CONCLUSIONS
Results of computations carried out on a model in which the samples are contained within cylindrical cavities in a highly conducting block, and in which centre temperatures are measured lead to the following conclusions. (1) The area under a DTA peak is directly proportional to the heat of reaction and mass of the sample and inversely proportional to the thermal conductivity of the material. The area, measured as AT cs. time is independent of the heating rate. (2) The temperature of a DTA peak increases for increasing radius of sample and the variation of sample peak temperature is less than that of the reference material. Hence DTA curves should use sample temperature as abscissa to reduce the influence of differing sample radius used by different experimental designs. The heating rate at the peak of the DTA curve is reduced from the nominal value in samples of large radius. Where the reaction is governed by an equation which is heating rate dependent, then erroneous peak temperatures will be recorded. Further, large radius samples distort the peak shape, and hence small radius samples are important to reduce or eliminate these two effects.
(3) The DTA peak reference temperature increases with decreasing sample conductivity and increasing density and specific heat. The peak sample temperature is sensibly independent of the physical properties. The physical properties do, however, influence the peak temperature shift with heating rate leading to erroneous values of activation energies when using techniques outlined by Kissinger (12). The positioning of measuring thermocouples is not so critical in samples of small radius as it is in samples of large radius. (4) Heat loss uia the measuring thermocouples causes a reduction in the area under the DTA peak and also lowers the peak temperature. Large heat losses can reduce the actual rate of heating of the centre of the sample even though the rise in block temperature is closely controlled. This fact must be borne in mind when thermocouples are changed, as a change of heat loss will affect the actual rate of heating of the sample. The heat loss may be minimized by the use of thin thermocouple leads, with the limitation that the wires should be conveniently handled when making connections.
RECEIVED for review August 29, 1968. Accepted April 28, 1969.
Theory of Curved Molar Ratio Plots and a New Linear Plotting Method Kozo Momoki, Jun Sekino,‘ Hisakuni Sato, and Noboru Yamaguchi Laboratory .for Industrial Analytical Chemistry, Faculty of Engineering, Yokohama National University, Ooka-machi, Minami-ku, Yokohama-shi, Japan By Introducing an idea of “normalized absorbance” into the familiar molar ratio method, theoretical general equations are shown to describe curKed plots encountered with weak complex systems. Normalized absorbance” means the degree of complex formation, but may also be called the degree of undissociation, as opposed to the degree of dissociation. At the same time, “the effective concentration factor of ligand used” is introduced for practical use of the method, as if formally corresponding to activity coefficient. Several important features of the curved plots are examined qualitatively, and a new linear plotting method i s used to obtain quantitative information. Experimental data and applications of the new technique are discussed. The discussion suggests that the purities of ligand reagents are important in complex formation studies.
Present address, Research Laboratory, Yarnanouchi Pharmaceutical Co., Tokyo, Japan.
ponents of a complex when the concentration of one component is varied as the concentration of the other is held constant. If the system forms a stable complex which does not show appreciable dissociation, such a plot gives a sharp break. The molar ratio at the sharp break indicates the composition of the complex. If, however, a weak complex is formed, only a curved plot results. Harvey and Manning (3) remarked that such a weak system should be treated to give a sharp break by changing the ionic strength of the solution. In spite of this, practical applications of the method have been intuitively made to a number of weak systems showing no sharp break by drawing convenient break points on such curved plots. Linear extrapolation of the curve is made where the curved plot becomes nearly parallel to the molar ratio axis after an excess of the variable component is added. However, no theories exist to justify such an extrapolation. Harvey and Manning (3) tried to explain the curvature of the plot by using “the degree of dissociation” of the weak complex formed, and showed that the formation constants of some complexes can thus be obtained. However, their treatment described qualitatively the shape of the whole curve and did not indicate how quantitative linear approximation could be made on a curved plot. The method, though seemingly
(1) P. Job, Ann. Chim. (Paris), 9, (lo), 113 (1928). (2) J. H. Yoe and A. L. Jones, IND. ENG. CHEM., ANAL.ED., 16, 111 (1944).
(3) A. E. Harvey and D. L. Manning, J. Am. Chem. Soc., 72,4488 (1950).
THEMOLAR RATIO METHOD, a counterpart of the continuous variation method (1) for spectrophotometric studies of complex formation systems, has been used frequently since its introduction by Yoe and Jones (2). The method, which has lacked a firm theoretical background, is based on plotting measured absorbances against molar ratios of the two com-
1286
ANALYTICAL CHEMISTRY
useful, is thus without the theoretical background needed to decide if extrapolation will give a reasonable result. One reason why the molar ratio method has lacked theoretical equations so far may be in the possible difficulties encountered in applying Beers’ law. Although the use of molar absorption coefficients is necessary, their exact values are not usually known. To avoid this difficulty, experimental conditions have been adopted (3-6) in which the molar absorption coefficients can be eliminated and/or the concentration terms can be approximated. In such modified molar ratio methods, the log-ratio method by Diehl and Lindstrom (6) is unique in eliminating molar absorption coefficients by using absorbance ratios, although this technique is usually employed in evaluating pK from pH-absorbance curves as well as in treating photometric titration curves (7). However, their method employed conditions for appoximating other concentration terms and does not seem to be generalized. It is obviously not logical to use absorbances measured where the concentration of one component is far larger than that of the other to obtain information about the middle range of molar ratios. A new theoretical and generalized approach to the molar ratio method is given herein in which no approximations are necessary. General equations which do not contain molar absorption coefficients are given for describing the curved plots of weak complex formation systems. Absorbance ratios are used in a procedure unlike that used by Diehl and Lindstrom (6). “Normalized absorbance” is introduced for absorbances measured in the middle range of molar ratios. In addition, “the effective concentration factor of ligand used” is introduced for practical cases. Some theoretical analyses of the molar ratio curves have become feasible, and a new linear plotting method has evolved. This new treatment opens several possibilities for interpretation of the familiar molar ratio method. THEORETICAL EQUATIONS OF MOLAR RATIO CURVES
Consider a general form for complex formation systems at equilibrium as
mM
+ nL e MmLn
(1)
where M represents a metal species, L a ligand, and M,Ln is a complex formed in a molar ratio of rn :a for M and L . The concentration formation constant explains the complex formation of Equation l at an appropriate temperature as shown below:
K
=
[MrnLn][M]-m[L]-n
= [M,L”I(Cnr
- m[MmL”I)-m(CL- n[MmLnl)-n
(2)
where CM is the total concentration of the metal and CL,that of the ligand. In the molar ratio method, absorbances are usually measured at a wavelength where the complex formed shows an absorbance maximum. For such cases, measured absorbances increase with increasing concentration of one component. Sometimes the measurements are at the absorbance maximum of the ligand, yielding curves in which absorbances decrease with increasing concentration of the metal, as done by Diehl and Lindstrom (6). Since the latter case is used less frequently and can be treated quite similarly to the former (as (4) H. A. Benesi and J. H. Hildebrand, J. Am. Chem. SOC.,71,2703 (1949). ( 5 ) H. McConnell and N. Davidson, Ibid.,72, 3164 (1950). (6) H. Diehl and F. Lindstrom, ANAL.CHEM.,31, 414 (1959). (7) T. Higuchi, C . Rehm, and C . Barnstein, Ibid.,28, 1506 (1956).
