Examples of input data preparation for CLINP 2.1
Example 1. Spectrophotometry
Example 2. pH-metric titration
Example 3. Solubility method
Example 4. Sorption equilibria
Example 1. Examination of equilibria of complex formation in solutions by spectrophotometry
Source
: Hartley F.R., Burgess C., Alcock R.M. Solution Equilibria, Ellis Horwood, 1980Method
: spectrophotometryNumber of wavelength
L : 6Chemical species
postulated in aqueous solution:Cu2+, En, Ox2-, H+, CuOH+, EnH+, EnH22+, HOx-, CuEn22+, CuEnOx and CuOx22-,
where En – ethylenediamine, Ox2- – oxalat ion.
Experimantal conditions
:
Total concentrations, mol/l |
|
|
pH range
: 5.12£ pH£ 11.21 (21 solutions)Preliminary information
: Equilibrium constants were preliminary determined for a number of reactions with Cu2+, En and Ox2-.Known equilibrium constants:
Reaction |
Logarithm of equilibrium constant |
|
-8.0 |
|
15.69 |
|
10.18 |
|
17.67 |
|
3.85 |
Stability constants for complexes CuEnOx è CuOx
22- are unknown.Representation of reactions
:The choice of components for this system is not unique. When complex formation is concerned, complexing metal ions, ligands and hydrogen ions are included as a rule into the set of components.
Set of chosen components: Cu2+, Ån, Ox2- and H+.
Reactions in the canonical form are listed in Table 1.
Note: Water participates in considered reactions and also serves as a medium in which chemical interactions take place. If concentrations of reagents are small, its activity is practically invariant. In this case water is taken into account neither in material balance nor in action mass law. Accordingly, there is no column for water in the stoichiometric matrix. The same is true for reactions in other media: solvents are not represented in the stoichiometric matrix..
Table 1. An example of representing reactions
No |
Species |
Matrix ||nij||: stoichiometric coefficients of components |
Reaction |
lg bi |
|||
|
|
Cu2+ (B1) |
En (B2) |
Ox2- (B3) |
H+ (B4) |
|
|
1 |
CuEn22+ |
1 |
2 |
0 |
0 |
Cu2+ + 2 En = = CuEn22+ |
15.69 |
2 |
CuEnOx |
1 |
1 |
1 |
0 |
Cu2+ + En + Ox2- = = CuEnOx |
? |
3 |
CuOx22- |
1 |
0 |
2 |
0 |
Cu2+ + 2 Ox2- = = CuOx22- |
? |
4 |
Cu2+ |
1 |
0 |
0 |
0 |
Cu2+ = Cu2+ |
0 |
5 |
En |
0 |
1 |
0 |
0 |
En = En |
0 |
6 |
Ox2- |
0 |
0 |
1 |
0 |
Ox2- + H+ = HOx- |
0 |
7 |
H+ |
0 |
0 |
0 |
1 |
H+ = H+ |
0 |
8 |
OH- |
0 |
0 |
0 |
-1 |
(H2O) - H+ = OH- |
-14.0 |
9 |
EnH+` |
0 |
1 |
0 |
1 |
En + H+ = EnH+ |
10.18 |
10 |
EnH22+ |
0 |
1 |
0 |
1 |
En + 2 H+ = = EnH22+ |
17.67 |
11 |
HOx- |
0 |
0 |
1 |
1 |
Ox2- + H+ = HOx- |
3.85 |
Return to the choice of components
Go to the discussion of the second choice of sought-for parameters
Let the objective be to determine parameters
and also molar absorptivities for complexes
CuEn22+, CuEnOx and CuOx22-.
