Data for CLINP 2.1 program

Model of equilibria

Calculations: flow and results

 

The input information falls into two groups: obligatory information required in all cases and optional information depending on peculiarities of an examined system and/or experimental method.

OBLIGATORY INFORMATION

OPTIONAL INFORMATION

Input information is arranged according to templates in separate sheets of a Microsoft Excel file (extension .xls).

Sheets for obligatory information: “Parameters”, “ Stoichiometry”, “Logarithms of constants”, “Transformation”, “Measured properties”, “Total concentrations”, “Weights”, “Intensity factors”.

Sheets for optional information: “Phase pointers” and “Volume ratios”.

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MODEL OF EQUILIBRIA. MEASURED PROPERTIES. DATA PREPARATION

Definitions

Format for chemical reactions

Meaningful model

Parameters to be found

Measured properties

Experimental methods

Material balance

Statistical weights and model of experimental errors

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SHEETS OF MICROSOFT EXCEL FILE, FLOW AND RESULTS OF CALCULATIONS

Arrangement of information in Microsoft Excel sheets

Computation Flow

Results of Computations

Cross-Validation

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Definitions

N – the number of studied mixtures;

L – the number of analytical positions (for instance, a number of wavelength in spectrophotometry);

W – the number of reagents with unknown intensity factors;

Li – i-th species;

bi – stability constant of species Li;

LKu – u-th parameters to be determined;

ali – intensity factor of species Li for the l-th analytical position.

Format for chemical reactions

Chemical reactions are represented in the usual way (see Examples):

where Bj is the subset of independent components. Their number Y is determined as Y=S-r, where S is the total number of reagents and r is the number of reactions between reagents. None of independent components may be transformed into another one in the course of reactions. Trivial reactions in which an independent component is obtained from itself must be included into the set of reaction with equilibrium constants 1 (logarithm of equilibrium constants 0). Stoichiometric coefficients nij may be both integer and fractional (positive, negative or zero). They form a S´Y stoichiometric matrix ||nij||.

Reactions leading to substances for which intensity factors are to be determined must occupy upper W rows of the stoichiometric matrix.

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Meaningful model

Activity coefficients of reagents are assumed to be constant in examined systems (e.g. due to the excess of a background electrolyte). This makes it possible to use concentrational equilibrium constants. In these terms the mass action law (MAL) reads

where square brackets denote equilibrium concentrations, b i is the concentrational stability constant for species Li. If an experimental activity is known for a component Bk, corresponding equilibrium concentration [Bk] is replaced by the estimation of activity a(Bk); if for a species Li stoichiometric coefficients , its stability constant is mixed rather than concentrational.

In the chemistry if complexing silicas this model is termed ”the model of chemical reactions

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Parameters to be found

  1. logarithms p of unknown equilibrium constants;
  2. intensity factors W of reagents (e.g. molar absorptivities in spectrophotometry) for L analytical positions (e.g. wavelength in spectrophotometry).

Total number of parameters to be found z = p + L ´ W.

For complexes (species) Li logarithms of stability constants to be estimated are considered as functions of parameters LKu to be computed:

Here lg bi0 is the invariable contribution into lg bi;

LKu are parameters to be found (their number must be equal to the number of unknown stability constants for Li),

tiu are elements of a p´ p matrix T transforming parameters LKu into lg bi.

The way in which information about parameters to be estimated is organized enables flexible modification of their list. In particular, one may easily pass from total to stepwise stability constants.

See sheets Logarithms of Constants and Transformation.

See Examples

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Measured properties

Any property A of an equilibrium system that is sensitive to the variation of its initial composition is suitable. It is measured for N different initial compositions, the number of analytical positions being L  ³  1 (an example of the analytical positions is the wavelength in spectrophotometry):

Alk = f(ll, n*k),

where k is the number of equilibrium mixture (1 £  k £  N), l is the analytical position, l is its number (1 £  l £  L), n*k is the vector of initial quantities of reagents mixed to give the k-th examined mixture.

These Alk form a L ´ N matrix of measured properties.