C M ( CL = c o n s t a n t 1
1 v)
0
a
0.5 0
E0 C
Ai
molar r a t i o ( C M / C L ) Figure 1. Molar ratio plot for metal-changing method (MCh)
both cases are mathematically the same), the present paper deals mainly with the former. It is obvious that the absorbance, A,, actually measured at a given wavelength is for both the complex and the ligand, as
As
+ 4MrnLnI + EL(CL- 4MrnLnI) = - tte~)[MrnLnI+ ELCL - n4[MrnLnI + (AIL =
~dMrnLn1 edL1
=
( ~ c
E
(ec
(3)
where e, and E L represent the molar absorption coefficients of the complex formed and ligand, respectively, at the wavelength being measured. The sample path length is assumed to be 1 cm in Equation 3 as well as in the following treatments. Also, ( A J L means the absorbance exhibited only by the whole concentration of the ligand. Absorbances of free-metal species may not necessarily be taken into account here, since they are comparatively small. If it is assumed that only species belonging to Equation 1 are responsible for the absorbances measured, the latter are expressed generally by Equation 3, but change with changing molar ratios. Molar ratios are changed experimentally either by changing the concentration of ligand with constant metalion concentration (ligand-changing method, LCh) or by changing that of metal with constant ligand concentration (metal-changing method, MCh) giving different curves for the same system. In the following systematic treatments, the difference between LCh and MCh is shown to be only a matter of form which makes the treatments attractive and useful. Consider the case of changing metal-ion concentration, when CL is kept constant, and, (AS)Lis thus constant. This (A,)L is made as the intercept of the curve of A,, where the absorbance axis is at zero molar ratio and no complex is yet formed with [M,L,] = 0 in Equation 3. When C.li is added A , is increased with increased [M,L,], making a curved plot, as in Figure 1 where the whole curve is elevated vertically from the zero origin by the constant ( A J L . The absorbance (A&, measured at (C&, corresponds to the concentration of the complex form, [MmLnIi,as [ M m L n I i = ((A,)< -
(As)L)/(€c
- EL)
VOL. 41, NO. 10,AUGUST 1969
(4) 1287
CL ( Cu = constant 1
i
Q
0 C
0
e 0
In n 0
-0 W L
3 v)
0 0,
E U
molar r a t i o
Figure 3. Relation between x and
Figure 2. Molar ratio plot for ligand-changing method (LCh) as derived from Equation 3. If Equation 4 is inserted directly into Equation 2 (as is often done) the newly obtained equation includes ec and E L , causing the difficulties as mentioned before. Since, however, [M,L,], cannot exceed [MmLnlmax, and the latter is the concentration when the whole CL is complex, ( A & will reach (As)max at the [MmL,Ima,,with no further increase at higher molar ratios. The flat portion of a molar ratio curve (Figure 1) is thus usually observed, where
WmLnlmax= ((As)max - ( A s > ~ ) / ( ~ nc 4
(5) Dividing Equation 4 by Equation 5, these terms of molar absorption coefficients can be eliminated, as
molar ratio curves be generally expressed by Equation 9 for the cases of changing metal-ion concentration. This type of normalized absorbance is useful even for the cases of changing-ligand concentration, where ( A S ) Lin Equation 3 is variable with ( C L ) ~ .If Beers’ law is obeyed (for as)^) in the experimental range, a plot will be obtained (Figure 2) which has zero origin with a linearly increasing portion after [MmZn]treaches [MmLnlmax, because of linearly increasing (A&. For such a curve, the normalization of Equation 6 is obtainable, only if the variable as)^ is measured each time and subtracted from the measured (A&. Further, if this normalization is employed, it is easy to obtain a similar general equation for LCh to Equation 9, even from the latter by making mere transformations between m and n as well as from C L to CM there, as XLCh =
(n/m) x f (mK)-”mCu-(m+n-’)in
where the notation xt is conveniently used for the absorbance term, which can be measured experimentally. Equation 6 can also be rewritten; if [MmLnlrnax = CL/n is applied
W m L n l t= (CLjn) . X ,
(7)
At the same time, the changing (C& is written with molar ratio ( X M C h ) i and the constant CL,as (cM)i
= (xMCh)l ‘
CL
(8)
Inserting Equations 7 and 8 into Equation 2, and removing the subscript i for generalization, the following equation is obtained : XXCh
=
(m/fi) ‘
H
-k (nK)-l/mCL-(m+n-l)/m
. xl/m(l
- ~)-n/”
(9)
In Equation 9, m,n, and K take definite values for a given complex formation system, while CL is obtained by experiment. On the other hand, x means a “normalized” absorbance of that which is measured, where the whole curve is lowered to zero origin by subtracting the constant ( A S ) Lfrom each ( A s ) (measured, and then scaled again only on the ordinate to simply assign 1 for the flat maximum portion, as in Figure 1. Only through such a simple normalization can the relationship between molar ratios and absorbances on the 1288
ANALYTICAL CHEMISTRY
CY
*
xl/n(l
- %)-m/n
(10)
where X L C h now equals CL/C,w. It should be noted here that the “normalized absorbance” x has a physical meaning of “the degree of complex formation” of such a weak complex formation system, as evident from the left-side term of Equation 6. If the normalized scale of absorbance, x is thus connected simply with a, the degree of dissociation (3), 1
-
CYt
= xi
(11)
(see Figure 3). Because the absorbances in the method are actually measured after the complex formation process is completed, the complex formation can naturally be explained either by the dissociated part of the resulted complex with a or the “undissociated” part with, as in the above relation, x . This situation for x may better be expressed by “the degree of undissociation,” as a simple substitute for “the degree of dissociation” in the new treatments, than above “the degree of complex formation.’’ In fact, the use of x and not a seems more reasonable to explain such curved plots, because the direction of increasing absorbances coincides generally with increasing x and decreasing CY. Although x might better be called still “the degree of (complex) formation,” the latter has been used for different meanings. To avoid confusion we present “the degree of undissociation” for x . The usefulness of employing “undissociation” instead of “dissociation” to explain the situations as mentioned above can also be stressed by this. We believe
that the descriptions may sometimes be simplified without losing nuance by using more such words as “dissociation” and “un- or non-dissociation”. Experimentally, the normalization of Equation 6 is simpler if all absorbances on a plot are always measured against a reference solution containing only the same CL. Since CL is kept constant in MCh while varying in LCh, the latter procedure will be more tedious and sometimes more erroneous than the former. However, it is necessary to compare the two methods both experimentally and theoretically. Thus, Equations 9 and 10 are the general equations of molar ratio curves for the complex formation systems shown in Equation 1, and are systematically introduced for the first time without any approximations. Since the treatments do not require a choice for K between true and conditional constants, these equations can cover almost all cases of the molar ratio method. Although the normalization needs a measured maximum absorbance for each curve, and still requires excess concentration of one component, it is possible to do without the maximum absorbance. as will be seen. EFFECTIVE CONCENTRATION FACTOR OF LIGAND USED
In the ordinary molar ratio method or other methods for complex formation studies, the concentration of ligand often presents more difficulties than metal-ion concentrations. Although the latter are almost always checked by analytical techniques, such as chelatometric titration, the determination of the former species cannot usually be compared with those of the latter at least in accuracy problems. Since most ligands used for such studies are organic compounds having rather high molecular weights as well as often being unstable, many of them have been subjected to neither suitable checking nor purification (8, 9). Many people measure the weight of a reagent ligand with or without purification, to prepare the solution in which its molar concentration is simply calculated using a supposed molecular formula. This primitive procedure may be questionable even with purified reagent without checking the concentration. Even alkalimetric titrations are sometimes as doubtful as the above procedure, because they measure directly only acidic protons, not complex-forming species. Photometric titrations have been used but only occasionally (10). It is the authors’ opinion that the concentrations of ligands such as metal-indicators should be reexamined more carefully, even if difficulties are expected (11). In the present treatment, the fundamental Equations 9 and 10 are derived using CL for a ligand, which should have the same effectiveness as C,! for the complex formation of Equation l. In considering the above, the actually effective CL is assumed from the experimentally weighed CL’ by “the effective concentration factor of ligand used,” fL, as CL = fL
.
