Christian Ekberg, Artem Matyskin - Hydromet 2016/Appied... · 2018-05-24 · Aquatic chemistry:...

Applied thermodynamics of solutions

Christian Ekberg, Artem Matyskin

Nuclear Chemistry / Industrial Materials Recycling group

Chemistry and Chemical Engineering Department

Chalmers University of Technology

Commercial

Outline

Basic complexations chemistry

Determination of stability constants

Uncertainties in stability constants

Chemical activity coefficients

Example using BaOH formation

Eaxmple using pH measurements

Basic complex formation reactions

34 ThOHOHTh

OHTh

ThOH4

3'

1

2

2

4 )(2 OHThOHTh 24

2

2'

2

)(

OHTh

OHTh

3

4 )(3 OHThOHTh

4

4 )(4 OHThOHTh

34

3'

2

)(

OHTh

OHTh

44

4'

2

)(

OHTh

OHTh

OHThsOHTh 4)()( 4

4 44 OHThKs

Nuclear Chemistry

1. Several methods for determining stability constants exists

2. They all have different pros and cons but also different validity regions

3. In principle it is possible to apriori select the optimal method using a fishbone

structure for uncertainty analysis.

4. The impact of uncertainties in hydrolysis constants have an important effect on

speciation diagrams.


Nuclear Chemistry


Potentiometric titrations

Solubility

Ion exchange

Spectrophotometry

Solvent extraction

Nuclear Chemistry

Potentiometric titrationsRationale:

As hydrolysis progresses, hydrogen ions are released. By titrating the solution

with and without metal, the difference in the hydrogen ion concentration at

different hydroxide additions permits calculation of the hydrolysis constants by

fitting.

Pros

Rather sensitive

Simple to perform

Cheap instrumentation

Indication of ratios e.g. using

Bjerrum’s method

Cons

Difficult for highly hydrolyzing metals

Only the first couple of constants obtainable

using simple methodologies

Sensitive to model input

No identification of species

High metal concentrations may be needed

Potentiometric titrations

The hydrogen concentration of a solution in absence of metal (squares)

and the hydrogen ion concentration in the presence of metal (rings)

plotted against the amount of titrant added.

The difference between the free hydrogen ion concentration in a

titrated solution without metal and the free hydrogen ion concentration

in a solution containing metal as a function of titrant added

0 100 200 300 400

0.000

0.002

0.004

0.006

0.008

[H+]

Titrant added (L)

0 100 200 300 400

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

[H

+]

(m

ol L

-1)

Titrant added (L)

][M(OH)2][MOH][OH][H][H 2-z2

1ztit0 2z

2z

1tit0 ]][OH[M2]][OH[M][OH][H][H

Nuclear Chemistry

Solubility

Rationale:

The total dissolved metal concentration is measured as a function of hydrogen ion

concentration. The set of hydrolysis constants are then fitted to the solubility curve.

Pros

Simple to perform

Solubility slope gives and indication

of the dominating species

Cons

Imprecise



Nuclear Chemistry

Solubility

r

r

q

qtot ][ML][M(OH)[M][M]

*Ks10 = [Mz+] / [H+]z

q

qqz-

s10tot [OH]β1

][OH

K[M]

0 2 4 6 8 10 12 14-12

-10

-8

-6

-4

-2

0

ZrOH3+

Zr4+

Zr(OH)2

2+

Zr(OH)3

+

Zr(OH)6

2-

Zr(OH)4(aq)

log

[Z

r]

-log [H+]

Nuclear Chemistry

Ion exchange

Rationale:

The distribution between an aqueous solution and an ion exchange resin is measured

as a function of hydrogen ion concentration. Constants obtained by fitting.

