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

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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, 1980

Method: spectrophotometry

Number of wavelength L : 6

Chemical 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 CuOx22- 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,

tiu 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+) (marked in red) for all examined solutions. This information is given in bold face.

Number of solution

Number of independent component Âj

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.

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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:

Stoichiometric matrix:

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, t 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.

Back to the index

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 Â1+) activities à(Í+) = 10-ðÍ have been measured.

Component Â2 (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) and

LK2 = 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

g2

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.

Back to the index

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 = (n0(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

k1 : k2 : k3 = 6.9× 1011 : 2.3× 1011 : 1

show 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

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