solid state electrochemical characterization of ......2.6.2 thermodynamics of the electrochemical...
TRANSCRIPT
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Max-Planck-Institut für MetallforschungStuttgart
Solid state electrochemical characterization ofthermodynamic properties of sodium-metal-oxygensystems
Md. Ruhul Amin
Dissertationan derUniversität Stuttgart
Bericht Nr. 166Mai 2005
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Solid state electrochemical characterization of
thermodynamic properties of sodium-metal-oxygen systems
Von der Fakultät Chemie der Universität Stuttgart
zur Erlangung der Würde eines Doktors der
Naturwissenschaften (Dr.rer.nat.) genehmigte Abhandlung
Vorgelegt von
Md. Ruhul Amin
Aus Jaipurhat, Bangladesh
Hauptberichter: Prof. Dr. F. Aldinger
Mitberichter: Prof. Dr. J. Maier
Tag der mündlichen Prüfung: 13 Mai, 2005
Institut für Nichtmetallische Anorganische Materialien der Universität
Stuttgart
Max-Planck-Institut für Metallforschung, Stuttgart
Pulvermetallurgisches Laboratorium
ramin2005
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Dedicated to my parents and my beloved daguhter, Ananna.
“I want to know how God created this world. I am not interested in this or thatphenomenon, in the spectrum of this or that element. I want to know Histhoughts; the rest are details.”
Albert Einstein.
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Acknowledgements This doctoral thesis was done from November 2001 to March 2005 in Max-Planck-
Institute für Metallforschung, Stuttgart, supported by a scholarship of the Max-Planck-
Society. I would like to express my deep gratitude to my advisor Prof. Dr. F. Aldinger for
giving me the opportunity to investigate the topic of this work in his department. He
encouraged and supported me with much kindness throughout this work. In particular, I
would like to thank him for giving the opportunity to present my results at international
conferences. During my stay at MPI, I learned not only the methods for scientific
research but also was trained to become an independent scientist, which may be of
more value.
I wish to express my heartfelt thank to Dr. H. Näfe for the initiation and subject of this
work, the excellent scientific support, lively discussions and assistance during the whole
period. Most importantly, I learned much from his serious attitude toward scientific work,
making scientific reports and seminars.
I want to thank Prof. Dr. J. Maier who accepted to become the ‘Mitberichter’ for my final
examination. I also want to thank Prof. Dr. E. J. Mittemeijer who, together with my
advisor and Prof. Dr. J. Maier gave me final examination.
My sincere thank goes to Mrs. Gisela Feldhofer for her technical support of the
experimental work and also for her encouragement during the difficult times.
My thanks are given to all colleagues in Powder Metallurgical Laboratory (PML) and
engineers of the service groups of the MPI for Metal Research and the MPI for Solid
State Research, in particular, to Mrs. S. Paulsen for administration service, to Mr. H.
Labitzke, Mr. G. Kaiser, Ms. M. Thomas and Mr. U. Kloch for their sincere cooperation
for materials analysis and Mr. E. Bruckner for computer service.
I would like to thank my friends and colleagues from “Functional Ceramics Working
Group”: Subasri Raghavan, Bogdan Khorkounov, Krenar Shqau, Vladimir Plashnitsa,
Devendraprakash Gautam and Natalia Karpukhina.
There were many other people who made my working and living easier and full of fun.
Here, I would like to thank all of my colleagues who have ever helped me. I would like to
remember Professor S. A. Akbar for his constant help and the affection of my parents
who had passed away but live in my heart and encourage me from behind the scene.
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Contents
Acknowledgements i
Contents ii
List of Figures vi
List of Tables xiii
1. Introduction 1
2. Theoretical background 3
2.1 Electrochemical phenomena 3
2.2 Structure of solid electrolyte 3
2.2.1 Structure of beta alumina 3
2.2.2 Structure of stabilized zirconia 7
2.3 Defect chemistry 8
2.4 Conductivity 14
2.5 Electrolytic domain 15
2.5.1 Ionic domain of sodium-beta-alumina 15
2.5.2 Ionic domain of yttria-stabilized zirconia 17
2.6 Thermodynamic fundamentals 18
2.6.1 Thermodynamics of the chemical equilibrium 18
2.6.2 Thermodynamics of the electrochemical equilibrium 18
2.7 Galvani voltage 19
2.8 Cell voltage with electronic transference 20
3. Literature survey 24
3.1 General description 24
3.2 Elemental sodium 25
3.3 Sodium alloys 26
3.4 Ternary system 27
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iii
3.4.1 Sodium-metal-oxide system 27
3.4.2 Previous knowledge on the thermodynamics stability of Na-Me-O
systems (Me = Mo, Ti, Nb)
28
3.4.2.1 Na-Mo-O system 28
3.4.2.2 Na-Ti-O system 31
3.4.2.3 Na-Nb-O system 33
4. Experimental 36
4.1 Measuring principle 36
4.1.1 Cell configurations 36
4.1.2 Sodium chemical potential of the carbonate/gas electrode 37
