family of least-square curves. T o compute the concentration of each sample, a curve from this family is chosen which gives equal or better precision than a conventional least-squares curve using all calibration data. The procedure is not useful (but not damaging) with small numbers of calibration standards. Its major application is for highly automated, high-throughput analyses, in which multiple standard measurements can be made at little extra cost. For that purpose, it is relatively simple and effective. More rigorous mathematical techniques, such as using a weighted least-squares approach to compensate for variable variance data or identifying more appropriate equations than first- and second-order linear models, require considerably more development. The multi-curve procedure has several important advantages. (a) Calibration curve calculations can be automated with less risk of gross error than in conventional procedures. (b) Data quality is improved, particularly for calibration over wide dynamic ranges and with nonlinear curves. Alternatively, with initially adequate data quality, the multicurve procedure allows use of wider dynamic ranges, often with increasing analytical efficiency.
(c) If data quality is inadequate for a particular application, confidence band statistics will suggest ways t o improve precision. (d) Confidence bands and minimum reportable concentrations are meaningful and should be reported to the data user-along with a warning that they do not account for less easily quantified errors such as systematic analytical error and unrepresentative sampling. (e) The procedure requires additional computation but no additional experimental work. As with all statistical techniques, the final result is only as good as the experimental data. The analyst should not allow the use of this sophisticated mathematical technique to induce overconfidence in the quality of the data.
LITERATURE CITED ( 1 ) N. R . Draper and H. Smith, “Applied Regression Analysis”, J . Wiiey and Sons, Inc., New York, N.Y., 1966. (2) M. G. Natrella, Section 17.3, “Experimental Statistics”, National Bureau of Standards Handbook 91, (1963). (3) R. G. Miller, “Simuhneous Statistical Inference”, McGraw-Hill, New York, N.Y., 1966.
RECEIVED for review November 18, 1976. Accepted July 13, 1977.
Solution of General Equilibrium Systems by the System Equation Jan Blaha National Textile Research Institute, Centre for Research and Application of Ionizing Radiation, 6647 1 Vev. 132~%ka,Czechoslovakia
The general qualitative and quantitative computational simulation for solution of equillbrium systems is described. The qualitative testing simulation set of equations consists of a quantitative simulation equations set extended by experimentally measurable criterion quantities. The quantitative simulation equations set consists of a set of experimentally nonmeasurable quantlties. A general system is not solvable by a mathematical simulation equation set directly. The solvability is conditioned by forming a sufficient number of partial simulation equations set transformable into a single algebraic equation of the nth degree. This transformability of the simdation equations set was formalized by the general system equation. This equatlon determlnes a general law of qualitative and quantitative relationships of indivldual subsystems of the general equilibrium system. The system equation was applied to a system of cyanide-heavy metals in aqueous solution.
The arrangement of systems based on system theory was first demonstrated in Biology ( I ) and later applied to other variable natural systems (2). Solutions of compounds involving complex ion equilibria have until now been studied from a quantitative point of view using a non-systems approach. The qualitative composition of the system (specification of the components formed in the system) was determined either by estimation (3) or by an iterative calculation ( 4 ) . The most probable estimate ( 3 )of the system composition was determined by the concentration 1660
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
of separated components and pH of the solution. The iterative calculation ( 4 ) presupposes systematic exhaustion of all possibilities of solution composition followed by completing the necessary calculations to find the single available combination which reproduces the actual qualitative composition. These methods ( 3 , 4 )are either of low accuracy ( 3 )or are at the expense of a fairly long and elaborate program which is difficult to apply ( 4 ) . The number of possible combinations of the qualitative composition of the system increases in a power series with an increasing number of complex forming system members. Therefore, it is not possible to feasibly apply these methods to more complicated equilibrium systems in a reasonable calculation time. The system scheme proposed here uses a simple method, from the point of view of formal mathematical formulation, enabling the calculation to be simplified. With increasing numbers of system members, the increase in the number of necessary calculations is additive. Thus the computation time for more complicated systems does not become burdensome. This study presents a general analysis of the principles of formulation of the system equations used in the solution of general equilibrium systems. The basic viewpoint in the classification of the systems is their separation into qualitative and quantitative characteristics. The analysis of preconditions enables the formulation of both partial qualitative methods of solution and a single quantitative one. The precondition for the formation of a solvable system is the ability to transform the system into a single algebraic equation of the nth degree. This is formalized as the general system equation which arranges the general equilibrium system according to