Measured properties must be representable as a linear combination of equilibrium concentrations of reagents:

where Li are reagents, S is their number, [Li]k is the equilibrium constant of Li in the k-th mixture, ali is the intensity factor of reagent Li, either known or to be determined for the analytical position ll.

Intensity factors form a L ´ S matrix E (sheet Intensity Factors).

See Examples

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Experimental methods

The program is adapted for treating data obtained by spectrophotometry, solubility method, distribution method, potentiometry, etc.

If solubility method is used (see Example 3), elements of matrix A are values obtained by measuring the solubility of a precipitate.

If spectrophotometry is used (see Example 1) matrix A consists of absorbance values divided by the length of absorption layer.

Distribution method (see Example 4) consists in examining the distribution of a component X between two phases. Phase I is the solution of substances in a solvent I (its volume is VI); phase II may be either solution of substances in a solvent II (volume VII) or sorbent (mass mII). VI and VII or mII need to be known exactly at each point of the "composition-property" dependence.

Also, at least two quantities of the following list need to be known experimentally: n0(X); RD; t[X]; t{X}, where n0(X) is the total content of X in the system (moles); RD is the distribution coefficient of X between phases

RD = t{X} / t[X],

is the total concentration of X-containing reagents in phase II (mole per gram of phase II in the case of sorption equilibria; mole per liter of phase II in the case of extraction);

is the total concentration of X containing components in phase I (mole/l); equilibrium concentrations in phase I are given in square brackets, in phase II – in braces.

If two of the above quantities are known, two more may be easily calculated. If, for example, n0(X) and RD are known as a result of sorption experiments, then

t[X] = n0(X) / (VI + RD× mII),

t{X} = n0(X) ´ RD / (VI + RD× mII).

In simulating equilibria either t[X] or t{X} are chosen to be described by the model. In the first case

À = t[X],

and this quantity is used directly by CLINP 2.1. In the second case A is determined as follows:

 

When potentiometry is used (see Example 2) the result is logarithms of either activities or equilibrium concentrations of a reagent X depending on the way in which potentiometric chain was calibrated. If the result is lg [X], then 10lg [X] are specified in the CLINP 2.1 program as values of A.

If, alternatively, activities a(X) have been estimated experimentally (e.g. by measuring pH: a(H+) = 10-pH) then equilibrium concentrations [X] = a(X) / gX need to be calculated using activity coefficients gX. They are used as input data for CLINP 2.1.

This calculation is absolutely necessary when pH is measured in the ranges ðÍ < 3.5 or pH > 10.5. In these cases contributions of [H+] or [OH-] into total concentrations t(H+) cannot be neglected (in examining equilibria in solutions these contributions are of the order of 0.001–0.1). Required activity coefficients may be calculated at 298 K using the formula

,

where I is the ionic strength of a solution, coefficients a and b may be found for variety of background electrolytes in [Komar' N. P. Chemical Metrology. Homogeneous Ionic Equilibria (in Russian). Kharkov: Higher School, 1983].

The table below provides some values for calculating activity coefficients of hydrogen ions in solutions of several salts.

Electrolyte

a

b

NaCl

4.20

0.22

KCl

6.10

0.113

KNO3

8.02

0.092

NaClO4

9.30

0.16

By passing from activities to equilibrium concentrations the calculation of concentrational equilibrium constants by CLINP 2.1 is ensured, otherwise constants of equilibria in which H+ ions participate will be mixed.

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Material balance

Invariants of a system are the contents of independent components. If reactions proceed in solutions the volume of which is not changed then the total (analytical) concentrations of components are also invariant:

,

where c(Li) are initial concentrations of Li. This system of equations determines the material balance (see sheet Total Concentrations)

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Statistical weights and the model of experimental errors

Proceeding from the information about experimental errors statistical weights wlk are assigned by the user.

It is reasonable to assume that all experimental quantities except property A are measured exactly.

If all Alk are measured with the same error s (this model is often used for spectrophotometric data) then the same weights

wlk = 1 / s2

are assigned to all measurements.

If all measurements Alk are characterized by the same relative error sr, the following weights are recommended

These ways of assigning weights, as well as other possible ways (a number of which may be found, for example, in [Hartley F., Berges K., Olkock R. Solution Equilibria, 1980]) are easy to realize using means of Microsoft Excel.

