THE DETERMINATION OF STABILITY CONSTANTS OF COMPLEX INORGANIC SPECIES IN AQUEOUS SOLUTIONS R. STUART The Ohio State University, Columbus The Royal Institute of Technology, Stockholm, Sweden D U R I N G the past few years, there has been a resurgence of interest in the study of the equilibria of inorganic complexes in aqueous solution (see Pokras 1, 2, 5 ) . This research has led to a much better understanding of the physical chemistry of aqueous solutions. As more accurate values of stability constants become available through the study of complex equilibria, quantitative theories can be developed explaining the interactions between the metal ion, ligand, and the solvent. However, the agreement of different authors on the values of complexity constants is such as to leave doubt about the reliability of much of the presently available data. This is the problem that remains to be solved before satisfactory quantitative theories can be developed. The purpose of this paper is, therefore, to discuss the presently accepted methods for the determination of stability constants, i.e., the methods that are deemed to be the most reliable for indicating just what ionic species exist in solutions of "salts" of multivalent metal ions with various ligands. This discussion is offered as a supplement to Volume 2 of the tables of stability constants (4) compiled by Bjernun, Schwarzenhach, and SillBn. This volume tabulates all available data on stability constants of complexes with inorganic ligands together with solubility products of inorganic substances. Unless the chemist has some familiarity with the methods for the determination of these constants, it is difficult to judge which values are likely t o be the most accurate; the procedure of selecting the most recently determined values is not a reliable one. The fundamental process to be discussed here is the formation of complexes by reactions of the general type,
q M(HaO),
+ p X = M,X,(H,O),
(1)
Ionic charges are ignored in the equation above. Particular attention will be paid t o the several criteria which must he followed if the measurements of stability constants are to be accurate enough to be of value in attempts to formulate quantitative theories of complex formation. I n considering reactions of type (I), we shall not be concerned with changes which may occur in the number address, Department of Chemistry, University of North Carolina, Chapel Hill. This work was made possible by a National Science Foundation Post-Doctoral Fellowship awarded to the author for study at the Department of Inorganic Chemistry, Royal Institute of Technology, Stockholm, Sweden.
' Present
592
of solvent molecules bound to the species, since equilibrium measurements can give no indication of these numbers. According to reaction (I), one or more complexes M,X,(H20), are formed when reaction occurs between M and X. In this paper the discussion will be limited to the case where M is a multivalent metal ion and where X is a simple anion like CI-, a complex anion like S04c2, or the quantity -H+ . All of these cases are treated together, since the experimental measurements and the analysis of the data are quite similar. It should, perhaps, he noted that while mathematical treatment of the data to calculate the stability constants is in principle identical, there are certain fundamental differences in the mechanisms of reactions producing, e.g., the complexes CdCI+, CdOH+, and Fe2(OH)2+4. For the complex CdCI+, the reaction may he written:
+
C ~ ( H X O ) , , + ~C1-
$
CdC1(H20),+, log Kt, = 1.59 ( 5 ) (2)
The second complex, CdOH+, appears to be formed according to the Br6nsted concept of aquo-acidity. Cd(H,O!"+P
+
=
H.0 CdOH(H%O).,+
+ HzOt, log Kit = -9.0
( 8 ) (3)
In the third example, Fez(OH)2+4is formed by the non-Br6nsted reaction describing the aquo-acidity of the Fe(II1) ion.