the experimentally measurable quantities describing it. An application of the system equation is shown for an example system involving the equilibrium of cyanide and heavy metals in aqueous solution. The detailed application of this calculation is apparent from the complete flow chart and examples of actual computer calculations. The flow chart is transferable to any programming language without additional modification. This proposed approach is applicable to more complicated equilibrium systems. Generally, it is applicable to inorganic and organic systems of complex ion equilibria involving metal central atoms and coordinated ligands; it is also applicable to systems of physicochemical or physical nature (phase equilibria, pressure and volume equilibria of real gases) or to other natural equilibrium systems. GENERAL THEORY While solving the equilibrium chemical system (the system cyanide-heavy metals in aqueous solution) from the nonexperimental (mathematical) point of view in order to determine the type and number of subsystems (components) that are formed in the systems after equilibrium is reached, general laws valid for every natural system which reaches equilibrium were found. It is necessary to describe the general system from two basic points of view: Qualitative and quantitative. The description of the quantitative properties of the system is simpler and presupposes knowledge of all of the subsystems (components) which are formed in the system. A general model of the system does not, however, assume any previous knowledge of either quantitative or qualitative properties. I t is necessary to know only the number of components that theoretically could exist in the system. In the general equilibrium system, the existence of individual subsystems (components) depends on the arrangement and concentration of the other participating (interacting) subsystems (components) and on the properties of the medium in which the system is contained. Without system analysis of the equilibrium state of the system, the existence of none of the subsystems can be presupposed. Experimental measurement of the presence of individual subsystems is not possible because of the complexity of the systems studied. T h e determination of the qualitative properties of the system necessitates finding suitable conditions for mathematical simulation and thus determination of the components formed in the system. The determination of the qualitative properties of the system by means of a mathematical simulation is called testing. The quantitative simulation set is composed of a number of equations with independent variables which correspond to the number of subsystems (components) formed in the system. The qualitative simulation set is composed of the quantitative simulation set to which conveniently chosen variables have been added (Le., criterion quantities in the quantitative simulation set are constants obtained by measurement). They are parameters obtained from the experimental measurements by which the system is specified. A chemical or physicochemical system is determined by the inlet subsystem (amounts of substances), by a generally known functional relationship (e.g., laws of chemical dissociation), and by properties of the system environment (medium) insofar as the medium properties relate directly to the arrangement of the given subsystems (components). A qualitative testing presupposes that criterion quantities are available in sufficient number. The sufficient number of criterion quantities is specific for each individual system. More than one equivalent testing simulation can be used. Every experimentally measurable quantity could be chosen as a criterion quantity. Either a single subsystem (a com-
plexing reagent) or all other subsystems (substances) entering the system simultaneously (e.g., central atoms) can be chosen if it interacts with a single subsystem only. The change of properties which determine the interactions of the medium and the system is specified by another independent criterion quantity. From the point of view of formal mathematical formulation and calculation of the properties of the system, the procedure depends on the limiting assumption that the set of equations describing the system can be transformed into a single algebraic equation of the nth degree. This seems to be a radically limiting assumption. However, if only one subsystem of the given system fulfills this condition, the condition must be fulfilled simultaneously by the n subsystems that are assumed t o exist. Although this assumption appears as a limitation, it is actually only a formally prescriptive (theoretical) approach to the methodology of equation solution. Specific real systems can be described by the choice of a very small set of equations (fundamental chemical or physical laws such as dissociation equations) from among the total number of conceivable equations. In this circumstance the use of this small set of equations is not a limitation. On the contrary, these real systems are solvable by system analysis by just such a small set of equations. I t is possible to assume that this requirement is a general requirement of system theory. The failure to obtain a solution in systems studied in this way may be due to incorrect formation or selection of basic equations describing the system. These failures are due primarily to the fact that the system of equations chosen cannot be transformed directly into a single algebraic equation. Another transformation method-an indirect one-can be presumed to exist. In spite of the anticipated knowledge of suitable basic equations, it may not be possible t o transform the systems studied into the single algebraic equation required. However, a method was found to express the qualitative simulation by means of a sufficient number of mathematical partial systems (equation systems) to permit solution. These are partial subsystems which are formally independent of each other. In this case, the partial subsystems do fulfill the transformation requirement. This method of solution is based on the possibility of using the criterion quantities which are variable in a qualitative mathematical simulation. In the system under study, we had to form a t least nine partial subsystems (one quantitative subsystem and eight qualitative ones). Apart from the quantitative subsystem there are also two qualitative testing subsystems describing the relationships between the subsystem medium (solvent) and the other subsystem components (metal ions), one general and one special subsystem, as well as four further independent subsystems describing mutual relationships between subsystem components (e.g., metal cyanide complex and metal cyanide precipitate) which arise from two fundamental types of subsystems (components) in the system being tested. Two of the four subsystems are formalized in a general and two in a special form. T o solve both of the special subsystems, it was necessary to form an auxiliary and a solution subsystem. The method of solution of the system by means of partial simulation subsystems (equation subsystems) can be demonstrated by an example consisting of the formation of the subsystems of both special qualitative tests. In the case of partial separation of a single component from the system under study (Le., considering the possibility of simultaneous precipitation of the component’s central atom with one common interacting system component (complexing reagent) and with the solvent medium), it was necessary to form an auxiliary partial simulation subsystem of equations ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