(mln)fL
’
x
+
.
(nK)-l/mCL‘-cm+n-l,/mf -V-l)/rn xl/m(l . L
Figure 4. Relation between a dissociation molar ratio curve and its corresponding nondissociation line where x I M C h is newly taken as CM/CL‘for MCh, and
where X ‘ L C h is also taken as C ‘ L / C for ~ LCh. These are the final equations to be treated here as fundamental for actual molar ratio plots, describing relations between experimentally obtainable molar ratios and normalized absorbances. Although fL formally corresponds to “activity coefficient” against CL to “activity” in Equation 12, j i has different meanings from the usual “activity coefficient.” The introduction of fL into these equations is found to be useful for practical cases, although its true meaning is not so clear yet because of its sensitive nature. THEORETICAL ANALYSES OF MOLAR RATIO CURVES
Equations 13 and 1 4 can be used for theoretically analyzing the molar ratio plots belonging to Equation 1 to indicate several interesting characteristics for them, most of which have not been clear or have sometimes been intuitively misunderstood. Even a plot which appears as a nonintegral molar ratio can be explained as a member of the curves having an integral molar ratio. These equations are now conveniently divided into additive parts, respectively, as X’MCh
=
(X’Mch)I
+
(X’.wh)z
(13)
(8) D. C . Olson and D. W. Margerum, ANAL.CHEM.,34, 1299
(1962). (9) H. Diehl, Ibid.,39, 30A (1967). (10) B. F. Pease and M. B. Williams, Ibid.,31, 1044 (1959). (11) K. Momoki, H. Sato, and J. Sekino, presented at the 13th Annual Meeting of the Japan Society for Analytical Chemistry, Sendai, Sept 1964.
=
+
(x‘LCh)L
(X‘LCd2
(15)
where (X‘MCh)l
= (m’*)fL
.H
(16)
(X’MCh12 =
limeL ’-(m+n- l)/mfL-(n--l)/m, x 1iy 1 (X’LChh
- )O-n/m
and X’LCh
CL’
(12) Inserting Equation 12 into Equations 9 and 10, the following equations are obtained: X’MCh =
molar r a t i o
=
-
(n/m>/fL . x
fm
(17) (18)
(X’LCd2 =
(mK)-I/nCM-(m+n-l)/nfL-l . xI/n(l
- X ) - m / n (19)
all for x from 0 to 1. Now, it may be easily considered that Equations 16 and 18 only represent the lines by corresponding nondissociation systems, respectively, which are expressed with the normalized coordinate and assigned the same m, n, and fL as in the present VOL. 41, NO. 10,AUGUST 1969
1289
Table I. MCh m = l,n = m = l,n = m = l,n = m = 2,n =
m
=
Inflection Points in Molar Ratio Curves H LCh 1 None m = l,n = 2 None m = 2,n = 3 None m = 3,n = 1 1 i(= 0.250) m = l,n =
2,n = 2
m = 2,n = 3 m = 3,n m
=
3,n
=
1
= 2
-3+22/3
(= 0.155)
-
-2+1/6
( = 0,112)
1 ,(=0.333) -
-2+d6
( = 0.225)
m = 3,n = 3
1 1 1
2
m = 2,n = 2 m = 3,n = 2 m = l,n
=
3
m = 2,n = 3 m = 3,n = 3
dissociation systems. And also, it will be obvious, as in Figure 4, that (x ’ M ~ Jand ~ (x I L C d 2 represent displacements of these curved plots by dissociation from the above nondissociation lines along the molar ratio axis to higher molar ratios, at each same value of x. Equations 17 and 19 are thus related with characteristic curved shapes by dissociation systems, while Equations 16 and 18 with seep rising slopes appeared in smaller molar ratio ranges. The displacement in Equation 17 or 19 has a form of a product between two terms, where one does not include x while the other is an explicit function of x. The first term can be given a certain value, only if a system is given for m, n, and K as well as an experimental condition for f L and CL’ or C M , where either of the latter two is kept constant during a molar ratio plot. On the other hand, the increase of the displacement with increasing absorbance, which is also meant by “curved” plot in the method, is obviously governed primarily by the second term which is decided solely by the combination of m and n. In other words, the displacement of each point on a plot, particularly as its characteristic shape, is determined primarily by the second “x-function” term decided only by the system, and then prolonged or shortened as a whole by the first “cclefficient” term decided once by the system and experiment. As another x-function will appear later, “x-function-anal” is here proposed. The above situation is applied quite similarly to both MCh and LCh. Thus, curved shapes by dissociation systems are characterized by analyzing further their x-function-anal which is not affected at all by K, C(CL’ or CM)or even f L . From this analysis, a quite interesting fact-that each of some molar ratio curves should theoretically be given an inflection point only by the combination of m and n-has been found unexpectedly as in Table 1. The new finding, which agrees well with the experimental results shown in Figure 9, is revealed by differentiating x-function-anal twice with x and equating it to zero. In Table I, the interchangeable nature of m and n between MCh and LCh is clearly shown while the inflection points are seen only at smaller absorbances. Each curve should have a downward curvature before the inflection point and an upward curvature after it, whereas a curve having no inflection gives only a simple upward curvature. Although such parts of the molar ratio curves have been given little attention, careful observation of these absorbances may serve for checking the system for possible m and n. However, such smaller absorbances may not often be sensitive enough for much quantitative deductions. 1290
ANALYTICAL CHEMISTRY
molar r a t i o
Figure 5. Separation of a degenerate curve by fL for a system (m = 1, n = 1)
#
1
A . f~= 1.00 MCh & LCh degenerate B. fI 0.80 MCh separated C . f~ 0.80 LCh separated All curves belong to KC = 10
The effect of f L can be considered both in the slope of the nondissociation line part and in the coefficient term of the displacement part, either in MCh or LCh, where f L = 1 obviously simplifies the equations. The simplest (m = 1, n = 1) system must show the same curve by MCh as by LCh when CL’in the latter is taken as equal to CMin the former, only if fL = 1. When fL # 1, the MCh curve of the above system should be separated from the LCh curve as if the f L = 1 curve is “degenerate,” as in Figure 5. On the contrary, a system ( m = 1, n = 1) must have a f L different from 1 if its MCh and LCh curveb measured with the same concentration for Cawin the former as for CL’ in the latter are obtained differently. If the same curve is obtained either by MCh or LCh, the f L must equal 1. Such a “separation test” on f L for the ( m = 1, n = 1) system has been found to be unexpectedly useful, as will be shown in the experimental part. For systems other than (m = 1, n = l), the effects of , f L are very complicated and will not be treated. The effect of K or C o n the curve is included in the coefficient term which is also affected by m and n even if f L = 1. In a system ( m = 1, n = 1) in which f L = 1, the curvature of the degenerate plot is determined only by KC, as similarly shown for photometric titration curves (12, 13), to be larger by the smaller KC. When KC exceeds 100, the curve rapidly approaches the nondissociation line, with almost a sharp break at about KC = 500 or greater. A group of curves having different KC values can be drawn for systems (m = 1, n = 1) in advance, as in Figure 6, where experimentally obtained curves are assigned for appropriate KC from their fitness. Since C is usually taken definitely in a range around 10-4 or lO-jM, such a group of curves can also be used for (12) R. F. Goddu and D. N. Hume, ANAL.CHEM., 26, 1679 (1954). (1 3) A. Ringbom, “Complexation in Analytical Chemistry,” Interscience Publishers, New York, N. Y., 1963.
