816
Karoly Vadasdi
tions and atomic populations for CuII-OH- and C U ~ ~ - O ~ -(2) (a) A. S. Hay, H. S. Blanchard, G. F. Endres, and J. M. Eustance, J. Amer. Chem. SOC., 81, 6335 (1959); (b) H. S. Blanchard, H. L. are similar to each other. Finkbeiner, and G. A. Russell, J. Polym. Sci., 58, 469 (1962). In conclusion, we can relate the experimental and cal(3) T. Yonezawa, S. Tsuruya, and T. Kawamura, J. Polym. Sci., B6, 447 (1968); T. Yonezawa, S. Tsuruya, S. Tsuchiya, and T. Kawamculated results on the catalytic activities of the oxidation ura, Kogyo Kagaku Zasshi, 71,1007 (1968). reaction of phenol derivatives as follows: the weaker the (4) H. Finkbeiner, A. S. Hay, G. F. Blanchard, and G. F. Endres, J. copper-ligand bond, the more facile is the substitution of Org. Chem., 31, 549 (1966). (5) E. C. Stathis, Chem. lnd. (London), 633 (1958). phenolate anion as ligand, though more factors inherent in (6) A. S. Hay, Japanese Patent S 39-29373 (1963); S.Tsuruya, T. Shithe catalytic function of the present copper complexes, rai, T. Kawamura, and T. Yonezawa, Makromol. Chem., 132, 57 (1970). must, of course, be present.
Acknowledgment. S . T. wishes to give special thanks to Dr. K. Kawamura for his helpful discussion and also to Dr. H. Konishi for the computer program used for the MO calculations. The calculations were carried out on a FACOM 230-60 computer at the computation center of Kyoto University. References and Notes (1) Department of Chemical Engineering, Faculty of Engineering, Kobe University, Nada, Kobe, Japan.
(7) A. V. Heuvelen and L. Goldstein, J. Phys. Chem., 72, 481 (1968). (8) H. Yokoi andT. Isobe, Bull. Chem. SOC.Jap., 41, 2835 (1968). (9) R. D. Gillard, J. lnorg. Nucl. Chem., 26, 1455 (1964). (IO) H. Beinert, “The Biochemistry of Copper,” Academic Press, New York, N. Y., 1966, p 213. (11) B. Bleaney and K. D. Bower, Proc. Roy. Soc., Ser. A, 214, 451 (1952). (12) E. Ochiai and H. Hirai, Kogyo Kagaku Zasshi, 72, 1785 (1969). (13) A. H. Maki and B. R . McGarvey, J. Chem. Phys, 29, 31 (1958). (14) D. Kivelson and R. Neiman, J. Chem. Phys., 35, 149 (1961). (15) D. Wolfsberg and L. helmholz, J. Chem. Phys., 23, 853 (1955). (16) R. Hoffman, J. Chem. Phys., 39, 1397 (1963). (17) J. C. Slater, Phys. Rev., 36, 57 (1930). (18) E. Clementi and D. L. Raimondi, J. Chem. Phys., 38, 2686 (1963).
On Determining the Composition of Species Present in a System from Potentiometric Data Karoly Vadasdi Research Institute for Technical Physics of the Hungarian Academy of Sciences, Budapest, Ujpest 1, Pf. 76, Hungary (Received August 17, 1973)
The mass balances of chemical systems and derivatives or integrals of them have been written as matrix equations in the case of suitably chosen mixtures. The appropriate combination of these equations may be put into Jordan’s normal form giving the stoichiometric coefficients as matrix eigenvalues. On the basis of the derived equations a computation method has been developed for determination of the number and composition of species in two-component chemical systems using emf data. The application of the method requires the knowledge of the free concentration of each component. Selecting matrix elements by interpolation the computation basically consists of a “pseudo-inverse” computation and finally an eigenvalue determination.