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SHEETS OF MICROSOFT EXCEL FILE, FLOW AND RESULTS OF CALCULATIONS

Sheets:

Parameters

Stoichiometry

Logarithms of Constants

Transformation

Meàsured Properties

Total Concentrations

Weights

Intensity Factors

Volume Ratios and Phase Pointers

 

Sheet “Parameters”

This sheet shows default values of parameters that control computations and enables users to change these values. See detailed description of parameters.

Default value

Text in Microsoft Excel sheet

Comments

100

The maximum number of iterations

Maximum number of iterations when LKu is determined at fixed s

0

Per cent of outliers

Per cent of outliers 0 £  d  £  99

0 – least squares method,

99 – least moduli method.

0

Method of calculation

0 – quasi-Newton,

1 – Gauss-Newton)

5,00E-06

Minimum relative value of gradient

The main criterion according to which computations are interrupted

Recommended: (1-10)*10-6

1,00E-06

Minimum step

Recommended 10-6

1

Maximum step

Recommended 0.5–1

10

The typical value of parameters to be found

Recommended 1 – 10.

25

The typical value of criterion function

Recommended 1 – 200.

1,00E-06

Minimum relative step

Criterion to stop computations.

Recommended (1-10)* 10-6.

0,0001

The parameter of Armijo condition

Adjust the step l for linear search.

Recommended 10-4.

0/1

Cross-validation

1 – cross-validation is required,

0 – cross-validation is not required

0/1

Transposition

0 – matrices of measured properties, of weights and of intensity factors will be introduced in the conventional form,

1 – these matrices will be introduced in the transposed form

0/1

Information about calculation process

0 – information is not given; 1– at each iteration the values of the minimized criterion, parameters, first and second derivatives appear in the additional window.

 

 

Sheet “Stoichiometry”

Contains the stoichiometric matrix. See Examples

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Sheet “Logarithms of Constants”

Along with the stoichiometric matrix a column is arranged for the values of lg bi. Instead of unknown values zero approximations for parameters LKu to be found are entered (they are shown in red). If there is at least one non-zero invariant contribution lg bi0, another column for lg bi0 is added; for exactly known lg bi this additional column contains zeros.

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Sheet “Transformation”

Elements of matrix T are entered into this sheet.

Sheet “Meàsured Properties”

This sheet contains measured properties of equilibrium systems. Row numbers are those of examined mixtures k, k =1, 2, ..., N;

column numbers are those of analytical positions l l, l = 1, 2, ..., L.

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Sheet “Total Concentrations”

Total concentrations tj of independent components Bj for all experimental points are specified in this sheet.

Row numbers are those of examined mixtures k, k =1, 2, ..., N;

column numbers are those of independent components Bj, j = 1, 2, ..., Y.

For each experimental point, total concentrations tj (mol/l) are interrelated with initial reagent concentrations ñ(Li) by the following formulae:

We realize that an experiment may be organized in a variety of ways, and an attempt to foresee all of them is hardly reasonable. Required by the program concentrations tj may be easily prepared using Excel tools for data processing.

If the complex formation in a two-phase system is under study, all tj must be recalculated with respect to the volume of phase I and must be given in mol/l.

The design of an experiment may be such that equilibrium activities (concentrations) for one of independent components are registered at all experimental points in addition to the chosen property of a system (e.g. absorbance of a solution or solubility of a precipitate). Usually these are activities of hydrogen ions à(Í+) = 10-ðÍ estimated on the basis of pH data. In this case total concentrations t(H+) do not required by the program. Instead, the corresponding column of the "Total Concentrations" sheet is filled with à(Í+) values, and these values are highlighted in red.

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Sheet “Weights”

The matrix of statistical weights is specified in this sheet. Row numbers are those of examined mixtures k, k =1, 2, ... , N;

column numbers are those of analytical positions l l, l = 1, 2, ... , L .

Statistical weights are determined by users proceeding from the adopted model of errors. If necessary, Excel tools for formulae processing may be used.