1.0
1.0 0.9C
x
0.75
x 0.5
0.5
I
0
0
I
1.0
ratio
molar
2.0
Figure 6. Theoretical molar ratio curves for system ( m n = 1) and fL = 1
=
1,
Scales are shown to assign the measured curves with the values indicated in KC A . KC
=
B. KC
=
C. KC
=
D. KC
=
E. KC
=
100 K
=
1 X lo6, if C = 10-4M K = 1 X lo7, if
c = 10-5~ 10-4MK = 1 X lo6, if c = 10-5~ 5 K = 5 X IO4,if C = lO-*M K = 5 X lo5, if c = 10-6M 2 K = 2 X lo4, if C = 10-‘MK = 2 X lo5,if c = 10-5~ 1 K = 1 X lo4, if C = 10-4M K = 1 X lo5,if c = 10-5714
10 K = 1 X
lo5,if
C
=
assigning, usually conditionally, K , if drawn with a definite C. Similar curves can also be drawn for various fL. Scales drawn 0.90 are used to determine more precise at large x in 0.75 KC or K values between the curves drawn as in Figure 6. Higher systems than ( m = 1, n = 1) involve more complicated situations. Theoretical curves for the systems ( m = 1, n = 2) are shown in Figure 7, where the curvatures are now determined by 2KCLI 2 f L for MCh while by K1I2CMfL for LCh. It may better be noted here that the curves for (m = 1, n = l), (m = 2, n = 2), and (m = 3, n = 3) are different from each other and theoretically distinguishable although all the three must give the same intersection at the molar ratio 1. Each system will be given the same curve either by MCh or LCh only iff6 = 1, while different curves if f L # 1. Thus, an intersection at 1 , even if obtained, is not necessarily taken as the system being (m = 1, n = 1). Such theoretical analyses of the curves can be useful also for other practical understanding of the method. By assuming that x = 0.99, the equation for (rn = 1, n = 1) with f L = 1 is reduced to
-
KC
=
1
+ lOO/x,,,‘
= lOO/x,,,’
(20)
where xmax’can be thought as a molar ratio reaching (A,),ax experimentally. Equation 20 means that, in the system, KC will have the order of 1 if (As),,, is reached at about 100 in molar ratio, 10 if around 10, and 100 if 1. Since the use of so large a molar ratio to attain (AJmaxis often impractical and further doubtful for the real meaning of such (As)max,reasonable molar ratio plots may have a limit for small KC, or rather for K
Figure 7. Theoretical molar ratio curves for system ( m = l , n = 2 ) a n d f ~ = 1. A. K
=
B. K
=
1010 and Clr = lO-*M(K
=
1010 and C’L = 10-4M(K
=
101*ifCAI= 10-5M),
LCh
lo1*if C’L = 10-5M), MCh C. K = 1Ol0and Cnr = 10-5M ((K = 10l2if C>i = 10-6M), LCh D . K = 1010 and C’L = 10-5M (K = lo1*if C‘L
E.
K =
lO*I
10-6M),LCh CAI= 10-5M (K = loi3if CM = 10-6M),
F . K = 1011 CIL = lO-SM(K =
lOI3
LCh if C’L = 10PM), MCh
as known C is taken for the experiment. This sets a minimum, whereas a corresponding maximum will be expected from the approach to the nondissociation line, for K available for reasonable molar ratio curves. With similar consideration, a rough idea on K available for the molar ratio method has been obtained for two levels of C usually used, for MCh and LCh, as well as for several combinations of m and n, as in Table 11, where some arbitrary assumptions are made as shown. On the other hand, whether MCh or LCh is better to be applied to a system becomes a practical problem. In addition to the experimental consideration on the backgrounds of the measured absorbances as described earlier, the relations between C and K i n Table I1 may also be taken into account for the situation. In this, the problem depends mainly on for a system, besides which method can attain smaller xmax1 whether the available ranges for K in Table I1 can cover the system’s own K. Thus, LCh will be noticed to be generally preferred to MCh if m > n, MCh to LCh if m < n, and either if m = n, for smaller xmax1.This is a qualitative deduction which may be applied to both cases of f L = 1 and f L # 1, sincefi may not usually be separated so much from 1 even if f L # 1. These analyses give some understanding of the molar ratio curves as well as the method, but are still qualitative and do not allow one to deduce m, n, K or f L directly from experimental plots. The shapes of obtained curves in comparison with assumed ones suffer from critical interactions between effects which may be caused indiscriminately by many paramVOL. 41, NO. 10,AUGUST 1969
1291
.? '
........
__.___... J
0.6
-z 0.4 -.-a
X'
-I
x
I
u)
- 0.2 0
Figure 8. New linear plotting method in MCh
A NEW LINEAR PLOTTING METHOD
(mln)fL i(nq-l/mCL'-(m+n-
I)/mfL- (n-l)/m
.
x-(m-l)/m(l - x ) - n / m
6.0
A . Cu(II)-PAN MCh (As)max - ( A ~ ) L= 0.353 B. Cu(II)-PAR MCh - as)^ = 0.430 C. Fe(II1)-PAR LCh (A,),,, - as)^ = 0.645 D. Fe(I1)-Ph LCh (A,),,, - (A*)L = 0.440 E. Fe(II1)-TR LCh (Ashax - ( A ~ ) L= 0.390
by dividing both sides by H. Since m, n, and K can be assumed to take definite values for a complex formation system being studied, while CL' and f L are also given definitely and not changed during the experiment, Equation 21 indicates that a linear relationship exists between two experimentally measurable quantities as
Equation 13 is now rewritten as =
4.0
molar ratio
Figure 9. Ordinary molar ratio curves of complex formation systems in Tables I11 and IV
eters. In the following part of this paper, a new linear plotting method evolved as a more quantitative deduction of the present analyses will be described.
X.WCh'/H
2.0
(21) X&fch'/X
= aMCh
+
bMch
'
?f-(m-l)'m(l - x)-n'm
(22)
where Table 11. Ranges of Conditional Concentration Constant, K, Available for Molar Ratio Method.
