirring at the slowest practical speed of titration. Esumb-On propose une extension de la mdthode de Gran pour determiner les points d%quivalence dam les titrages potentiometriques acide-base. On dtduit des equations en supposant que les constantes de stabilite des reactions sont connues. Les equations rendent possible la localisation du point d’&privalence avec une assez grande precision lorsqu’on titre des acides dont les constantes de stabilite se situent entre K$ N 10’ et KS N lOlo. La limite de KL au-dessus de laquelle les equations de Gran ne donnent pas de resultats satisfaisants est d’environ 10’. On utilise. des constantes conditionnelles et des coefficients a atin d’etendre la mtthode pour l’appliquer a des systemes plus complexes, oh plus dune esp&ce reagit avec l’agent de titrage. Zusammeafassnng-Es wird eine Erweiterung der Gran’schen Methode zur Bestimmung von Aquivalenzpunkten bei potentiometrischen +%ure-Basen-Titrationen gezeigt. Unter der Voraussetzung, daB die Stabilitltskonstanten der Rcaktionen bekannt sind, werden Gleichungep abgeleitet. Die Gleichungen ermbglichen die Lokalisierung des Aquivalenzpunktes mit ziemlicher Genauigkeit, wenn Stiuren mit Stabilitiitskonstanten von K~A - 10’ bis K& - lOlo titriert werden. Die Grenze von K&, tiber welcher die Gran’schen Gleichungen keine zufriedenstellenden Ergebnisse liefem, ist ungefahr 10’. Mediumabhangige Stabilittbskonstanten und a-KoefIizienten zur Erweiterung der Methode gebraucht, urn sie auf kompliziertere Systeme anzuwenden, wo mehr als eine Spezies mit dem Titranten reagiert.
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F. INGMANand E. SELL REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
G. Gran, Acta Chem. &and., 1950,4,559. G. Gran, Analyst, 1952,77,661. P. Sprrensen, Kern. Maanedsblad, 1951,32,13. A. Ringbom, Complexation in Analytical Chemistry, Wiley/Interscience, New York/London, 1963. I. M. Kolthoff and P. J. Elving, Treatise on Analytical Chemistry, Part I, Vol. 1, p. 242. Interscience, New York, 1959. A. E. Martell and L. G. Sillkn, Stability Constants of Metal-ion Complexes, Special Publication No. 17. The Chemical Society, London, 1964. J. A. Goldman and E. Meites, Anal. Chim. Acta, 1964,30, 28. E. W&n&en, Tukzntu, 1963, 10,221. B. Eklund and A. Ringbom, Finska Kemists. Medd., 1962,71, 53.