+
= + ~H,o+;log Ks? = -2.91
2Fe(H,0).+S 4H,O Fe2(OH)q(H20),+4
(7) (4)
HISTORICAL
The bmic experimental methods for the study of complex formation equilibria certainly cannot he considered to be recent developments; in fact,, most of these methods were available before 1900. The stepwise formation of complexes !.as first proved by Morse (8) and Sherrill (9) during the period 1902 to 1904. This was shown for the formation of the Hg(I1) complexes HgCl+, HgCI1, HgCI3-, and HgC14-2. Bodlander's method (10) for finding the composition of the predominant complex in solution from e.m.f. measurements was introduced in 1901 'nd first applied to the study of Ag+ and Cu+ complexes. Luther (11) and Abegg and co-workers (Morse (8) and Sherrill (9)) applied e.m.f. measurements, using a mercury redox electrode to detemine the central ion concentration, in their investigations of the Hg(I1) con~plexes. Euler (12) also employed e.m.f. methods to study complexes of NH3 and CN- with Ag+, Zn+2,and Cd+2, using Ag, Zn, and Cd electrodes t o measure the central JOURNAL OF CHEMICAL EDUCATION
ion concentrations. M o r ~ e(8) and Sherrill (9) used distribution measurements in the study of Hg(I1) halide complexes by determining the distribution of HgX2 between the aqueous phase and a benzene phase. Morse (8) used solubility measurements for the study of the Hg(I1) halide complexes, and Bodlander and Fittig (18) applied solubilities to the study of Ag' - NHBequilibria. Unfortunately, after this early beginning, progress in these studies of ionic equilibria slowed greatly during the 1920's and 1930's. Two of the deterrents to the study of complicated equilibria seem strangely enouph to have been the introduction of the Lewis concept cf activity and the Debye-Huckel relation for activity coefficients. Since the Dehye-Hiickel theory was quite successful in describing the hehavior of very dilute solutions of electrolytes, it became common practice t o consider I electrolytes as completely dissociated. Deviations from ideal behavior a t high concentrations mere attributed to nm-specific electrostatic interactions. I n order to conform to the generally accepted standard state for free energy measurements, the hypothetical 1 molal solution in the solvent water, equilibrium constants were usually extrapolated in some way to give the value of the constant for zero ionic strength. This is certainly a desirable practice, since data from the equilibrium measurements could then he used together with other thermodynamic data for a wide variety of calculations. It is necessary to consider the evidence, however, t,o see if it is generally possible to obtain values for complexity constants a t infinite dilution, since these complex formation equilibria are studied at relatively high ionic concentrations. In the case of certain simple equilibria involving only one complex formed from univalent ions, it is relatively easy to measure accurately the equilihrium concentrations by e.m.f. studies and to perform a reasonable extrapolation to infinite dilution. For example at very low concentrations of chloride ion, the follon-ingequilibrium exists: AgCl= Ag+
+ C1-
(5)
By making only a few logical assumptions, it is possible to extrapolate the data to give the value of the equilibrium constant a t zero ionic strength, e.g., as was done in works of Owen (14) and Popoff and Neuman (15). In systems where several complexes are formed, t,he steps are generally so close together that several complexes exist simultaneously. As an example of such a system, one may exa.mine the constants for the complexes of Hg(I1) wit,h C1- (16).
+ C l e HgCI+ Hgiz + 2 C l e HgCl, Hgt2
log K , ,
=
6.74 =t0.02
log Kz, = 13.22 f 0.02
+ 3C1- = H g C k log Ka, = 14.07 + 0.06 Hg+* + 4C1- = HgCI4r2 log K4, 15.07 + 0.06 Hg+z
=
(6)
(7) (8) (9)
These values of the equilihrium constants are valid for 25'C., ionic strength = 0.5, and [H+] = M. The distribution of mercury among the various complexes may be illustrated in a simple way by plotting the per cent of the total mercury in the form of HgClaC2, HgC1,-2 HgC1,-, HgC14c2 HgC1,HgCI?, and HgCI,-? HgCI,HgCL HgCI+ as a
+
+
+
+
VOLUME 35, NO. 12, DECEMBER, 1958
+
+
-IW~CI-I
rime1
Distribution of oampleaes ss a funotion of log [Cl-I in the HptLCI- syatern. conoentrations are in mo1e,1. If a vertical Line is drawn foraeiven value of [CI-1, the segment of the line falling in a given ares, e.g.. H g C l s . represents the per oent of the HE present as that complex.
function of the log [Cl-1. Since only mononuclear complexes are formed, the diagram is independent of the total mercury(I1) concentration. This is illustrated in Figure 1. If a vertical line is drawn a t a given ligand concentration, the segments of the line indicate the per cent of Hg(I1) in the various complexes. I t is obvious that even in a system such as this where only mononuclear complexes are formed, it is impossible to perform satisfactory extrapolations to zero ionic strength. I n order to form the complex HgClhC2, concentrations of C1- in the range 0.1 to 1.0 M are necessary. In certain systems, particularly those involving comnlexes of multivalent metal ions with hvdroxide ion, the investigation is further complicated by the formation of polynuclear complexes, e.g., in the hydrolysis of the tin(I1) ion (17). The following reactions were found to describe the hydrolysis in this system.