Introduction A number of different methods have been proposed for the determination of the compositions of species formed in equilibrium mixtures.1-3 A common feature of all these methods, which have found wide practical application, is that they can be applied only under restricted conditions. The usual conditions are, e.g., only a single species present in the system, or in the chosen experimental range; the system follows some special mechanism, etc.4-7 Those systems which do not satisfy the necessary conditions are treated by “least-square” computations, testing various “guessed” mechanisms, and the best “least-square fit” corresponds to the “most probable” mechanism.8.9 During the past 10 years several authors1°-13 have published methods for the determination of the number of species present in a system from optical data, applying matrix rank analysis. The use of these methods facilitates The Journal of Physical Chemistry, Vol. 78, No. 8, 1974
the application of different “least-square” programs to the systems, giving an upper limit for the number of possible combinations. No similar method is known in the literature for the treatment of data from other types of measurements, e.g., potentiometric or extraction. In this paper a general computation method is given that in principle allows the determination of the number and composition of the species present in an equilibrium system, using potentiometric data. It goes without saying that in a very complicated system, consisting of several components and several species, experimental errors may frustrate the aim. Discussion of the circumstances leading to such a shortcoming, however, does not constitute the subject of this paper. Potentiometric D a t a For the sake of simplicity, let us consider a system con-
a i7
Potentiometric Determinations of Composition TABLE I: Some Fitting Parameters in the Pb-OH System. The Range Numbers Correspond to Figure 2 Range
No. of data points
P
4
1 5 13 20 29 35 38 39 41
14 13 14 9 9 7 8 8 9
3.4075 3.9514 3 .5930 3 .6102 3.1929 3.7533 4 .7334 3.0503 4.7860
3 .5538 4.0282 3 3912 4.1718 4.1482 4.9558 6 .0667 4.1191 6.3439
-
Log
Residual error square sum
s
8.40 x 1 . 2 7 x 10-4 2 . 3 2 x 10-4 8.72 X 1 . 0 2 x 10-4 8 . 4 4 x 10-5 3 . 7 7 x 10-4 2.35 X 2 . 1 5 x 10-4
- 17.5332 - 19 .5274 - 19.4924 - 21 .5071 - 22.8734 - 26.8092 - 31.8827 - 23 .5015 - 34.0204
sisting of two components14 A and B, in which a series of species are formed with the compositions ApkBqk. Pk and qk are the stoichiometric coefficients of the components in the kth species and are therefore usually small positive integers. In the case of potentiometric measurements one measures the equilibrium concentration of one or both components in different mixtures of A and B. If the activity coefficients of all various species are taken as unity, in every mixture the mass balance condition for each component may be written
A
= a
+
N
xPhPhaPhbqh
(1)
k=l
'f
N
B
=
b
+ xqkPkaPhbqh
(2)
k= 1
Here A and B denote the total concentrations, while a and b are the equilibrium concentrations of the two components; N is the number of the species and denotes the stability constant of the Kth species, i.e. Ph
AP,B,, = ,ph.b9h
If one measures only the equilibrium concentration of one component ( e . g . , b ) , the equilibrium concentration of the other Component can be computed by the Hedstr6m-McKay equation without a knowledge of the actual mechanism.15 Let us suppose that the equilibrium concentrations a and b are determined in a series of different mixtures, in which the total concentrations A and B are varied Figure 1. Prescribing a set of equilibrium concentrations: al, a2, . . ., a,, and bl, bz, . . ., b,, one can determine possibly by means of interpolation the total concentrations A and B of the mixtures relating to the equilibrium concentrations a, and b,. The total concentrations A and B of the mixtures at the prescribed concentrations a, and b, will be denoted by A,, and B,,. For these selected mixtures the mass balance conditions for the two components are h'
N
H,,
B,, - b,
= c qh b k a L P h b J 9 h
(24
h -1
G,, is the difference of the total and equilibrium concentrations of component A at the ith a and the j t h b concentration. Similarly H,, is the difference of the total and equilibrium concentrations of component B in a mixture corresponding to the ith a and j t h b concentration. By in-
*2
'i
a
Figure 1. The principle of the selection of matrix elements. troducing the matrices
Ri,
6hsPsPh;s h s
-
E
6hsqsPh;ElkE aiPh; F,,
E
b,'.
where &s is the Kronecker symbol, we can write ( l a ) and (2a) as matrix equations
ERF
G
=
H
= ESF
(3)
(4) It is clear that number of selected a's and b's should be greater than the suspected number of species, i.e., m > N, n > N . Since the rank of a matrix product is always equal to the rank of its factor of smallest rank,17 the ranks of matrices G and H are equal to N provided that m I N; n I N , For the determination of the ranks of G and H we refer, e.g., to ref 13, where procedures are given for finding the rank of a matrix composed from experimental data. Now let us focus our attention on the problem of stoichiometric coefficients. If the matrices G and H are square and can be inverted,l* then from the matrix eq 3 and 4
CH-1
=
RS-'
=
ERS-IE-1
(5)
where
JhsPs 1 4 s GH-l can be put into diagonal form, and its eigenvalues must be equal to the elements of the diagonal matrix RS-l, and therefore the ratios of stoichiometric coefficients for each species are determined. Similar equations may be written for the derivatives or integrals of material balances, e.g.