In this case lgb2 = LK1; lgb3 = LK2. In accordance with equation
,
in which lgbi0 is invariant contribution into lgbi,
LKu are sought-for parameters,
t
iu are elements of p x p matrix T transforming parameters LKu into lgbi,one may write :
It follows that invariant contributions are lgb20 =lgb30 = 0 and matrix T is diagonal:
For zero approximations LK1 = 10 and LK2 = 5 sheets "Logarithms of constants" and "Transformation" are as follows:
"Logarithms of constants"
15,69 |
10 (marked in red ) |
5 (marked in red) |
0 |
0 |
0 |
0 |
-8 |
-14 |
10,18 |
17,67 |
3,85 |
"Transformation"
1 |
0 |
0 |
1 |
The situation is different if the objective is to find equilibrium constants of substitution reactions
It is not difficult to see from Table 1 that reaction
according to which complex CuEnOx is formed from components (reaction 2) is the linear combination of the following reactions
1´ |
Cu2+ + 2 En = CuEn22+ |
lg b1 |
+ |
|
|
1´ |
CuEn22+ + Ox2- = CuEnOx + En |
LK1 |
|
Cu2+ + En + Ox2- = CuEnOx |
lg b2 |
and consequently the logarithm of stability constant of CuEnOx is
lg b2 = lg b1 + 1× LK1 + 0× LK2.
Reaction according to which complex CuOx22- is formed from components
Cu2+, En and Ox2- (reaction 3 from Table 1) may be considered as the linear combination of three following reactions:
1´ |
Cu2+ + 2 En = CuEn22+ |
lg b1 |
+ |
|
|
1´ |
CuEn22+ + Ox2- = CuEnOx + En |
LK1 |
+ |
|
|
1´ |
CuEnOx + Ox2- = CuOx22- + En |
LK2 |
|
Cu2+ + 2 Ox2- = CuOx22- |
lg b3 |
The logarithm of stability constant of complex CuOx22- is
lg b3 = lg b1 + 1× LK1 + 1× LK2.
Therefore, invariant contributions are lgb20 = lgb30 = lg b1 = 15.69 and
the transformation matrix has the form
For sought-for parameters
with zero approximations LK10=-5 and LK20=-7 sheets "Logarithms of Constants" and "Transformation" are as follows.
Logarithms of Constants
15,69 |
0 |
-5 (marked in red) |
15,69 |
-7 (marked in red) |
15,69 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-8 |
0 |
-14 |
0 |
10,18 |
0 |
17,67 |
0 |
3,85 |
0 |
Transformation
1 |
0 |
1 |
1 |
Sheet "Measured Properties" contains absorbances divided by the layer thickness (shown in bold)
Number of solution |
Number of wavelength |
|||||
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
0,273 |
0,46 |
0,285 |
0,265 |
0,119 |
0,03 |
2 |
0,256 |
0,46 |
0,294 |
0,271 |
0,124 |
0,029 |
3 |
0,247 |
0,459 |
0,296 |
0,275 |
0,127 |
0,03 |
4 |
0,235 |
0,461 |
0,315 |
0,293 |
0,147 |
0,039 |
5 |
0,224 |
0,461 |
0,32 |
0,298 |
0,155 |
0,039 |
6 |
0,202 |
0,46 |
0,333 |
0,312 |
0,172 |
0,047 |
7 |
0,187 |
0,46 |
0,346 |
0,326 |
0,192 |
0,056 |
8 |
0,162 |
0,46 |
0,365 |
0,349 |
0,207 |
0,069 |
9 |
0,14 |
0,455 |
0,38 |
0,364 |
0,23 |
0,08 |
10 |
0,112 |
0,45 |
0,402 |
0,387 |
0,26 |
0,096 |
11 |
0,087 |
0,432 |
0,4 |
0,39 |
0,271 |
0,107 |
12 |
0,07 |
0,414 |
0,4 |
0,392 |
0,288 |
0,119 |
13 |
0,045 |
0,376 |
0,39 |
0,386 |
0,31 |
0,142 |
14 |
0,034 |
0,327 |
0,369 |
0,37 |
0,329 |
0,17 |
15 |
0,02 |
0,285 |
0,343 |
0,345 |
0,327 |
0,181 |
16 |
0,014 |
0,25 |
0,32 |
0,327 |
0,332 |
0,195 |
17 |
0,015 |
0,225 |
0,302 |
0,312 |
0,336 |
0,202 |
18 |
0,012 |
0,179 |
0,275 |
0,285 |
0,337 |
0,225 |
19 |
0,007 |
0,142 |
0,25 |
0,265 |
0,335 |
0,235 |
20 |
0,008 |
0,135 |
0,246 |
0,264 |
0,341 |
0,25 |
21 |
0,01 |
0,129 |
0,244 |
0,256 |
0,345 |
0,244 |
Sheet "Total Concentrations" shows total concentrations of components
B1 (Cu2+), B2 (En), B3 (Ox2-) and activities a(H+) = 10-pH of component Â4 (Í+) (marked in red) for all examined solutions. This information is given in bold face.