Nondissociation MethodY Minimum Kb Limit of Kc
System"
c = 1x
10-4~ m = 1, n = 1 m = l,n = 2
...
m = l,n = 3
...
m = 2, n = 2 m = 3,n = 3 = 1 x 10-5~ m = 1, n = 1 m = l,n = 2
c
... m
=
1, n
=
3
... m = 2, n m = 3,n
= =
2 3
MCh, LCh MCh LCh MCh LCh MCh, LCh MCh, LCh MCh, LCh MCh LCh MCh LCh MCh, LCh MCh,LCh
x 107 x 1013
1 X lo5 1 x 1011
1
3
107
2 . 5 X 1011
10'7
1
x
1x
5 x 109 5 X 1013 3 X IOz2
1 X 106
x 1013 x 109 1 x 1020 5 x 1012 I 3
5 X 1016 3 X lon
1
x
10'0 1015 X 1017 X 10%
x
1 5
3
x x x 1 x 1 x 1
I 2.5 '
5 3
108 1015 1013 1022
1018
X 1020
X
Values derived qualitatively for fL = 1; may be applied for Thus. system or method is interchangeable for m and n. * K gives x = 0.99 at 10 times molar ratio, as for each corresponding nondissociation intersection. K more than the value in this column and less than in the next will give a curved plot. K gives x = 0.99 at a molar ratio which is increased 10% from each nondissociation intersection. K more than this value will show a break for each system. a
f~
# 1.
1292
ANALYTICAL CHEMISTRY
aMCh
bMCh E (n$')-
= (m/n>fL
l / m C L ' - ( m + n - l ) / m .L f -(n-l)/m
(23) (24)
for the given system and experimental condition. On the right side of Equation 22, the variable part is called here as "x-function-plot" against the previous x-function-anal. A linear plot can be achieved between the two variables shown as X M C h ' and corresponding H are measured experimentally while rn and n are assumed properly, to obtain the intercept a M C h as well as the slope b M C h of the plot, as in Figure 8. The linearity of the plot should be used, first of all to ensure that the assumed values for m and n are correctly chosen, since incorrect sets of m and n will give nonlinear plots even if x and X.WCh' are measured precisely. Characteristic nonlinearities which are sometimes met in the plots will be discussed later. In the second place, the intercept aYChthus obtained under a correct set of m and n will simply decide f L in Equation 23. The value of K will then be calculated in Equation 24 from the slope bMchby applying the assumed m and n, the aboveobtained f L , and CL' taken for the experiment. The K is usually a conditional concentration constant for the system being studied. The same is also applied to the cases of LCh. In the latter, the x-function-plot for a system (m = a,n = /3) by MCh can be directly employed as the x-function-plot for a system
I
20
I
i
'
I
( f o r B and C )
x % I-lo+
6.0 X' x
4.0
2.0 I
I
I
0
5.0
I
1
I I-X
10.0
Figure 10. New linear plottings for systems ( m = 1, n in Table I11
=
1)
I)/%fL--l
I
( for
A)
Figure 11. New linear plottings for systems ( m = 1, n = 2) and ( m = 1, n = 3) in Table IV A . Fe(II1)-PAR ( m = 1, n = 2) LCh B. Fe(I1)-Ph (rn = 1, n = 3) LCh C . Fe(1II)-TR ( m = 1, n = 3) LCh
( m = P , n = a ) by LCh against measured xtCh'lx because of the interchangeable nature between these two functions. The intercept and slope thus obtained in the plot now become
(mK)- l/nC*-(m+n-
I
4.0
{XCI - u$
A . Cu(II)-PAN MCh B. Cu(II)-PAN LCh C . Cu(II)-PAR MCh
bLCh
I
I
2.0
0
(26)
From these, the values of m, n, fL, and K are also evaluated as in MCh. The new linear plotting method can thus be carried out almost automatically with normally obtained experimental data to give unique information for complex formation systems. The method is unique because these evaluations are based on almost all absorbances measured at the curved part of the plot except one for (As)maxand need no approximations for even concentration terms involved. The possibility of eliminating even the need for a measured (As)maxwill be discussed later along with several interesting features of the present new method. EXPERIMENTAL
To investigate the applicability and problems of the plotting method, several systems known to form ML, ML2, or ML8 with certain metal ions and metal-indicators were tested. In addition, several new systems were studied. All metal ion solutions were prepared from their appropriate salts and standardized with EDTA titrations. The solution of Fe(II1) was also checked gravimetrically. The metal-indicator solutions were prepared from manufacturers' reagents with or without crystallization. The weighed molar concentrations were usually taken to be CIL. PAN was checked by photometric titration (10) and found to have a purity of 84.0z which agreed well with f~ evaluated in the plotting method as will be shown later.
Spectrophotometric absorbance measurements were made with 1-cm cells with Shimadzu's QV-50 instrument. Complete complex formation was ensured by allowing metalligand mixtures to stand 2 or 3 hours at 25.0 f 0.1 "C. Absorbances were measured within 1 "C less than 25 "C or as indicated. Accessories attached to Hirama's 11-B filter photometer were used for the photometric titration of PAN, where 3-cm glass cell was employed. Measurements of pH were made with Tda-Denpa's HM-5A as well as Beckman's Zeromatic 96 pH-meters. Alkalimetric titrations were carried out in a 4-hole flask at a temperature of 25.0 f 0.1 "C ( I ] ) . RESULTS AND DISCUSSION
Figure 9 shows some ordinary molar ratio plots newly measured in this experiment. Most of the plots showing curvatures do not give definite integral molar ratios at their intersections as shown by the dotted lines in the figure. Also, previous theoretical analyses are verified by the expected downward curvatures shown for ( m = 1, n = 2) and ( m = 1, n = 3) systems. Figures 10 and 11 show the results of applying the new linear plotting method. The experimental conditions and the evaluated values off& and K are tabulated in Tables I11 and IV. The effective concentration (C,),,,of the complex formed at a molar ratio showing (As)msx is determined, iff^ is determined, by : (Cc)max
f~
*
CL
(27)
for systems which use Equation 1. Thus, the molar absorption coefficients ec of the complexes can also be calculated from (A&,= measured and (C&, obtained as explained above (see Tables I11 and IV). Systems with a K which is too large are plotted by experimentally reducing the conditional K . This procedure is specific to the present method and contrary to that of Harvey VOL. 41, NO. 10, AUGUST 1969
1293
Experimental Conditions Method PH Buffer Ionic strength adjusted by Concentration of the constant species Temperature, OC Wavelength, mp References Plotted Results
Table 111. Experimental Conditions and Results for (rn = 1, n = 1) Systems Cu(II)-PANa Cu(II)-PANa Cu(II)-PARb MChc LChc MCh 2.39 2.39 4.80 Citrated Citrated Acetatee 0.20 NaC104 0.20 NaC104 0.20 NaC104 cL'= 2.0 x 1 0 - 5 ~ CM= 2.0 X 10-6M cL'= 2.0 x 1 0 - 5 ~
f L
K , obtained as conditional
K', calculated K',in literature ec, obtained ec, in literature
5 50
25
25 550
25 510
(10)
(10)
(14-16)
0.85 1.4, x 106 1.3 X 1016 ca. 1016 (10) 2.1 x 104 2.36 X 104 at 550 mp
0.84 1.48 X lo6 1.3 x 10'6 ca. 1018 (10) 2.1 x 104 2.36 X 104 at 550 mp (IO)
0.90 5.3, x 105 1 . 2 x 1017 3.2 X 1017(16) 2.4 x 104
(10) a
b
...