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transformable into a single algebraic equation (this equation expresses the amount of precipitated component with medium). This equation was used as one condition for formation of the ensuing solution subsystem of equations, so that two simulation subsystems of equations have been established. Both simulation equation subsystems (auxiliary and solution) cannot be combined directly. It was found, however, that a t least one variable takes different numerical values in each of the two simulation subsystem equations. In the auxiliary equation subsystem, this variable is used as an auxiliary variable, while in the solution equation subsystem as an independent variable, in spite of the fact that this variable determines the relationships of a single system component at a specific state of the system in both subsystems of equations (auxiliary and solution). The auxiliary equation subsystem expresses a quantitative description (calculation of the amount of separated component) and the solving equation subsystem expresses the qualitative testing of the system. Under these circumstances the latter equation subsystem can also be solved by means of a single algebraic equation. The correctness of the assertion that the ability to solve the assembled simulation equations set after its transformation into a single algebraic equation depends on the proper choice and arrangement of basic equations can be demonstrated by describing the assembly of a partial equation simulation set. Both selection and formation of basic conditions formalized as equations describe the system under study. When constructing the partial simulation subsystems of equations, the simplest physical and chemical laws are used (e.g., dissociation laws, solubility product equations, mass balance). These are generally known and may be expressed by simple mathematical relationships. In the testing simulations, after certain components (e.g., a metal) were eliminated from the system by precipitation (the component with medium but not with complexing reagent) the mathematical formalization became simplified, contrary to expectation. This simplification is a basic characteristic when the condition of transformability into a single algebraic equation is met (it holds for every analogous situation). The assertion above can be demonstrated by consideration of a simulated interaction between subsystem components (e.g., a metal and the solvent medium). The testing simulation set consists of the resulting conditions expressed as independent equations. If a complicated phenomenon such as the formation of a new type of component (precipitate formation) occurs, then one condition which was previously true is no longer valid and must be left out. The equation which would have to change owing to the separation of a component is, for example, the mass balance equation. It is thus necessary that the system not be determined by superfluous relationships but that it can be solved by a single algebraic equation. Each of these partial simulation equation systems was solved for a sufficiently representative number of independent variables and corresponding independent equations (amounting to a total of nearly sixty). Irrespective of the number and combination of the variables, the above systems with different mathematical terms were transformable into one algebraic equation. The form and degree of the resulting equation depended only on the type of the model of the solution which was selected. The justification for assuming the transformability of each real similar system (mathematical simulation equation set) into a single algebraic equation is supported by the fact that the various different partial simulation equations can be united in a single relationship. First, qualitative simulation systems were united into a single relationship; then quantitative and qualitative simulations were combined and formally expressed by a single equation. 1662
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The formalized equation is termed the general system equation. The form and the rules for use of the system equation both express the requirement of transformability of a general system into a single algebraic equation. Real equilibrium systems can be expected to be arranged in accordance with this requirement. Any real equilibrium system, irrespective of its number of subsystems, may be formalized by means of a suitable system equation which is thus transformable to permit its solution. The general system equations can be expressed as a system of equations (Ao), (Ad, ..., (A,):
t
=
(A,)
cx
Subsystem of Testing Equations (A,) - (A,) where
E
P
=
i i ? CX = i
i=l
~
cxi l
E = sum of all experimentally nonmeasurable quantities
(system components). ti = subsystem for ith kind of the system components for the nonmeasurable system quantities. C,y = sum of experimentally measureable quantities which were used us a criterion. n. = number of subsystems for ith kind of the system components. p = number of used criterion quantities. Equation (Ao) is basic and describes quantitative relationships between the system components. Equations (A,) - (A,) are specific and correspond to the qualitative tests. The method of system equation application to the solution of real systems is based on an omission rule. Both Equation (&) or Equations (Al)-(An)and subsystems or subsubsystems (subsystems members) comply with the omission rule. The form and method of system equation application is demonstrated by introducing a real chemical system to be solved.