Number of solution |
Number of independent component  j |
||||||
1 |
2 |
3 |
4 |
||||
1 |
0,01 |
0,1 |
0,1 |
6,17E-12 |
|||
2 |
0,01 |
0,1 |
0,1 |
5,25E-08 |
|||
3 |
0,01 |
0,1 |
0,1 |
7,24E-08 |
|||
4 |
0,01 |
0,1 |
0,1 |
1,20E-07 |
|||
5 |
0,01 |
0,1 |
0,1 |
1,41E-07 |
|||
6 |
0,01 |
0,1 |
0,1 |
1,86E-07 |
|||
7 |
0,01 |
0,1 |
0,1 |
2,19E-07 |
|||
8 |
0,01 |
0,1 |
0,1 |
2,75E-07 |
|||
9 |
0,01 |
0,1 |
0,1 |
3,39E-07 |
|||
10 |
0,01 |
0,1 |
0,1 |
4,27E-07 |
|||
11 |
0,01 |
0,1 |
0,1 |
5,50E-07 |
|||
12 |
0,01 |
0,1 |
0,1 |
6,46E-07 |
|||
13 |
0,01 |
0,1 |
0,1 |
8,32E-07 |
|||
14 |
0,01 |
0,1 |
0,1 |
1,26E-06 |
|||
15 |
0,01 |
0,1 |
0,1 |
1,55E-06 |
|||
16 |
0,01 |
0,1 |
0,1 |
1,91E-06 |
|||
17 |
0,01 |
0,1 |
0,1 |
2,45E-06 |
|||
18 |
0,01 |
0,1 |
0,1 |
3,16E-06 |
|||
19 |
0,01 |
0,1 |
0,1 |
5,13E-06 |
|||
20 |
0,01 |
0,1 |
0,1 |
6,31E-06 |
|||
21 |
0,01 |
0,1 |
0,1 |
7,59E-06 |
Sheet "Weights" contains statistical weights of measurements. In estimating the standard deviation for absorbances s(A) = 0.0035, all weights were equal:
wlk = 1/(0.00352) = 8.16× 104.
Sheet "Intensity factors" contains intensity factors of reagents. Information is given in bold face; intensity factors to be calculated are marked in red.
Number of reagent in stoichiometric matrix |
Intensity factors (molar absorptivities) for analytical positions |
|||||
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
3 |
0 |
0 |
0 |
0 |
0 |
0 |
4 |
0 |
0 |
0 |
0 |
0 |
0 |
5 |
0 |
0 |
0 |
0 |
0 |
0 |
6 |
0 |
0 |
0 |
0 |
0 |
0 |
7 |
0 |
0 |
0 |
0 |
0 |
0 |
8 |
0 |
0 |
0 |
0 |
0 |
0 |
9 |
0 |
0 |
0 |
0 |
0 |
0 |
10 |
0 |
0 |
0 |
0 |
0 |
0 |
11 |
0 |
0 |
0 |
0 |
0 |
0 |
Sheets "Volume Ratios" and "Phase Pointers" remain empty.
Example 2. Simulation of equilibria of complex formation in a system studied by pH-metric titration
The volume of examined (titrated) solution is v0 = 50 ml.
It contains:
unhydrolized metal ions (M), c0(Ì) = 0.01 mol/l,
monoprotic acid HA, c0(ÍÀ) = 0.02 mol/l
(here and further charges are not shown).
This solution is titrated with NaOH solution, cx(NaOH) = 0.05 mol/l.
To keep the constant ionic strength, strong electrolyte NaNO3 (0.10 mol/l) was added to both these solutions.
Equilibrium values of pH were registered at each point of titration curve with error » 0.05.
A part of this curve containing 10 point was selected for simulation; range of titrant volume is 5£ vk£ 15 ml (4<pH<7).