+ H.0 = SnOHi + H t log K , , = -3.92 + 0.15 2Snta + 2H10 = S ~ S ( O H )+ ~ +2H+ ~ log KZP= -4.45 * 0.15 3Sn + 4HaO = Sn8(OH),+' + 4H Snt2
lP
(lo) (11)
+
log K d 3= -6.77 f 0.03 (12)
These constants are valid under the conditions of 25'C. and the constant ionic medium with [C1Q4-] = 3 M . From the values of the constants given above, it is zeen that hydrogen ion concentrations of up to 0.05 A1 are necessary to repress the hydrolysis of the Sn(I1); therefore, if complexes of Sn(I1) with other ligands besides OH- are to be investigated, the concentration of H + alone must he maintained at a minimum of 0.05 M if mixed complexes with hydroxide ion are to he reduced to a negligible concentration. Since the ionic strength of the solutions being measured frequently approaches unity, it is very difficult to perform extrapolations of the data which will give two or three different constants at infinite dilution. In 1938, the results of a detailed study of the equilibrium Pb+Z
+ Cl- e PhCI+
were published by Giintleberg (18). This diseertation is relatively inacce~sihlefor most Americans, since it was published in Danish, but the results are very interesting in the light of the previous discussion. This equilibrium presents an almost ideal case for study, since the concentrations of Pb+%in the equilibrium
solution can he measured using lead amalgam electrodes and the [Cl-] can he obtained with Ag/AgCl electrodes. E.m.f. measurements were made t o 0.01 mv., and an attempt was made to extrapolate the data to infinite dilution using various forms of the DebyeHiickel equation for the activity coefficients. Values of the equilihrium constant were obtained which ranged from log K1, = +1.0 to -1-1.5 at infinite dilution. Although Giintleberg felt that the most accurate value (log KI1 = +1.40 to 1.44) was obtained using the Guggenheim equation for the activity coefficients, he concluded that too little was known about the behavior of activity coefficients to permit an extrapolation to be performed confidently, even in this simple case. Because of our limited knowledge of the physical chemistry of ionic solutions, it appears necessary to abandon the calculation of equilihrium constants a t infinite dilution for reactions which are as complicated as the Hg(I1)-C1- complexes or the hydrolysis of the Sn(I1) ion; however, it is certainly of interest to the inorganic chemist to know just what complexes are formed in these systems. The first person who appears to have developed an accurate method for studying the stepwise formation of complexes in such a system over a wide concentration range was Leden (6, 19). In his work on the complexes of Cd+2 with various ligands, the reactions were studied in a solvent of rather high ionic strength, 3 M NaCIOI. It was assumed that activity coefficients would remain conetant in this solvent and that the law of mass action could be applied directly t o determine just what species were present and t o calculate the successive equilibrium constants. I n this way, Leden succeeded in establishing the types of complexes present and the distribution of Cd among these species as a function of the ligand concentration, all in the solvent 3 M NaC104. Although the constants obtained in this way only apply exactly to the solvent 3 M NaCIOa, one can be reasonably certain that the same species exist in other similar solvent systems. EXPERIMENTAL METHODS FOR STUDYING COMPLEX FORMATION EQUILIBRIA
Since it is impossible to separate the question of what species exist in the solutions from the determination of the equilihrium constants for the species, the first step in the investigation of an unknown system is the determination of the values of p and q for all of the species M,X, in the system. If one assumes an incorrect set of reactions, the equilibrium constants calculated are of no significance. I n the past, for example, the aquo-acidity of cations has been generally assumed to arise from reactions of the Br6nsted type: Many acidity constants have been calculated assuming this type of reaction when in reality the main products of the hydrolysis were polynuclear species. The data in these cases were obtained over such a short concentration range that they could he approximately fitted by assuming one or two mononuclear constants; however, more careful measurements show that the products are actually polynuclear and therefore explanation of the data in terms of mononuclear constants is useless.