Pros

Simple to perform using radioactivity

All constants may be obtained

Cons

Distribution can be tricky to obtain

Radioactive tracers are preferred

Many parameters to fit



Nuclear Chemistry

Ion exchange

)(M)(M exaq )(M

)(Mk

aq

ex

..)()(

..)()(1

1

exMOHexM

exMOHexMD

....)()(

....)()(1

1

1

aqMOHaqM

aqMOHkaqMkD

0 2 4 6 8 10 12

1

2

3

4

5

6

7

8

9

Kd

[OH-]

Nuclear Chemistry

Spectrophotometry

Rationale:

The shift in a spectrum is measured as a function of hydrogen ion concentration.

Constants obtained by fitting.

Pros

Possible identification of species


Shifts may be pre-calculated

Cons

Slightly tricky to perform

High metal concentration needed

Nuclear Chemistry

Spectrophotometry

A = ecl

%A = 100 A/A0

350 400 450 500 550

0,00

0,05

0,10

0,15

0,20

0,25

pH [U(VI)]

4.275 5.20.10-3

4.209 4.10.10-3

4.252 4.75.10-3

3.939 4.84.10-3

3.746 4.69.10-3

3.503 4.53.10-3

3.843 2.50.10-3

3.884 2.04.10-3

4.718 6.62.10-4

4.512 5.79.10-4

2.947 4.27.10-4

absorp

tion / [cm

-1]

wavelength / [nm]

Nuclear Chemistry

Solvent extraction

Rationale:

The distribution between an aqueous and organic solution is measured as a function

of hydrogen ion concentration. Constants obtained by fitting.

Pros

Simple to perform using radioactivity

Indication of dominating species

from slope


Cons

Distribution can be tricky to obtain

Radioactive tracers are preferred



Nuclear Chemistry

Solvent extraction

phaseaqueoustheinmetalofionconcentratTotal

phaseorganictheinmetalofionconcentratTotal

[M]

[M]D

(aq)T

(org)T

i j

jYj,

iAi,

zAz,z

i j

ji

org

[Y][A]1

[A]

][MY][MA[M]

[MA]D

2 4 6 8 10 12-4

-3

-2

-1

0

1

2

3

log

D

-log [H+]

Nuclear Chemistry

Multiple Methodologies

Rationale:

Allows the pros of the different methodologies to be combined. The cons of certain

methods can be eliminated by using another methodology.

Pros

Different methods can be used to

identify different species

Multiple methods can decrease

uncertainty in constants

All constants may be obtained by

careful selection of methodologies

Cons

Few methodologies identify species

leading to uncertainty

Large amount of data is needed to

adequately define all species

May still be sensitive to model input

Nuclear Chemistry

Uncertainty analysis

Fishbone structure:

Allows the user to identify the contributors to the uncertainty. It also helps future

reader to reevaluate the uncertainty in the experiment.

Contributor Description Uncertainty

t < 0.5 %

pHElectrode,

buffersNegligible

I Balance, volume Negligible

[NaClO4]Purity, volume,

balance< 1 %

[HAa] Purity, volume < 1 %

[ISA-]Purity, volume,

balance< 1 %

AKUFVE

Mixing, phase

separation

Sample volume

< 1 %

each Negligible

Detection Counting < 1 %

purity

phase

sep.

mixing

volume

purity

balanc

e

volume

purity

countin

gvolume

balanc

e

buffers

electro

de

t pH I Detectio

n

[NaClO4

]

[HAa][ISA-] AKUFVE

D

Nuclear Chemistry


Statistical methods:

Several different ones exist. A simple one called the Chi-square method is recommended

How to do:

1. Fit your parameters using e.g. the sum of least squares as the minimization

function

2. Change one of them up so the sum of least squares increase by 1.41 (

SQRT(2)

3. Change the same one down until the sum of least squares increase by 1.41

4. Repeat for all fitted parameters

5. Now one standard deviation is obtained for each fitted

Nuclear Chemistry


0.0

0.2

0.4

0.6

0.8

1.0

2 3 4 5 6 7pH

Th+4

ThOH+3

Th(OH)2+2

Th(OH)3+

Th(OH)4

Species log(*)