Chosen components
: H+, HA and metal ions M.Known constant
: dissociation constant of HA; lg Ka = -5.1.Objective
: to calculate stepwise stability constants for complexes ÌÀ è MA2:
No |
Species |
Matrix ||nij||: stoichiometric coefficients of components |
Reaction |
lg bi |
||
Í +(B1) |
Ì (B2) |
ÍÀ (B3) |
||||
1 |
H+ |
1 |
0 |
0 |
H+ = H+ |
0 |
2 |
M |
0 |
1 |
0 |
M = M |
0 |
3 |
HA |
0 |
0 |
1 |
HA = HA |
0 |
4 |
OH- |
-1 |
0 |
0 |
(H2O) - H+ = OH- |
-13.8 |
5 |
A |
-1 |
0 |
0 |
-H+ + HA = A |
-5.1 |
6 |
MA |
-1 |
1 |
1 |
-H+ + M + HA = MA |
-5.1 + LK1 |
7 |
MA2 |
-2 |
1 |
2 |
-2 H+ + M + + 2 HA = MA2 |
2´ (-5.1) + LK1 + + LK2 |
Go to discussion of component total concentrations
It is not difficult to see that
lg b6 = -5.1 + 1× LK1 + 0× LK2,
lg b7 = -10.2 + 1× LK1 + 1× LK2,
invariant contributions are:
lgb60= -5.1; lgb70= -10.2,
and transformation matrix has the form:
Measurements of pH are carried out in solutions for which contributions of [H+] and [OH-] into material balance are negligible. With this in mind, quantity
is chosen as a measured property A to be simulated. Its relative error is accepted to be sr(A) = 0.02 which roughly corresponds to measurement error ðÍ ~0.05.
Values à(Í+) are arranged as a column in "Measured Properties" sheet.
In "Total concentrations" sheet total concentrations of components (tj)k (mol/l) need to be specified for all experimental points. After a successive portion of titrant has been added, they may be calculated in accordance with experimental conditions using formula
,
where v0 is the initial volume of examined solution, tj0, tjõ are total concentrations of components in examined solution and titrant respectively, vk is the added volume of titrant.
Our communications with users show that finding tj0 and tjx may rise some difficulties. In calculating them it is convenient to use material balance in the form
,
where c(Li) are initial concentrations of Li (known from conditions of reagent mixing). Taking into account that metal ions and HA acid are only reagents introduced into examined solution and using stoichiometric coefficients one arrives at
t10 = t0(H+) =
= n11× c0(L1)+n21× c0(L2)+n31× c0(L3)+n41× c0(L4)+n51× c0(L5)+n61× c0(L6)+n71× c0(L7)=
= 1× c0(H+)+0× c0(M)+0× c0(HA)+(-1)× c0(OH-)+(-1)× c0(A)+(-1)× c0(MA)+(-2)× c0(MA2) =
= 1× 0+0× 0.01+0× 0.02+(-1)× 0+(-1)× 0+(-1)× 0+(-2)× 0 = 0;
t20 = t0(M) =
= n12× c0(L1)+n22× c0(L2)+n32× c0(L3)+n42× c0(L4)+n52× c0(L5)+n62× c0(L6)+n72× c0(L7) =
= 0× c0(H+)+1× c0(M)+0× c0(HA)+0× c0(OH-)+0× c0(A)+1× c0(MA)+1× c0(MA2) =
0× 0+1× 0.01+0× 0.02+0× 0+0× 0+1× 0+1× 0 = 0.01;
t30 = t0(HA) =
= n13× c0(L1)+n23× c0(L2)+n33× c0(L3)+n43× c0(L4)+n53× c0(L5)+n63× c0(L6)+n73× c0(L7) =
= 0× c0(H+)+0× c0(M)+1× c0(HA)+0× c0(OH-)+1× c0(A)+1× c0(MA)+2× c0(MA2) =
= 0× 0+0× 0+1× 0.02+0× 0+0× 0+1× 0+2× 0 = 0.02.