The mathematical methods used to determine the equilibrium constants are too involved to be discussed in detail in this paper. For recent reviews of the methods used in systems with mononuclear complexes, the reader is referred to the papers of Sullivan and Hindman (20) and of Roseotti and Rozsotti (21). Methods for the study of systems with polynuclear complexes have been discussed in detail by Sill6n (22, 25, 24). The analysis of the data depends, of course, on just what quantities are measured, and the common experimental methods for the study of complex formation in solution may be divided into three different types. The most reliable method for the etudy of complex formation equilibria is the measurement of the concentrations of individual species. The e.m.f. of electrochemical cells is used to measure both the central ion and ligand concentrations with very high accuracy. Polarographic methods may also be used to obtain the central ion concentrations under favorable circumstances. Vapor pressure measurements may sometimes he used to determine the ligard concentration, as for example with NHI in ammines. In systems with only mononuclear complexes, measurement of the distribution of the neutral complex between immiscible solvents is employed to obtain data on the concentration of the neutral complex as a function of the ligand concentration. The analysis of these data may he summarized in the following way. For the sake of brevity, we may write down the following symbols: B
=
A b
= = = =
a
K,,
total concentrhtion of metal M total (analytical) ligand concentration equilibrium concentration of free metal ion %Iz+ eouilihrium concentration of free lieand X equilibrium constant for formation~ofspecies M,X,
The following equations are obtained directly applying the law of mass action. B
= b
A =a
+ Zq[M,X,] + Zp[M,X,I
=
b
= a
+ Zga'bqh,,
(15)
+ ZpapbQKni,,
(16)
When the ligand concentration is experimentally determined, it is most convenient to summarize the data in terms of the "average ligand number," i.e., the :n.rr>Jgrilumh~rof lignndc hound per rnit?l ?tom. l'liis i~u:~ilfitv 101. Z 111 .i\.itpms. " is er~writllvr~i)re.w11~~1 with polynuclear complexes and by il in systems with only mononuclear complexes. Using equations (15) and (lfi), u
"
A
Also, in addition to the average ligand number, it is useful to study the variation of the free metal ion concentration with the l i g a ~ d concentration. The function analyzed in this case is usually log Bb-'. Again using equations (15) and (16) me obtain Typical data for a system with polynuclear complex formation (the hydrolysis of the tin(I1) ion) are given in Figures 2 and 3. A and B are know1 from the analytical data and baried over as vide a range as JOURNAL OF CHEMICAL EDUCATION
Average number, 2,of OH- bound per Sn as a function of log [Hi]. B as a Conoentrations are in mole/l. Curveaoalculatedfoiloe K u = -3.921, I o g K n = - 4.45, LogKtr = - 6.7L B = 4 0 m M 0 , 2 0 n r M 0, 10 m M A. 5 mM 0 and mkd 2.5 mM E, .
parameter.
concentrations, B, will distinguish between mononuclear and polynuclear complex formation. It should be noted that if all species including the free central ion, M, are polynuclear and contain the same number of metal atoms, Z and log Bb-I will be independent of b also. For example Hg2fZwould exhibit mononuclear behavior upon complex formation so long as all complexes were of the type Hg&(2-n)+. This special case should cause no confusion. When distribution measurements are used to study systems of mononuclear complexes, a is calculated from the variation of the degree of formation, a,, of the neutral complex with the ligand concentration, cf., Rossotti and Rossotti (21). The degree of formation of the complex MX,, a,, is defined by
a.
possible while a and b are measured. a and b are measured as A varies from 0 to A,, while B is held constant. This is repeated for several different values of B. These measurements are discontinued upon the appearance of colloidal sols, i.e., complexes of too high molecular weight to be solvated. Under these conditions, rapidly reversible equilibria connot be obtained. The solid lines in Figures 2 and 3 were ohtained using the set of constants given in ( l o ) , ( l l ) , and (12). Using the functions Z and log Bb-I, the equilibrium constants occur as coefficients in the power series in a and b, equations (17) and (18),and may be calculated by a variety of methods without making assumptions about what products are formed. Since polynuclear complexes are formed by the hydrolysis of tin(II), the distribution of the tin among the various complexes is a function of the total tin concentration as well as of the ligand concentration. This is illustrated in Figure 4 which gives the distribution for two values of B, 40 and 2.5 d. When only mononuclear complexes are formed, equations (17) and (18) reduce to the more familiar forms given below.