ThOH3+

-3.30.2

Th(OH)22+

-8.60.1

Th(OH)3+ -14.3a

Th(OH)4 -19.40.5

Nuclear Chemistry


Species log(*)

ThOH3+

-3.30.2

Th(OH)22+

-8.60.1

Th(OH)3+ -14.3a

Th(OH)4 -19.40.5

0.0

0.2

0.4

0.6

0.8

1.0

2 3 4 5 6 7

pH

Activity coefficients: introduction

It is important to keep ionic strength constant when determining stability constant.

Background inert electrolyte is usually used (NaClO4, NaCl etc)

Stability constants are usually determined at specific ionic strengths, it means that

determined constant is valid only for specific ionic strengths and only for specific

electrolyte.

Why?

n

ii zcI1

25.0

Strong electrolyte – substance which exists in solution only as free ions (completely

dissociated)

Weak electrolyte – substance which exist in solution both in ionic (as free ions) and

molecular forms (partly dissociated)

In ideal solutions ions and molecules are assumed as point particles (without size,

shape etc).

In real solutions particles have size, volume, charge etc and move in solutions. In

case of neutral species Van der Waals and dipole-dipole interactions becomes

important in concentrated solutions (no chemical bond is formed). Ions can be

attracted to each other or repulsed due to electrostatic Coulomb forces even in

dilute solutions (long-range interactions).

Activity coefficients: introduction

To be able to use equations and laws for ideal solutions for description of real

thermodynamic system properties we need to include long-range electrostatic

interactions. This can be done by introducing activities and activity coefficients:

Activity coefficients: definition

cA

Thus we can describe chemical equilibrium of real thermodynamic systems using

activities instead of concentrations:

4

2

4 NaSOSONa

24

24

44

24

4

4

SOSONaNa

NaSONaSO

SONa

NaSO

NaSO cc

c

AA

AK

Experimental determination of activity coefficients

Practically it is impossible to determine activity coefficient of an ion, but activity coefficients

of solute and solvent depend on each other. Thus, it is possible to measure change of solvent

activity (at different concentrations and with or without solute) and derive mean activity

coefficient of solute.

1. Vapor pressure decrease (measurements of vapor pressure of pure solvent and with

solute)

2. Boiling point increase (measurements of boiling point of pure solvent and with solute)

3. Freezing point decrease (measurements of freezing point of pure solvent and with

solute)

4. Osmotic pressure measurements (osmotic pressure of real solution is higher than

osmotic pressure of ideal solution, this pressure difference can be measured. Thus

osmotic coefficient has the same meaning as activity coefficient - deviation from

ideality. Activity coefficient can be calculate from osmotic coefficient and vice versa

using Gibbs-Duhem equation)

5. EMF measurements (lead to experimental values of mean activity coefficient)

6. Solubility measurements (Measurements of solubility of sparingly soluble salts in dilute

solutions when activity coefficients are almost 1 and in concentrated solutions)

Expressions for activity coefficients

Law Equation Applicability, Ionic

strengths, M

Debye - Hückel ~0,001

Extended Debye -

Hückel

~0,1

Davies ~0,3

IzA ii 2)log(

IaB

IzA ii

1)log(

2

IIaB

IzA ii

2.0

1)log(

2

A and B are temperature/pressure dependent constants

a is “distance of closest approach” . Normally at 25 deg C we use A = 0.5085 and B =

0.3281. Often the term Ba = 1.5

rraStumm, W., & Morgan, J. J. (2012). Aquatic chemistry: chemical equilibria and rates in natural waters (Vol. 126). John Wiley & Sons.

Expressions for activity coefficients

The Davies equations does not take the specific shorter range interactions into

account so further corrections have to be made

Stumm, W., & Morgan, J. J. (2012). Aquatic chemistry: chemica equilibria and rates in natural waters (Vol. 126). John Wiley & Sons.