The only reagent introduced into titrant is ÎÍ
- ions. Accordingly:t1x = tx(H+) =
= n11× cx(L1)+n21× cx(L2)+n31× cx(L3)+n41× cx(L4)+n51× cx(L5)+n61× cx(L6)+n71× cx(L7) =
= 1× cx(H+)+0× cx(M)+0× cx(HA)+(-1)× cx(OH-)+(-1)× cx(A)+(-1)× cx(MA)+(-2)× cx(MA2) =
= 1× 0+0× 0.01+0× 0.02+(-1)× 0.05+(-1)× 0+(-1)× 0+(-2)× 0 = -0.05;
t2x = tx(M) =
= n12× cx(L1)+n22cx(L2)+n32cx(L3)+n42× cx(L4)+n52× cx(L5)+n62× cx(L6)+n72× cx(L7) =
= 0× cx(H+)+1× cx(M)+0× cx(HA)+0× cx(OH-)+0× cx(A)+1× cx(MA)+1× cx(MA2) =
= 0× 0+1× 0+0× 0+0× 0+0× 0+1× 0+1× 0 = 0;
t3x = tx(HA) =
= n13× cx(L1)+n23× cx(L2)+n33× cx(L3)+n43× cx(L4)+n53× cx(L5)+n63× cx(L6)+n73× cx(L7) =
= 0× cx(H+)+0× cx(M)+1× cx(HA)+0× cx(OH-)+1× cx(A)+1× cx(MA)+2× cx(MA2) =
= 0× 0+0× 0+1× 0+0× 0+0× 0+1× 0+2× 0 = 0.
The result is t10 = 0, t20 = 0.01, t30 = 0.02, t1x = -0.05, t2x = t3x = 0.
In preparing the array of total concentrations it is convenient to use built-in Excel tools for data processing.
Sheet "Weights" contains the column of statistical weights wk calculated according to the model of experimental errors as
wk = 1 / [Ak × sr(A)]2 = 1 / [0.02 × a(H+)k]2.
Sheet "Intensity Factors" contains the column of intensity factors of reagents:
1 for H+ and 0 for other reagents.
Sheets "Volume Ratios" and "Phase Pointers" remain empty.
Example 3. Calculation of equilibrium constants using data of solubility method
Solubility of copper hydroxide (II) was measured in dependence on pH of aqueous solutions. Relative experimental error is 10%. For stabilizing activity coefficients a strong electrolyte was added to solutions.
Dissolution is due to reactions
Cu(OH)2¯ + H+ = CuOH+ + H2O,
Cu(OH)2¯ = Cu(OH)2,solution,
Objective
: calculation of equilibrium constants LK1 and LK2 of these reactions.Chosen independent components Cu(OH)2¯ and H+.
Stoichiometric matrix
:
No |
Species |
Matrix || nij||: stoichiometric coefficients of components |
Reaction |
lg bi |
|
Í +(B1) |
Cu(OH)2 ¯(B2) |
||||
1 |
H+ |
1 |
0 |
H+ = H+ |
0 |
2 |
CuOH+ |
1 |
1 |
H+ + Cu(OH)2¯ = = CuOH+ + (H2O) |
LK1 = ? |
3 |
Cu(OH)2,solution |
0 |
1 |
Cu(OH)2 ¯ = Cu(OH)2,solution |
LK2 = ? |
Invariant contributions
are lgb20 = lgb30 = 0,transformation matrix
T is diagonal:Measured values of solubility are specified in "Measured Properties" sheet.
Information about composition of the system is specified in "Total concentrations" sheet.
For component Â
1 (Í+) activities à(Í+) = 10-ðÍ have been measured.Component Â
2 (Cu(OH)2¯ ) is an individual solid phase; its activity is 1 by definition.Accordingly, one of two columns in this sheet consists of values à(Í
+) = 10-ðÍ;the second column is unit one.
Elements of both columns are marked in red.
Statistical weights wk calculated as
wk = 1/ [sr(A) × Ak]2 = 1 / [0.1 × Ak]2,
(where sr(A) = 0.10 is the estimate of relative standard deviation for A) are specified in "Weights" sheet.
The column in sheet "Intensity Factors" contains coefficients that determine contributions of equilibrium concentrations of reagents into the measured property:
A = 0 × [H+] + 1× [CuOH+] + 1 × [Cu(OH)2,solution].
Therefore this column is
0 |
1 |
1 |
Sheets "Volume Ratios" and "Phase Pointers" remain empty.