SPECIAL THEORETICAL PART On solving the system: heavy metals-cyanide and univalent complexing reagents forming common complexes with cyanide, the system equation reads as follows:
t
=
cx
S,/y"tLt [OH]
= ff
Testing Equations for Complexing Reagents Forming Common Complexes with Cyanide and Heavy Metals (-43)-(Atl) In this case: n,p = 1 and a = C,/a2Lt,S,/y"lL, for cyanide complex and cyanide precipitate, respectively, Cx, = total analytic concentration of complexing reagent X I ,E, = sum of all chemical forms of complexing reagent Xiwith central atoms,
El
" I
=
Z Ai i=l
+
s2[OH]-'"iWj +
j=1
D
[OH]-' KCN
+ 1)y
for the cyanide test, for the hydroxyl test the member [OH] in is substituted by a according to Equation (A2), = C,al/u2 for metal-cyanide complexes, AI = m,(C, - Sl/yml)for cyanide precipitates of metals, W, = L,al,
n,, n2 = number of metals, r = number of ligands for one metal, i, j = not tested or tested metal, t = tested metal, y = concentration of the free univalent complexing component,
m = metal valency, C = metal concentration, L = solubility product of metal hydroxide, S = solubility product of metal-cyanide precipitate, K,,,, KcN = dissociation constants (metal-i-CN-complex and HCN, respectively), P = ionic product of water, [OH] = concentration of hydroxyl ions. In a special case, when metal-t occurs both in cyanide complex and hydroxide precipitate and metal-t-cyanide precipitation testing is done, it will be cy = St/y”‘L,. The test Equation (A,) replaced by the Equation (A2)and the member [OH] is substituted by the expression [OH] = a-ltrnfin Equation (Ao). In this case the replacement of the Equation (A,) and the member [OH], respectively, represents both the omission and simultaneous validity of the condition (metal-t-hydroxide precipitate), which corresponds to the testing of simultaneous existence of both the hydroxide and cyanide precipitate. However, if we consider simultaneous metal-1-hydroxide and -cyanide precipitation, then it is not possible to apply the system Equations (A,) - (A,) when the hydroxyl test for other metal-t is done. In this case more simple relations hold as follows; For metal-t in cyanide complex:
E
Kt,.[$.]‘’”‘
[OH]‘ -
r=l
C
Lt
[OHImt + 1 = 0
(B)
for metal-t in cyanide precipitate:
[2] p] =
l’mt
and Precipitation of
Metal-t-hydroxide at the Same pH as
Metal-1-hydroxide Precipitation (C) The application of the general system equation is based on the very simple omission rule. In the most general case, we use the general system equation in full, thus considering all theoretically possible chemical forms in the system. For specific application of the qualitative tests and the quantitative calculations, the omission rule is used. Basic Equation (A,) is used in both the qualitative and quantitative calculations. Specific equations corresponding to the tests are Equations (Al)-(An); e.g., Equation (Al) corresponds to the cyanide test, Equation (A*)to the hydroxyl test, etc. When testing is carried out, the equation corresponding to the test is used and other test equations are omitted. If testing is not being done, as when quantitative equations are being carried out, all test Equations (Al)-(A,) are left out. The omission rule is valid for all individual subsystems (composed of complexing reagents forming one kind of complex) and subsystems-subsystem members (metal or central atoms bound, e.g., in cyanide complex or cyanide precipitate) of which the basic Equation (A,,) consists. Individual central atoms form partial subsystems of metal complexes with a varying number of complexing ligands (a single metal can form one or, maximally, four cyanide complexes simultaneously with increasing number of ligands). When the whole subsystem of one complexing reagent (considering a system with more complexing reagents) or a subsystem (a member of a subsystem, a chemically bound metal) does not occur in the system or when a partial complex of the given metal does not exist (theoretical ligands can number from one to six), the corresponding members or groups of members will be left out of the basic Equation (Ao).