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Sheet “Intensity Factors”

This sheet contains intensity factors. Row numbers (i) are those of chemical species in the stoichiometry matrix, i = 1, 2, ..., S;

column numbers are those of analytical positions l l, l = 1, 2, ..., L .

Intensity factors to be computed are highlighted in red.

If an intensity factor of reagent Li is unknown for at least one of L analytical positions, CLINP 2.1 requires computation of all intensity factors of this reagent. It is not possible to specify a part of intensity factors and to compute rest of them for the same reagent.

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Sheets “Volume Ratios and Phase Pointers”

These sheets need to be filled only when the distribution of a reagent between phases is under study.

The single column in "Volume Ratios" sheet contains N ratios.

These are ratios of phase volumes VII/VI when both phases are liquids or alternatively mII/VI ratios where mII  is the mass of phase II (sorbent).

The single column in "Phase Pointers" sheet consists of S pointers. The pointer is 1 if a reagent is in phase I; the pointer is 2 if a reagent is in phase II.

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COMPUTATION FLOW

To run the program press CTRL+R

The "Running" window informs about the computation flow:

Status

Running / Success

Method

Newton / Gauss-Newton

Per Cent of Outliers

Specified value of d

Scale Iterations

The number of s iteration

Criterion Function

The current value of the function

Relative Gradient

The current value of the relative gradient

Scale Parameter

The current value of the scale parameter s

Iteration

The number of lg b iteration (s is fixed)

Criterion Function

The value of the function

Relative Gradient

The value of the relative gradient

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Results of Computations

The following information appears as a result of computations:

The global adequacy criteria are as follows:

Sheet “Results”

The first information in this sheet is the criterion according to which computations were stopped.

Under "Logarithms of Constants" one may find logarithms of stability constants and calculated parameters LKu (the latter are shown in red).

This is followed by the variance-covariance matrix for LKu.

Under "Intensity Factors" intensity factors of reagents are given. Calculated values are shown in red.

Under "Weighted Discrepancies" the matrix of weighted discrepancies is given; it contains values x =w1/2× D, where D  = Acalculated-Aexperimental. The minimum absolute value of x is shown in blue; the maximum absolute value of x is shown in red.

Mean residuals and residual means are given for each mixture, for each analytical position and for the whole array of weighted discrepancies.

Along with the residual variance its estimation is given which is obtained in calculating the scale parameter s according to Huber (Huber variance).

Then one may find statistics required for checking the model adequacy: skewness, kurtosis, robust Person's chi-square observed, 5% point of c 2 distribution for f degrees of freedom.

This is followed by multiple, partial and total correlation coefficients for parameters LKu.

Finally, results of the singular decomposition of Jacoby matrix J = A / q are shown, namely singular numbers k i and matrix V.

Under "Equilibrium Concentrations" calculated equilibrium concentrations (mol/l) are given for all reagents and all experimental points. (In the case of two-phase systems concentrations of chemical species of phase II are recalculated with respect to the volume of phase I). Concentrations of prevailing components are shown in blue.

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Cross-Validation

The cross-validation [Stone M. J. of the Royal Statistical Soc. 36 (1974) 111] finds ever-wider use in exploring the adequacy of models.

A set of N experimental points is subdivided into two subsets. One of them contains (N-1) points and is used for calculating parameters LKu and <ali>. The other one contains the only point with the number g and is used for the assessment of the model. For this point discrepancies are calculated at all analytical positions:

where Algcalculated is calculated for experimental conditions corresponding to the point g using parameters obtained from experimental measurements at all other points. This procedure is repeated for all N possible subdivisions of the set of experimental points and then a quantity

.

is calculated. This quantity is believed to measure the prediction ability of a model in contrast to conventional weighted sum of squares characterizing its approximation ability. If quantity

is close to unity, this is a strong factor for accepting the model.

Sheet “Cross Validation”

The following information is given in this sheet:

Discrepancy dlg for each experimental point with corresponding values of parameters (calculated for all points but considered one).

Cross-validation variance

.

Calculated parameters LKu arranged according to the stoichiometric matrix.

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See also:

Parameter calculations: Principles, Criterion Function, Robust Estimations

Examples

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