Interpretation of results obtained in Example 3
For a priori probability of outliers d = 0 % (LSM-estimation) the sampling kurtosis of weighted discrepancies is found to be g 2 = -0.93. This value gives no substantiation for rejecting the hypothesis about the normal distribution of weighted discrepancies x k (Table 2). Therefore, LSM-estimates are accepted for
LK1
= lg KS1(Cu(OH)2¯ + H+ = CuOH+ + H2O) andLK2
= lg KS2(Cu(OH)2¯ = Cu(OH)2, ð-ð).This decision is supported by insignificant (within experimental error) variation of lg Ks1 and lg Ks2 when d is varied from 0 to 99 % (Table 2 and Figure below).
Table 2. Results of the simulation of hydrolysis equilibria in which Cu(OH)2 participates
d , % |
Parameter |
||||||
lg KS1 |
lg KS2 |
|
g 2 |
f |
|
c 2(f; 0.05) |
|
0 |
1.884 |
-5.553 |
0.36 |
-0.93 |
8 |
9.5 |
15.5 |
10 |
1.881 |
-5.555 |
0.37 |
-0.94 |
12 |
15.2 |
22.4 |
30 |
1.870 |
-5.561 |
0.40 |
-0.96 |
13 |
16.3 |
22.4 |
50 |
1.866 |
-5.562 |
0.43 |
-0.97 |
13 |
16.8 |
22.4 |
70 |
1.866 |
-5.562 |
0.43 |
-0.98 |
13 |
16.9 |
22.4 |
90 |
1.865 |
-5.561 |
0.44 |
-0.98 |
13 |
16.9 |
22.4 |
99 |
1.863 |
-5.562 |
0.44 |
-0.99 |
13 |
17.3 |
22.4 |
Figure. Logarithms of constants of reactions in which Cu(OH)2 participates and sampling kurtosis of the distribution of weighted discrepancies (g2) as functions of parameter d . Vertical bars show 90% confidential intervals of parameters.
Example 4. Simulation of chemisorption equilibria
Liquid phase (phase I): CoCl2 solutions in dimethylformamide (DMF)
Sorbent (phase II): aerosil chemically modified with n-propylamine (Q)
Volume of phase I: VI = 0.025 l.
Weights of phase II: mII = 0.050, 0.100 and 0.200 g.
Specific concentration of Q: cQ = 0.58× 10-3 mol/g of sorbent
Measured property
: concentrations t[CoCl2] (mol/l) of CoCl2 in solutions after sorption (Table 3)Table 3. Isotherm of sorption of CoCl2 by chemically modified silica, n0(CoCl2) is the content in moles
mII = 0.050 g |
mII = 0.100 g |
mII = 0.200 g |
|||
n0(CoCl2), 10-6 mol |
t[CoCl2], 10-3 mol/l |
n0(CoCl2), 10-6 mol |
t[CoCl2], 10-3 mol/l |
n0(CoCl2), 10-6 mol |
t[CoCl2], 10-3 mol/l |
4.825 |
0.06 |
25.75 |
0.3 |
26 |
0.05 |
24.125 |
0.8 |
51.5 |
0.4 |
52.25 |
0.7 |
48.25 |
1.7 |
77 |
0.4 |
90 |
2.0 |
72.25 |
2.6 |
102.75 |
1.2 |
186 |
5.7 |
96.5 |
3.5 |
154.25 |
1.3 |
285 |
9.8 |
2.575 |
0.02 |
205.5 |
2.3 |
392.5 |
13.9 |
4.1 |
0.02 |
23.75 |
3.0 |
|
|
10.275 |
0.15 |
26 |
3.3 |
||
20.6 |
0.6 |
47.5 |
3.4 |
||
35.75 |
1.2 |
95 |
4.9 |
||
51.5 |
1.8 |
104.75 |
5.3 |
||
77.25 |
2.8 |
142.5 |
5.4 |
||
103 |
3.8 |
157 |
6.7 |
||
154.5 |
5.9 |
190 |
7.4 |
||
|
|
209.5 |
CoCl2Q, CoCl2Q2 and CoCl2Q3 are complexes assumed to form at the interface.