1-(2-Pyridylazo)-2-naphthol. 4-(2-Pyridylazo)-resorcinol.
Measurements were made in 20% dioxane aqueous solutions as in Ref. 10. Na2Hcitrate HCl. Sodium acetate acetic acid. Ammonium oxalate was added as 0,001M in the measured solution to make the plot curve. a of Cu(I1) by oxalate was thus used in the following calculation for K'.
+
+
Table IV. Experimental Conditions and P Results for (rn Experimental Conditions Method PH Buffer Ionic strength adjusted by Concentration of the constant species Temperature, "C Wavelength, mp References Plotted Results
K,obtained as conditional K', calculated K', in literature ecr obtained e,, in literature
1, n = 2) and (rn
=
1, n = 3) Systems
Fe(II1)-PAR m = l,n = 2 LCh 9.25 BoraxC 0.20 NaC104 CM = 1.0 X 10-6M
Fe(II)-Pha m = l,n = 3 LCh 2.60 Citrated 0.20 NaC104 c . =~ 4.0 x 1 0 - 5 ~
LCh 9.90 Ammoniae 0.10 KNOs CM = 6.5 X 10-'M
18 530 (17 )
25 510 (18)
20 480 (3, 19)
1
f L
=
.oo
1.00 x 1010 1.7 x 1084
...
6.5 x 104 6.04 X lo4 at 536 mp (17 )
1.oo 3.33
x 10'8
2.6 X 1020 2.0 X 1021(18) 1 . 1 x 104 1.11 x 104(20)
Fe(111)-TRb m = l,n = 3
0.88 2.39 x 1013 1.9 x 1047 7.9 x 1046 (19) 0.60 x 104 0.6 x 104atp = 0.60 (3)f
1,lO-Phenanthroline. Pyrocatechol-3,5-disulfonicacid (Tiron). Potassium hydroxide borax. Nitrilotriacetic acid (NTA) was added as 0.001M in the measured solution to avoid hydrolysis of Fe(II1). a of Fe(II1) by NTA was thus used for calculating K'. d NalH citrate HC1. e Ammonia ammonium chloride. EDTA wa3 added as 0.05M in the measured solution to avoid hydrolysis of Fe(II1). a of Fe(II1) by EDTA was thus used for calculating K'. f This value was calculated from Figure 5 of Ref. 3. a
b
+
+
+
and Manning (3), because absorbances measured in the curved parts are most useful in the present method. Thus, Cu(I1)-PAN was determined by adjusting the pH of the solution, while Cu(I1)-PAR by adding ammonium oxalate as a (14) F. H. Pollard, P. Hansen, and W. J. Geary, Anal. Chim. Acta, 20, 26 (1959). ( 1 5 ) T. Iwamoto, BUN. Chem. SOC.Japan, 14, 605 (1961). (16) M. HniliEkovh and L. Sommer, Collection Czechoslov. Chem. Communs., 26, 2189 (1961). (17) T. Takeuchi and Y . Shijo, Japan Analyst, 14, 930 (1965). (18) H. Irving and D. H. Mellor, J. Chem. SOC.,1962, 5222. (19) A. Willi and G . Schwarzenbach, Helv. Chim. Acra, 34, 528 (1951). (20) M. G . Moss, IND. ENG.CHEM.,ANAL.ED., 14, 862 (1942). 1294
ANALYTICAL CHEMISTRY
sub-coplex forming reagent. The complexes of Fe(II1) are measured and hydrolysis is avoided by adding suitable reagents. The conditional K thus obtained is multiplied by ~ L ( H ) and , a.w(Lt) if necessary, to get the corresponding true K'. Most values of a are taken from Ringbom's book (13), while ~ L ( H )for PAR were calculated from pK's given by Freiser et al. (21). The new technique gives reasonable values for K' and as shown in Tables I11 and IV. The success of the new plotting method depends, first of all, on the linearity of such a plot. Experimental data show that (21) A. Corsini, I. MaiLing Yih, Q. Fernando, and H. Freiser, ANAL.CHEM.,34, 1090 (1962).
I
0
1
I
4.0
2.0
I
6.0
I
I
1.0 I 0
I -%
Figure 12. The first type of unlinearity appeared in systems (m = 1, n = 1) by MCh A . Pb(I1)-XO: pH 5.35(acetate buffer), 580 mp, and C'L = 2.0 x 10-6M B. Cu(I1)-PV: pH 6.82(hexamine buffer), 650 mp, and C'L = 1.7 X 10-6M
I
2.0
I
4.0
I
I
I
{XC I-%)\T Figure 13. The second type of unlinearity for Fe(111)-TR by LCh at pH 5.70 (acetate buffer) as (m = 1, n = 2) Ckf = 1.0 X lO-'M. and 560 mp
there are at least two factors which cause two types of nonlinearity. The first type of nonlinearity is shown in plots of (m = 1, n = 1) systems, where steeply rising abnormally higher points at smaller molar ratios are observed as in Figure 12 for Pb(11)-XO (xylenol orange) (22) and Cu(I1)-PV (pyrocatechol violet) (23). The possibility of choosing incorrect sets of m and n for the systems to explain the nonlinearity does not seem likely, since such a misplotting must give a curved plot for all abnormal points, not just a part of them measured for one system. The fact that this nonlinearity is observed only in small molar ratio ranges and remarkably only for some special systems may suggest another possibility. At the smaller molar ratios by MCh used in Figure 12, there should be only small amounts of metal ions compared with each ligand. Therefore, some complexes which contain more ligands than metal ions as m < n in M,L, may be formed there even if each ML is to be formed as a main product for the most parts of their molar ratio plots. If, for example, MLz or ML3 is formed only in such small molar ratio ranges, the curve will start with a downward curvature, as the theoretical analyses showed, to be connected with the main curve for ML which must have an upward curvature. The absorbances measured as ML are thought as lowered somewhat at the small molar ratios and as giving there abnormally higher points in the linear plots for (m = 1, n = 1). Thus, the first type of nonlinearity may be caused by poly (ligand) species which can be formed in such smaller molar ratio ranges. Although small absorbances in the regions have not been allowed to indicate species involved, relations between systems and nonlinearity may be a subject of future studies. In our
(22) J. Korbl and R. Pribil, Chemist-Analyst, 45, 102 (1956). (23) J. Cifka, 0. Ryba, V. Suk, and M. Malat, Chem. Listy, 50, 888 (1956).