The long range interactions described by the Davies equations will affect the activity

coefficient like this:

0,8

0,85

0,9

0,95

1

0,0001 0,001 0,01 0,1 1

Act

ivit

y c

oef

fici

ent

Electrolyte concentration

Specific Ion Interaction Theory (SIT)

Iki

IaB

IzA ii )(

1)log(

2

e

Where ε is interaction coefficient between two ions of opposite charge.

Let’s take the following reaction in 1M NaCl media:

4

2

4 NaSOSONa

Stability constant for this reaction at constant ionic strengths:

24

4

424

24

44

4

0

SONa

NaSO

NaSO

SOSONaNa

NaSONaSO

NaSOK

cc

cK

)log()log()log()log()log( 24444

0

SONaNaSONaSONaSOKK

IIaB

IAKK

NaSOClNaNaNaSONaSONaSO

)(1

4)log()log( 2

4444

0 eee

Extended Specific Ion Interaction Theory

Iki

IaB

IzA ii )(

1)log(

2

e

)log(21 I eee

In this model second ion interaction coefficient is added. This makes possible to

describe thermodynamic systems at higher temperatures and ionic strengths.

However, according to this equation:

e when 0I

Grenthe, I., & Wanner, H. (2000). Guidelines for the extrapolation to zero ionic strength.

IaB

IzA i

1

2

- Debye Hückel term

Pitzer formalism

MXMX CmBmf

24)ln(

Can be compared with equation for real gasses:

...2CpBpTRVp

)1ln(

2

1Ib

bIb

IAf

IIII

B

2

1112

1

10 5.01)(exp(1

22

K. S. Pitzer, Thermodynamics 3rd ed., McGraw-Hill, New York, 1995

Pitzer, K. S. (1991). Activity coefficients in electrolyte solutions. CRC press.

- Debye Hückel term

Determination of weak complex formation

4

2

4 NaSOSONa

Stability constant for this reaction at constant ionic strengths:

IIaB

IAKK

NaSOClNaNaNaSONaSONaSO

)(1

4)log()log( 2

4444

0 eee

This equation is valid if ionic strengths was kept constant (background inert

electrolyte NaCl, NaClO4), only in this case activity coefficients or long range

electrostatic interactions are constant.

In case of weak complex formation it is necessary to introduce large quantities of

ligand (2 mole/L) and thus, substitute more than 10% of background electrolyte. It

means that it is necessary to separate two effects: weak complex formation (short

range) and activity coefficient change (long range).

Direct methods for weak complex formation determination

• Direct experimental determination of activity coefficients (vapor pressure,

boiling point, freezing point, osmotic coefficient, emf, solubility)

• Conductivity measurements of electrolyte solutions (change in conductivity due

to neutral species formation)

• Spectroscopic measurements: UV, IR, Raman, NMR (additional lines in

spectrum can indicate complex formation)

• Relaxation methods (direct relaxation spectroscopy)

Marcus, Y., & Hefter, G. (2006). Ion pairing. Chemical reviews, 106(11), 4585-4621.

Barium hydrolysis: ion interaction model

• Ba2+ + 2·NaRsolid → BaR2solid + 2·Na

2

][

][

2

2

2][][

Na

Ba

solid

BaBaR

2

][

][

22

][

][

2

2

222

][

1][

][

][

Na

Ba

Na

Basolid

Ba

Ba

Ba

BaRD

][][

])[(][][][lg

44

44 4

OHIOHDH

OHIOHDHClOOHDH

ClONaClONaOHNa

ClONaOHNaClONaOHNaNa

eee

eeee

][][4

])[(][4][][4lg

44

442 4

OHIOHDH

OHIOHDHClOOHDH

ClOBaClOBaOHBa

ClOBaOHBaClOBaOHBaBa

eee

eeee

)22(][22444410 ClONaOHNaClOBaOHBaClONaClOBa OHIIDH

D

eeeeee

][10 OHbaD ][)lg()lg( OHbaD

Spahiu, K., & Puigdomenech, I. (1998). On weak complex formation: re-interpretation of literature data on the Np and Pu nitrate complexation.Radiochimca

Acta, 82(Supplement), 413-420.