If CoCl2 and Q are chosen to serve as independent components, the stoichiometric matrix is as follows:
No |
Species |
Matrix ||nij||: stoichiometric coefficients of components |
Reaction |
lg bi |
Phase |
|
CoCl2 (B1) |
Q (B2) |
|||||
1 |
CoCl2 |
1 |
0 |
CoCl2 = CoCl2 |
0 |
I |
2 |
Q |
0 |
1 |
Q = Q |
0 |
II |
3 |
CoCl2Q |
1 |
1 |
CoCl2 +Q = CoCl2Q |
LK1 = ? |
II |
4 |
CoCl2Q2 |
1 |
2 |
CoCl2 +2 Q = = CoCl2Q2 |
LK2 = ? |
II |
5 |
CoCl2Q3 |
1 |
3 |
CoCl2 +3 Q = = CoCl2Q3 |
LK3 = ? |
II |
Sought-for parameters
: logarithms of total stability constants of complexes at the interface:LK1 = lg K(CoCl2 +Q = CoCl2Q) =
.
LK2 = lg K(CoCl2 +2 Q = CoCl2Q2) =
.
LK3 = lg K(CoCl2 +3 Q = CoCl2Q3) =
Here [CoCl2] is the equilibrium concentration of [CoCl2] in DMF (mol/l),
{} denote equilibrium concentrations of reagents at the interface (mol/g).
Invariant contributions are lgb30 = lg b40 = lgb50 = 0.
Transformation matrix
T is diagonal:
If t[CoCl2] is chosen to be described with the model, no difficulties arise in data preparation.
In the case of t{CoCl2} (equilibrium concentration of CoCl2 at the interface, mol/g) this procedure is not so involved also. Property A specified in this case in the program is
The values of t{CoCl2} may be found with the account of material balance for CoCl2:
n0(CoCl2) = t[CoCl2] × VI + t{CoCl2} × mII.
It follows that
t{CoCl2} = (n0(CoCl2) - t[CoCl2] × VI) / mII,
the property fitted by the model is:
À = (n
0(CoCl2) - t[CoCl2] × VI) / VI.These are values arranged as a column in sheet "Measured Properties".
Sheet "Total Concentrations" consists of two columns. The first one contains total concentrations of component Â1 (CoCl2) recalculated with respect to the volume of phase I:
t1 = t(CoCl2) = n0(CoCl2) / VI,
the second column contains total concentrations of component B2 (Q) recalculated with respect to the volume of phase I:
t2 = t(Q) = cQ × mII / VI.
The column of statistical weights
is arranged in sheet "Weights".
Sheet "Volume Ratios" is a column consisted of values mII/VI.
Two more sheets are as follows:
Intensity Factors |
Phase Pointers |
0 |
1 |
0 |
2 |
1 |
2 |
1 |
2 |
1 |
2 |
Interpretation of results obtained in Example 4
Modified Newton method was used.
Percentage of outliers was set to d = 0 %.
Statistical weights were assigned in estimating relative standard deviation sr(A) = 0.25.
Zero approximations were: LK10 = 0, LK20 = 12, LK30 = 24.
Iterations were stopped when the norm of gradient had reached the boundary value.
The result is LK1 = 1.95 (0.018), LK2 = -4.62 (1.1× 1010), LK3 = 10.19 (0.045)
(variances are given in parenthesis).
Obviously, parameter LK2 cannot be determined from the data obtained.
Singular decomposition of Jacoby matrix J leads to the same conclusion: ratios of singular numbers
k
1 : k2 : k3 = 6.9× 1011 : 2.3× 1011 : 1show that the rank of matrix J is incomplete. As to matrix V, elements of the column corresponding to small singular number
-3,3E-10 |
1 |
-1,4E-10 |
indicate that the minimized function is insensitive to parameter LK2.
The root of this redundancy of the model becomes clear from the analysis of calculated equilibrium concentrations: concentration of complex CoCl2Q2 is considerably (many orders of magnitude) smaller than error of property A and total concentrations of components at all points of "composition-property" dependence.
Values obtained for statistics used in checking the adequacy:
|
c 2(exper) = 31.5 < c2 (f = 32) = 46.2 |
|
|