The Journal of Physical Chemistry, Vol. 78, No. 8, 1974
Karoly Vadasdi
818
-log
I5
I #O
300
2,o
Schematic representation of the Pb-OH system on the basis of Olin's data as -log b = /(log a ) B where b 3 [Pb2+]and a [H+]. The analytical Pb concentrations 4,B2.. ., 6 7 range from 6, = 1.25 mM to 6 7 =- 80.0 mM. The dashed lines are the borders of each concentration range where the following function has been fitted: 21 = ppbrPa,-q/(b, i@b,Dal-P). -0 values were used in the exponent of hydrogen because the reaction had been defined as: pPb2+ pHOH e Pbp(OH),(*P - q ) + + q H + . The full lines represent the areas from which the matrix elements were taken. Figure 2.
+
TABLE 11: p , q, and q / p Values Computed from Matrices Indicated by Rectangles (Full Line) in Figure 2. Due t o t h e Factorization All Other Eigenvalues Were Less T h a n Range
Size
no.
of matrix
Rank
q/p
P
4
I
5 x 6
1
I1 I11
6 X 4
1
IV
3 X 3
3
1.00 1.02 0.96 1.02 1.31 1.27 1.27 1.28 1.35
3.95 3.95 0.77 3.99 6.21 3.12 3.60 6.38 3.05
3.95 4.04 0.59 4.05 7.86 4.09 4.59 8.07 4.12
~
~~~
V
value determination. In practical systems only a few species exist, usually far fewer than the number of rows or columns of matrices which may be built up from the data. We must therefore deal with singular matrices. The ranks of these matrices, i.e., the number of species, can be determined13 and then matrices of suitable size may be constructed. However, this method seems rather questionable because of the experimental error. We seek a method, therefore, which is a "best" solution in the "least-squares" sense. A solution may be obtained using the pseudo-inverses of the matrices. The condition of the "leastsquares" solution of a matrix equation
G
=
XH
(9)
where From these quantities matrices can be built up giving in appropriate combinations the p k and q k values separately. For example, eq 6 can be denoted by
c where Q k s
&&s.
=
ERQF
is that the euclidian norm of the residual matrix be a minimum
IIG - XHlI
(8)
In eq 9 the matrix H is rn x n, with rank N, where N < ( v ~ , n ) . The ~ O matrix H may be factorized in the form
The eigenvalues of the left side of (8) give the q k values. Inversion of Matrices The application of the derived equations in principle requires a matrix inversion, a multiplication, and an eigenThe Journal of Physical Chemistry, Vol. 78, No. 8, 1974
m-'E-1
(7)
Combining with eq 3
CG-' = EQE-'
X
H = UV where U and V are rn x N and N x n matrices, respectively, with ranks N . flu and V v are symmetric, N x N nonsingular matrices. The factorization of H is not unique, but all factorization may be written as
Potentiometric Determinations of Composition
81Q
TABLE 111: Comparison of Some Models by Equilibrium Constant Computation
Ionic species
Log 3, found by Olin
Our computation on Olin’s data (Run I, 3 M NaC101)
-7.9 - 19.25 -22.87 -42.14
-7.81 -19.27 - 22.. 90 -42.13
P b (OH) Pbc(OH)4 Pba (OW4 Pbs(OH)s Pbz(OW Pbz (OH),
... ...
Number of data points Residual error
-7.84 -19.26 -22.92 -42.12
...
.,.
...
79
procedure
(Run I
(Run I) 3.76
x
10-3
8.61
x
< 0.5; E
< 10.0mM
...
...
-19.29 -22.52 -50.02 -5.32
-19.32 -23.55 -41.65
...
*..
...