Barium hydrolysis: ion interaction model

][)lg()lg( OHbaD

Barium hydrolysis: ion association model

0])[][1(][][

][][][][][][][][][

4

44

4

4

ClOKOHKNaNa

ClONaKOHNaKNaNaClONaOHNaNa

NaClONaOH

total

aqueous

NaClONaOH

total

aqueous

0])[1(][][

][][][][][][

4

4

44

44444

NaKClOClO

ClONaKClONaClOClOClO

NaClO

total

aqueous

NaClO

total

aqueous

0])[1(][][

][][][][][][

NaKOHOH

OHNaKOHNaOHOHOH

NaOH

total

aqueous

NaOH

total

aqueous

total

aqueoustotal

aqueous

NaOH

BaOH

total

aqueous

OHNaK

K

ClOD

][)][1

1(1

][

1

410

Barium hydrolysis: ion association model fit

total

aqueoustotal

aqueous

NaOH

BaOH

total

aqueous

OHNaK

K

ClOD

][)][1

1(1

][

1

410

Barium hydrolysis: extrapolation to zero ionic strength

According to the plots logarithms of stability constants of NaOH and BaOH+ complexes at zero ionic

strength are 0,31 and 0,73 which is in agreement with literature data (-0.4 ± 0.2 for NaOH and 0.68 ± 0.07 for

BaOH+)

“The chief criterion for [classifying] an electrolyte [as nonassociated] is the absence of valid evidence for any

form of association. Since the validity of such evidence can be a matter of personal opinion...there can be no

general agreement.” – Robinson and Strokes

“However, if K<2 [logK<0,3], the mere existence of the ion pair may be questioned”

[EKB15] Ekberg, C. and Brown, P. (2015). Studies on the Hydrolysis of metal ions

[MAR06] Marcus, Y., & Hefter, G. (2006). Ion pairing. Chemical reviews, 106(11), 4585-4621.

BASIC ELECTRODE KNOWLEDGE”What you need to know when measuring pH”

Christian Ekberg 2013

Basic electrochemistry (light thermodynamics)

Consider the reaction:

Basic electrochemistry (light)Consider the reaction:

We also know that Gibbs energy is an equation of state which can

be related to other equations of state like the enthalpy and entropy

according to: ΔG= ΔH-T ΔS.

The energy change (possibility) during a reaction is given by how far

it is from its equilbrium at standard state:

ΔG= ΔGo + RTlnQ = -RTlnK + RTlnQ

Thus, at equilibrium at standard state the change in Gibbs energy is

zero

Basic electrochemistry: a side track


Basic electrochemistry (cells)

Anode

Always oxidation

Electrons donated

Anions accepted

Minus (GALVANIC CELL)

Cathode

Always reduction

Electrons accepted

Cations accepted

Plus (GALVANIC CELL)

A galvanic cell is goverened by a

spontanous reaction while an

electrolytic cell requires electric work

This reaction ca be divided into an oxidation and a reduction reaction:


Zn(s) Zn2+(aq) + 2e- (Oxidation) Anode

Cu2+(aq) + 2e- Cu(s) (Reduktion) Cathode

Zn(s) + Cu2+(aq) → Zn2+(aq) + Cu(s) Cell

If the contact between the reactants are limited we can build a cell


Cu(s) Cu2+(aq) Zn2+(aq) Zn(s)

Cathode Anode

Phase boundaries

Salt bridge


Zn(s) Zn2+(aq) + 2e- (Oxidation) Anode Eao = -0.76 V

Cu2+(aq) + 2e- Cu(s) (Reduction) Cathode Eco= 0.34 V

Zn(s) + Cu2+(aq) → Zn2+(aq) + Cu(s) Cell Eo=Eco – Ea

o=1.1 V

Please note the clear inconsistency here: The potentials are not added

when the reactions are added. The reason for this is that by convention

it is ALWAYS the reduction potential that is used. The oxidation

potential has a reversed sign and then the basic rules of thermodynamics

would apply (like that the Gibbs energy of two added reactions is the

addition ofthe respective Gibbs energies)

It is all about comparison.....