PRACTICAL APPLICATION Application of the proposed theory is simple. Analytical concentrations of total cyanide and individual metals with the p H are the only necessary input data. The computer input data can be fed through a direct interface. The analyzer suggested for this purpose supplies the computer with the input data automatically after having performed the analysis (5). When this analyzer is not available, the computer input data may be obtained by carrying out the analysis of a solution of interest according to the following analytical scheme. After analytical separation of the total cyanide from the solution of heavy metals in cyanide (e.g. by means of distillation from strongly acidified solution) the individual heavy metals and the total cyanide are determined by means of some proper analytical method (e.g., the former by polarography and latter by the dimedone method (6)). The distillation of Co3+cyanide complexes in sulfuric acid proceeds according to the equation (13): ~ C O ( C N ) ,+ ~ -22” + 13H,O = 2 C 0 2 ++ 12NH,+ + l l C 0 + CO,
From the data obtained and the known p H of the solution, the input data for the computer calculations are prepared. The calculation results form the qualitative and quantitative analysis of all solution components formed, including hydroxide and cyanide precipitates. The computer outputs the results indicating both the presence and amounts (concentrations) of individual system components in molar concentrations. In addition to the concentration input data, it is necessary to enter all required constants which correspond to the system studied (cyanide-heavy metals (7-22)) into the computer. Supplementary Material Available: Both the overall flow charts enabling the practical computer calculation and six typical examples of solution of the “cyanide-heavy metals” system that were used (6 pages). Microfiche of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from Business Operations, Books and Journals Division, American Chemical Society, 1155 16th Street, N.W., Washington, D.C. 20036. Remit check or money order for $2.50 for microfiche, referring t o code number ANAL-77-OCT.
LITERATURE CITED M. D.MesaroviE, “Systems Theory and Biology”, Springer Verlag, New York, N.Y., 1966;, M. D.MesaroviE, Views on General Systems Theory”, Proceedings of the 2nd systems Symposium, Case Institute of Technology, 1963, IV, Wiley, New York, N.Y., 1964. J. Blaha, Vom. Wasser, 34, 175 (1967). N. Ingri, W. Kakolowicz, L. G. SiilBn, and 6. Warnqvist, Talanta, 14, 1261 (1967). J. Biaha, Czechoslovak Patent 164153 i(197.5). V. Kratochvil, Collect. Czech. Chem. Cbmmun., 2 5 , 259 (1960). L. G. SillBn and A. E. Martell, “Stability Constants of Metal Complexes”, Chemical Society, London, New York, 1964. I. P. Alimarin and N. N. Uschakova, “Tables on Analvticai Chemistrv”. Moscow, 1960, p 24. K. B. Jazimirski and W. P. Wassiijew, “Instabilitatskonstanten von KomDiexverbindunaen”. Berlin. 1963. o 74. Gmeiins Handbuch-der an&g. Chemie, Co(A), 247, Fe, Verlag Chemie, Berlin, 1932. K. Masaki:&J. Chem. SOC. Jpn., 6, 146 (1931). H. Remy. Anorganicki Chemie”. Vol. 2, SNTL. Prague, 1962, p 465. R. Leschber and H. Schlichting, Fresenius 2.Anal. Chem., 245, 300 (1969).
RECEILXD for review January 5, 1976. Accepted June 27, 1977.
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