A. B.
- as)^ = 0.450 (actually measured) - (A@)L= 0.500 (assumed)
experimental data, most systems showed traces of this nonlinearity while the systems in Figure 12 were remarkable cases. This suggests that these systems are more likely to form polyligand species than others under favorable conditions. The second type of nonlinearity has been noticed in the system Fe(II1)-Tiron at pH 5.70 by LCh. Although this system is known to form ML2 at the above pH (3), the direct linear plotting as (m = 1, n = 2) gives a curve as shown dottedly in Figure 13. This curve was made almost linear by a procedure in which the value of (A,),,, - (A& used for calculating x was temporarily increased about 10% from the originally measured one and the linear plotting was redone by using the new x. The similar situation is also shown in Figure 14 for the system Pb(I1)-PAR at pH 6.20 by MCh. The system is treated originally as (m = 1 , n = 1) (14)and required about a 4% increase of (As)max - (&)L this time to make the plot linear. It will be obvious that such nonlinearities concern the absorbances measured at large molar ratios. Even if one component of a complex is added in excess enough to ensure the complete complex formation, the maximum absorbance thus obtained may sometimes be not the appropriate (As)max belonging to the same group as the absorbances measured in the middle range of molar ratios. If this is the case, some higher order complexes must be responsible for the situation. The above assumption seems to be verified by careful inspecting absorption curves at several large molar ratios. In the Fe(II1)-Tiron by LCh, the MLz peak at 560 mp is moved toward smaller mp by adding more Tiron than at molar ratio 4, where the maximum absorbances of the peak decrease slightly. This suggests that, by adding more Tiron than Fe(II1) in molar ratios, the peak naturally tends to be transferred from MLz to the more ligand-containing ML3which has VOL. 41,NO. 10,AUGUST 1969
1295
4.0
3.0
XI ?(
2.0
c /
I
1.0
I
0
,
I
5.0 I
I
10.0
I
15.0
-x
Figure 14. The second type of unlinearity for Pb(II)-PAR by MCh at pH 6.20 (hexamine buffer) as (m = 1, n = 1)
C’L = 2.0 X 10-%f and 525 mp. Measurements were made in 20 dioxane solutions. A . (As)msx- ( A ~ L = 0.397 (actually measured) B. (A&,&, - (A.)L = 0.418 (assumed) C. - ( A ~ L= 0.437 (assumed)
the peak at 480 mp (3). The two peaks cannot be resolved by the spectrophotometer and the continuous move of one peak is thus observed. Since the absorbances measured at the fixed 560 mp, can give only the shoulders of the absorption peaks at such molar ratios, the obtained molar ratio plot will not be normal and the curved plot in the linear plotting method might result (Figure 13). The same is almost true for the Pb(I1)-PAR by MCh with possible M2L (24, although the latter peak has not been measured. This M2L peak does not seem to be separated from the M L peak so much as in the former system, because the above increase factor is only 4% compared with 10% for the former case. Both types of nonlinearity suggest a verification for what the authors described as “not logical” in procedures employing too large or too small molar ratios in measuring absorbances. Absorbances measured at such extreme molar ratios may belong to different systems from the main one, but these various kinds of absorbances measured might still have been included into one molar ratio plot as a continuous curve in the ordinary simple method. Particularly the quantitative estimation of appropriate (As),,, for the main system is most important for completing the plot but may sometimes not be experimentally possible owing to the second type of nonlinearity. On the contrary, such points of different nature in a plot can be easily differentiated in the new method. Also in this method, (As),,, qualified for the complex formation system being studied need not be actually measured but can be (24) F. H. Pollard, P. Hanson, and W. J. Geary, Anal. Chim. Acta, 26, 575 (1962). 1296
0
ANALYTICAL CHEMISTRY
I
I
0 I
10.0
5.0
1
.ISSO
I
I-x Figure 15. New linear plottings of Zn(I1)-XO at 570 mp pH 5.55(acetate buffer), ionic strength 0.20 (NaCIO,), and 25 “C A . MCh C’L = 1.0 X 10-5M: (f0wd)fL = 1.45, K = 5-41 X 1W B. MCh C’L = 2.0 X 10-6M: (f0wd)fL = 1.45, K = 5-29 X 106 0.90 X 106 C. LCh CM = 1.0 X 10-6M:(f0wd)j-L = 1.33, K D. LCh CM = 2.0 x 10-5M: (foUd)fL = 1.33, K = 1.30 X 106
attained by successive approximations until a linear plot for the system is obtained as in Figures 13 and 14. Thus only absorbances measured in the middle range of molar ratios can be used for complex formation studies as the most logical technique. For successive approximations, computerized treatments (25) of measured absorbances may naturally be expected, although these do not seem to be possible soon. Difficulties in precision problem (25)as well as other problems will have to be solved before further treatments are developed. One of the most difficult problems in the new technique is the chemical interpretation of the meanings of fL and K thus obtained. Two cases need to be differentiated. Firstly, the values obtained as fL and K , respectively, of the Cu(I1)-PAN by MCh agreed well with those by LCh as in Table 111. Whether the obtained fL which did not equal 1 with the K is justified or not for the case was also checked with the “separation test” mentioned in Figure 5 . The test gave the expected result in which the theoretically drawn LCh curve by using fL and K obtained in the MCh-plot agreed well with the experimentally obtained LCh-plot. These values of fL obtained in the linear plots also agreed well with that by photometric titration. Thus there seem few questions for the fL obtained to show the “purity” of the reagent PAN used, although the remaining 16y0 of the latter is not yet identified. At least for
39, (25) K. Momoki, H. Sato, and H. Ogawa, ANAL. CHEM., 1072 (1967).
I
5.0
I
I
10.0
15.0
I
I-x Figure 16. New linear plottings of XO chelates by MCh Hexamine buffer and ionic strength O.l(NaC10,) were used throughout at 25 “C. Zn(I1)-XO was measured at pH 5.3 and 574 mr, while Pb(II)-XO at pH 5.0 and 570 mp. A.
B.
C. D. E.
Zn(I1)-XO with commercial XO, C’L = 1.0 X 10-5M.Found: f~ = 1.66, K = 2.3 X lo5 Zn(I1)-XO with recrystallized XO from A, C’L = 1.5 X 10-sM. Found: f~ = 1.45, K = 3.0 X lo6 Pb(I1)-XO with the same reagent as A, C’L = 1.0 X 10-6M. Found: f~ = 1.77, K = 1.8 X 106 Pb(I1)-XO with the same reagent as B, C’L = 1.1 X 10-6M. Found: f~ = 1.72, K = 2.8 X 106 A in Figure 15
this system, Equations 13 and 14 can define the fL and K as mentioned earlier. Secondly, however, the complexes of XO with Zn(I1) (26) and Pb(I1) gave very confusing results as shown in Figures 15 and 16. The reagents XO which showed only traces of semiXO in chromatographic as well as in spectrophotometric tests (8)were used for the experiments. In Figure 15 for Zn(I1)-XO, fL as well as K obtained by MCh disagree considerably with those obtained by LCh, respectively. And also, each fL or K by even one method (MCh or LCh) differed from another fL or K obtained by changing C, the constant concentration of one component. Furthermore, fL obtained in the Zn(I1)XO is shown to be different from that in the Pb(I1)-XO in Figure 16, although the same reagent of XO was used for both cases. The use of a reagent recrystallized further with ethanol (26) also gives a different fL and Kfrom those with the original reagent XO. Thus such fL obtained in the new method cannot be assumed as representing the “purity” of the reagent XO. Still it is very interesting that fL obtained by the linear plots are found to be quite different from those by alkalimetric titrations. Since the ordinates in Figures 15 and 16 are scaled with using x ’ corrected for alkalimetric titration values, values almost one and a half larger apparently have to be assigned as fL by the present method than as the “purity” by alkalimetric titration. The separation tests for these systems gave similar results in which, on the contrary to the previous Cu(I1)-PAN, the experimental curves did not agree with their corresponding (26) K. Studler and I. Janousek, Talanta, 8, 203 (1961).