173
By graphical
Using the data z
+ 11) 10-4
79
(Run I 8.69
x
-11.67
+ 11) 10-3
79
(Run I 2.61
x
+ 11) 10-3
square sum
H
=
ments of the experimental concentration space. -
(UY)(Y-’V)
where Y is a nonsingular N x N matrix.lg According to a theorem of Peters and Wilkinson, a unique “leastsquares” solution may be obtained by any UV factorization of matrix H. Referring the reader to the original paper, we cite only the result.21
x = CV(VV)-1(ZrU,-G
(10) The matrix v(Vv)-l(UU)-lU is called a “pseudo inverse” of matrix H. In this way we can compute the inverses of matrices which are necessary for our equations. Computation of Matrix Elements The matrix elements cannot be obtained directly from experimental data under the usual experimental circumstances. They can be computed by numerical interpolation or by some tedious graphical procedure. The emf experimental data are often presented as z = f(1og U)B or 17 = f(1og U)B, where z = ( A - u ) / B is the average coordination number and 17 = log ( B / b ) .A number of methods for the computation of equilibrium constants are based on the fitting of calculated z values (zcalcd) to the experimental data (zexptl) with a given set of compositionspk and qk, i,e., finding the minima of the function z(Z;exptl
- Z;a1cd(Ph)}*
(11)
L
or
k
In the present case we have no idea as to the compositions of the species. In principle, function l l a might be minimized in the form
and the best P k and q k values might be obtained without any further computation if the number of species can be guessed. Now the number of parameters is 3 times as many as in eq 11 and taking into account the slowness of nonlinear minimizing methods and the quantity of data involved, the computations would involve an enormous amount of time, even with high-speed digital computers, when N > 2. Accordingly, instead of fitting (12) on the full experimental range we tried to fit (13) locally, on small seg-
?{?
}
- b, ppalPbP + za,Fb,g
(13)
Equation 13 requires fitting with respect to three parameters. This procedure is fairly fast using the Simplex method of Nelder and Mead.22 The matrix elements are then calculated withi the,particularrp, 9, land16 values relating to the segments containing at and b,. Application and Conclusions The full computational procedure has been tested on the Pb-OH system using the data of Olin.23 Two Fortran programs were written. The first computes the matrix elements at prefixed points of the concentration space. The becond computes the stoichiometric coefficients after finding the ranks of the matrices by the method of Hugus and Awady. As can be seen in Figure 2, the original data were divided into 41 segments. Each range contained 5-20 experimental points and function 13 was fitted to each of them. The residual error square sum as a measure of the accuracy of fitting varied from to 5 x Some fitting parameters are listed in Table I. By graphical analysis, Olin found four ionic species with the following compositions: PbOH, Pb4( O H h , Pb3(OH)p, and Pbe(0H)s. Other authors mention the existence of further species such as PbZOH, PbZ(OH)z, and Pbg( 0 H ) 1 ~ . ’ ~ As can be seen in Table 11, our calculations led to the identification of the compositions Pb4(0H)4, Pb3(0H)4, and Pb6(OH)g and approximately’ PbOH. The reason for the relatively poor identification of PbOH is that it occurs in small concentrations in the investigated experimental range; too few experimental points can be found here, and their distribution is uneven as they are heavily weighted in the direction of the H+ axis, which decreases the accuracy of fitting. The existence of PbOH‘ has been proved by calculating the equilibrium constants of the species by a minimizing method similar to L e t a g r ~ p .These ~ computations are summarized in Table 111. The OR values are in good agreement with those found by Olin. If the residual error square sum is investigated separately in the range where PbOH, PbZOH, and PbZ(0H)z may be expected to predominate, the choice of PbOH gives the best agreement with these data.
Acknowledgment. The author is indebted to Dr. F. Beleznay, Dr. I. Gafil, and Dr. T. Geszti for many helpful disThe Journal of Physical Chemistry, Vol. 78, No. 8, 1974
a20
E. N. Sloth, M. H. Studier, and P.G. Wahlbeck
cussions and to Dr. L. Bartha for his interest in the course of this work. Thanks are due to Dr. D. Durham for revising the grammar.
References and Notes (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
F. J. C. Rossotti-H. Rossotti, "The Determination of Stability Constants," McGraw-Hill, New York, N. y., 1961. H. L. Schlafer, "Komplexbildung in Losung," Springer, Berlin, 1961, M. T. Beck, "Chemistry of Complex Equilibria," Van Nostrand-Reinhold, Princeton, N. J., 1970. L. G. Sillen, Acta Chem. Scand., 8, 299, 318 (1954). 8. S . Jensen, Acta Chem. Scand., 15,487 (1961). J. Grandeboeuf and P. Souchay, J . Chim. Phys., 56, 358 (1959). J. Lefebvre, J. Chim. fhys., 54, 553 (1957). J. Rydberg and J. C. Sullivan, Acta Chem. Scand., 13, 186 (1959). N. ingri and L. G. Sillen, Acta Chem. Scand., 16, 173 (1960). G. Weber, Nature (London), 190, 27 (1961). R. M. Wallace, J . Phys. Chem., 64,899 (1960).