When only mouonuclear complexes are formed, a and log Bb-I are independent of b and hence of B , so that measurements a t two or more different total metal
and
A and B are again chosen as the independent variables, and the concentration of the neutral complex is the measured dependent variable. When only mononuclear complexes are formed a, is independent of B;
Figu.. 4
Dist~ibutionof tin(I1) hydrolysis products ha a function of -log [ H i ] . Concentrations are in mo1e/l. The distribution is given for two vsluesof B, 40 m M and 2.5 mM.
therefore the distributiou diagram, e.g., Figure 1, is independent of B. I t is assumed in all of the above relations that the activity coefficients of all species remain constant in all of the measurements, and the experimental conditions must be such as to satisfy this condition. In general, knowledge of one of the sets of data (A,a,B), ([MX.],a,B), or (A,B,b) for the system as complexes are formed makes possible the calculation of a as a function of a. Once this function is obtained it is possible, in theory, to calculate all of the N mononuclear complexity constants for the system. The number of constants which actually can be obtained depends upon the accuracy of the measurements. When polynuclear complexes are formed it is necessary to have the data (A,B,a) for several values of B and desirable to have (A,B,a,b),since the number of possible products is very large. Type 2
- 61 rwi Fig"-
3
parameter.
Lop Bb-1 as h function of log [H*], B aa a Conoentrationa arein mo1e/1. Curvesoaleulated for thesame a e t of oonstsnts aa in Figure 2. B vhlues are the ahme as in Figure 2.
VOLUME
35, NO. 12, DECEMBER, 1958
In the measurements of this second type, instead of the determination of the concentration of individual species, a function 2k,et is determined where k, is a constant for each of the it' species with coucentratiou
I n addition to N unknown equilibrium constants, thcrr :In. now A' : ~ d d i t i o ~I I~I Id~ I I ( I W I C I IIIIS~~ kt.I I.~.kN. S, The funrtion 2l:,r, . .m:tv" be obtnincd hv sncvtro~hotornetry, magnetic susceptibility, conductance measurements, etc., the constants depending of course on the quantity measured. I n the case of spectrophotometry, one additional dimension is added to the measurements, since the optical density can be determined over a range of wave lengths. The number of constants which can be obtained and their accuracy depends upon how accurately the change in the physical quantity can be measured as complexes are formed in the solution. I n general the number of unknowns involved in the analysis of data of this type is too great to permit a solution of the problem when three or more mononuclear complexes exist or when the complexes are polynuclear. These methods are generally limited to systems with certain special properties. For example, the spectrophotometric method is particularly useful when the absorption of the complex(es) occurs in a different region of the spectrum from that of the free ligand. Measurement of the extinction of the solution as a function of a allows construction of the plot e / B versus a from which 7i may be obtained ($5,86). Conductivity methods are useful when a single uncharged complex is formed. Once the species present have been identified, e.g., with e.m.f. measurements, these methods are of value as confirmatory evidence. c,.
"
A
Type 3
Solubility measurements have often been used to establish the ionic species present in solution. Measurements of this type are, however, of limited value, since A and B (total concentrations in all solvated species) can no lmger be varied independently. If the end product on the stepwise formation of complexes is and insoluble species M,X,, a and b will always he related by the solubility product expression when measurements are made in the presence of the insoluble complex. Thus a"bm = K,,
and substitution of b = (a-nK,,)l/m into (15) and (16) gives A and B as functions of a alone. Therefore B = f(A), and they cannot be varied independently. This limits the information which can be obtained, making i t impossible to distinguish between mononuclear and polynuclear complexes. Solubility measurements only furnish information on the charges of the ions in the solution and not on the composition of the complexes. For example, the solubility, S, of SnO in dilute acid has been found to be described by an equation of the type I n this equation, the constant K2 is the of sum all of the equilibrium consttnts for the formation of species Sn,(OH)2,2+2 with the charges of +2. The hydrolysis studies on the tin(I1) ion (17) indicate that there are three species in the solutions with charges +2, Sn+=, Sn2(OH)z+2, and Sn3(OH)4+2; therefore, the constant K2 determiued by solubility measurements is not the equilibrium constant for the reaction producing Sn+2 as was of necessity assumed in measure-
ments of this type. Solubility measurements are particularly of value when used with measurements of Type 1 to give the ionic species present in a system over a wide range of concentrations of the reagents in both saturated and unsaturated solutions. There are various other methods used for the study of complex formation such as bends in simple titration curves, polarographic and conductance curves, Job's method, etc. These were not discussed above, since they are qualitative in nature and difficult to interpret correctly. CRITERIA FOR THE STUDY OF COMPLEX-FORMING EQUILIBRIA (1)