Basic electrochemistry (hydogen electrode)

Consider the cell:

By keeping the hydogen pressure at 1 atm and the proton activity at

1 M and define that under these conditions the reduction potential for

that half cell reaction =0 we can also define the potential of the other

half cell.

Now we can immerse a platinum wire (and bubble H2) in a solution

which is in contact with the reference by electric conduction and a

”salt bridge” and any H+ activity can be measured.

VIOLA We have built a pH electode!!

Pt H2, H+ KCl(sat), Hg2Cl2(s) Hg(l)

Basic electrochemistry (modernising)

Over the years both the hydogen and the calomel electrodes were

abandoned for the glass electrode and the Ag/AgCl electrode. The

principles are the same but the exact working of the glass electrode

is still a mystery. However, its properties are well investigated.

It is today rather unusual to have

separate refernce electrodes and

glass electrodes. Typically the

refernce electrode is within the

combined electrode while only

the glass electride is in contact with

the solution.

Ag(s) | AgCl(s) | KCl(aq) || 1×10-7M H+ solution || glass membrane || Test Solution || ceramic junction || KCl(aq) | AgCl(s) | Ag(s)

Basic electrochemistry (getting complicated)

So far everything has been rather straight forward (except the working

of the glass electrode). Matters get complicated when it is time to

consider which partial potentials actually build up the measured

potential. So far we have only considered the chemical potential

(ideally).

The most important additional potential is the liquid junction potential.

To maintain electrical contact between the reference electrode and the

pH electrode, there must be a relatively free diffusion of ions between

the reference fill solution and the process solution. Essentially we need

the negative and positive ions to be equitransferent which means that

they move (diffuse) equally fast through the solution and membrane.

If this is not the case we will have a charge imbalance over the

membranes which will add to the measured potential.

Basic electrochemistry (getting complicated)Evidently we can not use a filling solution which will form a solid when

in contact with the measured solution. A typical example is using a

KCl electrode when measuring in a ClO4- based solution. KClO4 will

then precipitate in the glass frit and destroy the electrode.

Please note that an electrode measures the H+ ACTIVITY and NOT

concentration. To get the concentration from a measurement either

calibrate using well determined concentration standards (in the same

ionic strength and medium) or calculate the activity coefficient for

the hydogen ion.

Remember that all reactions are temperature dependent. So are also

buffers and the auto-protolysis of water. Keep track of the temperature!

Most pH sensors are designed to produce a 0 mV signal at pH 7.0,

with a (theoretically ideal) slope (sensitivity) of -59.16 mV / pH at 25 °C.

Probably some items and important issues have been forgotten.

By considering the content of this lecture you are ready to make

”working” pH determinations. For more precise determinations and

evaluations other aspects have to be taken into account.

Basic electrochemistry (final words)

During this leacture only pH electrodes have been considered.

However, as you understand from the theory, almost any ions

activity can be measured this way by (home made) electrodes.

Measuring pH is an art and should be regarded with the respect it

deserves. Any detector can produce a value but what does the value

actually say?

This is the field of science!

Christian Ekberg, Artem Matyskin - Hydromet 2016/Appied... · 2018-05-24 · Aquatic chemistry:...

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Transcript of Christian Ekberg, Artem Matyskin - Hydromet 2016/Appied... · 2018-05-24 · Aquatic chemistry:...