S . Ainsworth,J. Phys. Chem., 65, 1908 (1961). 2 . 2 . Hugusand A. A. El-Awady, J. Phys. Chem., 75, 2954 (1971). This treatment may be generalized for a multicomponent system. B. Hedstrtim, Acta Chem. Scand., 9, 613 (1955). H. A . C. McKay, Trans. FaradaySoc., 49, 237 (1953). R. Bellman, "Introduction to Matrix Analysis," McGraw-Hill, New York, N. Y., 1960. (18) This condition is not necessary for the numerical calculations, as can be seen later. (19) B. Higman, "Applied Group-Theoretic and Matrix Methods." Clarendon Press, Oxford, 1955. (20) If N = (m, n ) , whichever is smaller, i.e. AH or HA is nonsinguiar, then the least-squares solution is given by the normal equation X =
(12) (13) (14) (15) (16) (17)
(AH)-' kZ5 (21) G. Petersand J. H. Wilkinson, Comput. J., 13, 309 (1970). (22) D. M. Himmeibiau, "Applied Nonlinear Programming." McGraw-Hili, New York, N. Y., 1972. (23) A. Olin, Acta Chem. Scand., 14, 130 (1960). (24) L. G. Sillel, and A. E. Martell, "Stability Constants of Metal-ion Complexes, The Chemical Society, London, 1964, 1971,
Effect of Anionic Constituents on the Surface Ionization of Lithium SaltslaYb Eric N. Sloth, Martin H. Studier, Chemistry Division, Argonne National Laboratory, Argonne, lllinois 60439
and P. G. Wahlbeck*
lC
Department of Chemistry, lllinois lnstitute of Technology, Chicago, Illinofs60676 (Received November 9, 1973)
Surface ionization using a double filament technique was studied using lithium and lithium compounds to determine the effects caused by anionic constituents. The validity of the Saha-Langmuir theory of surface ionization was demonstrated for the lithium-rhenium system over a temperature range 12002600°K. The effective surface-ionization work function obtained for rhenium was 5.21 f 0.01 eV, compared to the measured thermionic-emission work function of 4.98 f 0.03 eV. A t relatively high temperatures, the temperature dependence of the lithium surface-ionization current from all molecules studied was identical with that obtained from lithium atoms. Incomplete dissociation of lithium chloride and lithium bromide can account for the lithium ionization threshold temperatures being well above that for surface ionization of lithium atoms. Dissociation energies of Do"[LiCl(g)] = 4.8 f 0.1 eV and Do"[LiBr(g)] = 4.3 f 0.1 eV were obtained. Enhanced emission of lithium ions was observed in the temperature range where incomplete dissociation of lithium iodide was expected. Lithium sulfate decomposed to give beams of dilithium oxide and lithium atoms. At each temperature, the lithium surface-ionization current was the sum of the individual ion currents from lithium atoms and dilithium oxide. These results are discussed with regard to the overall mechanism of chemical and physical processes.
Introduction Experiments and measurements of ion emission from heated metal surfaces were performed by Richardson2 in 1903. A systematic study of surface ionization was performed by Langmuir and Kingdon3a and by Taylor and L a n g m ~ i r .When ~ ~ alkali atoms impinge on a tungsten surface, they found that the collisions resulted in the adsorption of the alkali atoms. Upon desorption alkali ions were formed. Taylor4 first applied surface ionization to the detection of a molecular beam. A review of atomic, molecular, and ionic impact phenomena on surfaces has been provided by Massey and B ~ r h o p .Reviews ~ of data The Journal of Physical Chemistry, Voi. 78, No. 8, 1974
have also been provided by Kaminsky,6 Hasted,' and Ehrlich .S Surface ionization has been used extensively in mass spectrometry for the determination of isotope abundances with solid samples. From a study of uranium by Studier, Sloth, and Moore,g the chemistry of the species on the surface ionization filament was shown to be important in determining the extent of ionization. They showed that the oxidizing or reducing conditions present on the filament influenced the production of metal ions, metal oxide ions, or metal carbide ions. Other chemical features which could be important to surface ionization are the nature of