The Constant Ionic Medium
A constant ionic medium is employed so that the activity coefficients of a certain ion or ions can be considered to remain constant in all of the solutions studied. I n general this is done by adding a large concentration of some cation like Li+, K+, or Na+ or some anion like C10,- or NO3-. These ions are supposed to be inert, i.e., not to form complexes with the ions being studied. If the activity factors are held constant, concentrations can be obtained directly from e.m.f. measurements, etc., and the law of mass action applied using concentrations. This elimination of activity factors is essential in the study of complicated equilibria, especially those in which the reaction products are not known a priori, since it reduces the number of variables to one half. The first use of the constnut ionic medium for the study of complex forming equilibria appears to have been Grossmann's study cf Hg+Z-SCN- complexes ($8). According to Grossmann, the idea originated with Rodlander sometime before his death in 1904. BMnsted (29, SO, 31, St) published several papers on studies of e.m.f. and simple ionic equilibria using an aqueous solution of various inert salts in place of water as a solvent. I n 1927 (52), he stated that the Nernst, equation, the solubility product, and the mass action law in their classical (concentration) form were applicable in these concentrated salt solutions, because the activity coefficients remained, practically constant. J. Bjerrum (55) used 2 M NH4N03as a solvent in his studies on the complexes between C U + ~and NHs, hut this was primarily to prevent precipitation, since activity coefficients were not a great problem with an uncharged ligand like NH,. Since the work of Leden in 1943 ( 5 ) , the constant ionic medium has been used frequently for the study of complex formation equilibria. I n most recent researches, C104- has been used to furnish the constant ionic medium; however, different. authors have often used different concentrations of perchlorate, which makes comparison of their data difficult. NaC104 has usually been added so as to make [C104-] = 2, 3, or 4 M. Some authors have also attempted to maintain the ionic strength constant (assuming no association) by the addition of NaCIO4. Since the work of Riedermann and Sillen (54) indicates that activity coefficients of cations are affected mainly by variations in the concentrations of anions and vice versa in these concentrated salt solutions, it appears best to maintain the [C104-] constant when measuring the concentrations of cations by e.m.f., etc. The JOURNAL OF CHEMICAL EDUCATION
concept of ionic strength has little significance in these concentrated solutions. To permit comparison of different experiments, the constant ionic medium with IC104-] = 3 M is to be preferred. An objection to the constant ionic medium has been raised since CIOn- may form complexes with certain cations, e.g., Fe+3 (36); however, it must be remembered that equilibrium measurements cannot distinguish between species with different amounts of solvent. For example, one cannot tell of the borate ion of - 1 charge is Bop-, or H2BOx-. In general some convenient formula is used which should, if possible, have structural justification. In NaC104 medium, the concentration of the dinuclear tin(I1) hydrolysis product which is written [Sn2(OH)z+2] actually means Z Z Z[Sn,(OH),(HpO),(Naf),(C104-),lBfu-sI+] = u s
It has also been said, cf. Young in "Annual Reviews of Physical Chemistry," 1952 ($6),that AF, AH, and AS values calculated from equilibrium measurements in a constant ionic medium are not "true thermodynamic values." The choice of the hypothetical 1 molal solution in a neutral salt medium, e.g., 3 M NaCIOa, instead of in the solvent water does not invalidate the laws of thermodynamics, and so these constants are as thermodynamically sound as those at infinite dilution. It is true, however, that these data cannot be combined in calculations with the bulk of available thermodynamic data where the standard state is defined in the solvent water. Although the constant ionic medium has proved to he very useful, it cannot be considered a panacea. Very little is really known about ionic activity coefficients in these concentrated salt solutions where the activity coefficients are defined as lim f, = lim a d c , = 1, c, -+ 0 in the concentrated salt solution. These difficultieswere pointed out by Leden in a study of the complexes between Cd+%and S04-2 (37). Values of the complexity constants were determined by measurement of [Cd+l] with amalgam electrodes while the C10,of the medium was replaced partially with S04c2. In this case the data obtained with the Cd amalgams could be fitted by assuming the stepwise reactions Cd+P
+
= CdS04(,,)
(24)
These reactions and the constants calculated gave agreement with data obtained from the measurement of [S04-2]only when IS04-2]