theoretical curves for all cases. Thus, for such XO chelates, the new method has not been able to give definite fL as well as K as defined in Equations 13 and 14. However, this does not seem to be ascribed to the method but rather to these systems themselves. More careful inspections of the molar ratio plots as well as the linear plots of these systems showed some further abnormalities. The molar ratio plots do not agree with any members of the exact (m = 1, n = 1) group especially at larger absorbances than about K = 0.7. The larger absorbances in the Pb(I1)-XO were somewhat more remarkable than in the Zn(I1)-XO to give abnormal upward displacements from the normal plot. Correspondingly, the linear plot of the former as (m = 1, n = 1) showed an upward curvature at the larger values of x-function-plot with downward displacements of the larger x ’ / x from the expected linear line, while that of the latter only a trace. The first type of nonlinearity was also remarkable in the Pb(I1)-XO than in the Zn(I1)-XO. In spite of the fact that the same reagent was used for both systems, the former system gave thus more abnormalities at small and large absorbances than the latter. These confusing results suggest that the ligand XO and/or the reagent are complicated and can form several complexes with the metal-ion on the molar ratio plot when the latter is measured by changing the concentration of one component holding the other constant. An “apparent” molar ratio plot similar to a (m = 1, n = 1) curve might be obtained by such several complexes formations mixed or averaged. Besides complexes by impurities such as semi-XO (8) in the reagent, higher order successive complexes including polymetal-polyligand species (27) and mixed complexes including hydrogenand hydroxy-complexes (28) by the ligand XO will probably be responsible for these XO-chelates systems. If, however, such several complexes formations can be averaged into a molar ratio plot, even fractions or decimals, besides integral numbers, may have to be assumed in assigning m and/or n. This problem seems to have a correlation with “fi” by Bjerrum, though cannot be discussed yet in the immature state of the present work. The present paper opens a door for future studies by giving a new simple technique to the field where available checking methods have been limited. Thus, even the applicability of alkalimetric titration values to spectrophotometric studies may have to be checked by considering what species are actually being measured in both methods. Especially important is that the purities and impurities of ligand reagents which are actually effective for measured data should be re-examined more carefully in such complexes. Actual absorbances measured are not necessarily limited to be ones as in Equation 3 but can include also absorbances effectively exhibited by other species if in the solutions. The use of two or more wavelengths for plots will be useful or necessary as suggested previously (21), while even careful inspection of a number of absorption curves will become necessary for obtaining only one plot. Purifications as well as checkings of ligand reagents should naturally be developed further, but even some inorganic component of the reagents cannot be ignored. For example, many ligand reagents are used as sodium salts which are thought to contain definite integral numbers of sodium ion as a component of the molecules. However, actual concentration of the sodium ion do not always seem to be definite as exactly corresponding to ones such as di(27) B. BudgSinskf, 2.Anal. Chem., 209, 379 (1965). (28) Ibid., 2W,247 (1965). VOL. 41, NO. 10,AUGUST 1969
1297
I
Cw ( CL= c o n s t a n t 1
0.0 I
I
0.5 I
If
3.0
2.0
1.0
'
0.0 0.0 Xi
molar r a t i o ( C w / C ~ ) Figure 17. The normalization of (m = 1, n = 1) molar ratio curve having decreasing absorbances with increasing molar
ratios
I
I
I
I
4.0
THE MOLAR RATIO PLOT OF DECREASING ABSORBANCES WITH INCREASING VARIABLE CONCENTRATIONS OF ONE COMPONENT
The method hitherto described concerns only the molar ratio plot in which measured absorbances increase with increasing variable concentrations of one component at a conANALYTICAL CHEMISTRY
I
I
6.0
7
I -1c Figure 18. New linear plottings of Calmagite chelates by MCh at 620 mp
+
sodium salt when commercial reagents are used with or without purifications. One verification for this is in alkalimetric titrations. By using different lots of reagents, or even by repeating purifications of a reagent, for a ligand, the detailed portions of resulting curves are not usually identical. Particularly initial pH in the titrations is often not given definitely but varies slightly from reagent to reagent even if the solutions are prepared and the main pH-jumps are treated alike. This is at least partly due to indefinite concentration of sodium ion, in addition to ligand impurities, in the reagents. Such indefinite concentrations of even sodium ion can affect purity factors measured by alkalimetric titrations as well as fL by the present method, of ligand reagents. If values for fL are thus affected and changed by the reagents, the values of formation constants and molar absorption coefficients are affected and one cannot give a definite value for each complex formation system. Without defining such sensitive problems in actual measurements as precisely and accurately as possible, vast amounts of data given in complex formation studies will thus be of less value. Even the continuous variation method which has been considered as giving more quantitative complexes data and advanced again recently (27-30) does not eliminate the problem. The present paper has not dealt with an important precision problem (25) involved in the new method. This problem will differ in systems being studied because of possible interactions between deviations in measurements and abnormal displacements of a chemical nature. It will be treated in more detailed studies by applying the present method to systems, including some presented here, in future publications. Data presented in this paper are thus mostly tentative and should be refined in the future. With these and other problems are still left for future studies, a more fundamental understanding of complex formation systems and the molar ratio method are promoted by present studies.
1298
I
2.0
+
pH l0.34(buf€er: monoethanolamine HCl KCN), ionic strength O.lO(KCI), and C'L = 2.0 X 10-5M. The Calmagite reagent used was recrystallized twice according to Ref. 32 = 0.80. K found = 1.9 X 105(16 "C) can be compared with 4.9 X 105(pH 10.00) (31) by continuous variation method B. Ca(I1)-Calmagite: f~= ? K found = 3.6 X 103(18 "C) can be compared with 4.7 X 103(pH 10.00) (31) by logratio method. Actually measured points are not shown as far out of scale
A . Mg(I1)-Calmagite: f~found
stant one of the other. Such plots are mainly used and, as previously mentioned, obtained usually by measuring the peak absorbances of the complex being formed. Sometimes, however, the plot of decreasing absorbances with increasing variable concentrations of one component is preferably measured mostly at the ligand peak, because of the latter's higher sensitivity for detecting the complex formation than the complex peak. Diehl and Lindstrom thus measured calcium and magnesium compleses of EBT (6) and Calmagite (31) at each ligand peak in their log-ratio plotting method. The present treatments with x can easily be extended to such a case as in Figure 17, since both cases are mathematically the same. As Equations 3, 4, and 5 are nearly applied except for some notations, an equation corresponding to Equation 6 for the normalization is derived as
where all the notations correspond to those in Figure 17. This equation shows that the normalization of the measured absorbances for the case can be carried out similarly, but in opposite direction concerning the absorbance axis, to that in Figure 1, as in Figure 17. Thus, normalized absorbances xt' can be used instead of x 4 in the equations that followed Equation 6 without any other modifications. (29) E. Asmus, Z . Anal. Chem., 183, 321 (1961). (30) K. S. Klausen and F. J. Langmyhr, Anal. Chim. Acta, 28, 335 (1963). (31) F. Lindstrom and H. Diehl, ANAL.CHEM., 32, 1123 (1960). (32) F. Lindstrom and R. Isaac, Talanta, 13, 1003 (1966).
Therefore, the generalized treatments described earlier in this paper become useful also for this case of the molar ratio plot. The linear plottings using %