4.1.3 Determination of the activity of sodium oxide dissolved in the phase
mixture
38
4.1.4 Determination of the sodium activity using cell configuration (III) 40
4.1.4 Cell voltage measurement using sodium-beta-Al2O3 as solid electrolyte 42
4.2 Materials and Measurements 43
4.2.1 Techniques for characterization of the materials 43
4.2.1.1 Chemical analysis 43
4.2.1.2 X-ray diffractometry 43
4.2.1.3 Scanning electron microscopy and X-ray microanalysis 44
4.2.1.3.1 Sample preparation for crystallographic analysis 44
4.2.1.4 Differential thermal analysis (DTA) 44
4.2.2 Solid electrolytes 45
4.2.3 Electrodes 46
4.2.3.1 Preparation the Na-Mo-O system 46
4.2.3.2 Preparation of phase mixture of the Na-Nb-O system 47
4.2.3.3 Preparation of single phases of the Na-Nb-O system 47
4.2.3.4 Preparation of Na2Ti3O7 and Na2Ti6O13 48
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4.2.3.5 Preparation of the bi-phasic solids 48
4.2.3.6 Preparation of the carbonate electrode 49
4.2.4 Gas atmosphere 49
4.2.5 Cell fabrication 49
4.2.6 Cell assembly 54
5. Results and discussion 56
5.1 Na-Mo-O system 56
5.1.1 Composition characterization of the eutectic phase mixture Na2MoO4 +
Na2Mo2O7
56
5.1.2 Thermodynamic stability of the Na2MoO4 + Na2Mo2O7 phase mixture 58
5.1.2.1 Results on cell configuration (I) 58
5.1.2.2 Results on cell configuration (II) 62
5.1.2.3 Results on cell configuration (III) 64
5.1.2.4 Results on cell configuration (IV) 69
5.1.2.5 Comparatively discussion of obtained results 72
5.2 Na-Ti-O system 74
5.2.1 The thermodynamic stability of the phase mixture Na2Ti3O7 + Na2Ti6O13 74
5.2.1.1 Results on cell configuration (I) 74
5.2.1.2 Results on cell configuration (II) 76
5.2.1.3 Results on cell configuration (III) 80
5.2.2 The thermodynamic stability of phase mixture Na2Ti7O13 + TiO2 81
5.2.2.1 Results on cell configurations (I) and (II) 81
5.3 Na-Nb-O system 85
5.3.1 Characterization of the eutectic phase mixture NaNbO3 + Na3NbO4 85
5.3.2 Determination of the sodium oxide activity dissolved in the eutectic
mixture Na3NbO4 + NaNbO3
87
5.3.3 Determination of the sodium activity of the eutectic mixture NaNbO3 + 89
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Na3NbO4
5.3.4 Investigation on the thermodynamic stability of the phase mixture
NaNbO3 + Na2Nb4O11, Na2Nb4O11 +NaNb3O8 and NaNb3O8 + Na2Nb8O21
using cell configuration (I)
93
5.3.5 Investigation of the phase mixture NaNbO3 + Na2Nb4O11 by cell (III) 99
5.3.6 Investigation on the thermodynamic stability of phase mixture NaNb3O8 +
Na2Nb8O21 using the cell configuration (II)
101
6. Conclusions and Outlook 106
7. Summary 108
8. Zusammenfassung 113
9. References 118
Curriculum Vitae
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vi
List of Figures
Figure Contents Page
2-1 Structure of Na-β -Al2O3 (left) and Na- ''β -Al2O3 (right) 4
2-2 Oxide ion packing arrangement in ß-Al2O3 (left) and 'ß' -Al2O3 (right)
(letters refer to stacking arrangement where ABC represent face-
centred cubic packing while ABAB represents hexagonal packing)
5
2-3 Ideal structure of the conducting plane of beta alumina. Solid circles
are column oxygen ions; open circles are mobile cations on BR sites
unoccupied hexagon vertices are aBR sites; and sites between
neighbouring BR and aBR are mO sites. A mobile cation in ideal
structure is in a deep potential well indicated by dotted lines
6
2-4 Structure of the cubic, tetragonal and monoclinic ZrO2 phase 8
2-5 Brouwer diagram for undoped Na-β -Al2O3 10
2-6 Brouwer diagram for yttria-stabilized zirconia 13
2-7 Vacancy mechanisms for transport of ions 14
2-8 Interstitial mechanism for transport of ions 14
2-9 Interstitialcy mechanisms showing the two possible locations of ions
after movement
14
2-10 Limits of the ionic domain of Na-beta-alumina indicated by dotted
lines. The upper part of the sodium activity scale is defined by the n-
electronic conduction parameter a0 and the lower by the p-electronic
conduction parameter ⊕a (1: [29], 2: [30], 3: [31], 4: [32], 5: [33], 6,7:
[5], 8: [34], 9: [35], 10,11: [2], 12: [36])
16
2-11 Limits of the ionic domain of YSZ. The upper part of the oxygen
partial pressure scale is defined by the p-electronic conduction
parameter /p and lower part by the n-electronic conduction para-
meter p0 [37]
17
2-12 Generation of galvani voltage between two phases 20
3-1 Phase diagram of {(1-x1-x2)Na + x1Mo + x2O} at 673-923 K [90, 91] 29
3-2 Schematic phase diagram of the MoO3-Na2MoO4 system based on 30
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vii
[93] constitutes eutectic phase mixture Na2MoO4 + Na2Mo2O7
3-3 Gibbs energy of formation of Na2MoO4 as a function of temperature
from different sources. (1: [96], 2: [75] and 3: [83])
30
3-4 Phase diagram of Na2O-TiO2 system [101] 31
3-5 Standard Gibbs free energy of formation of Na2Ti3O7. (1: [96] and 2:
[103], 3 and 4: [102])
32
3-6 The standard Gibbs energy of formation of Na2Ti6O13 from different
sources plotted against temperature. (1, 2: [66], 4: [2 ] and 3: [103])
33
3-7 Schematic binary phase diagram of the system Na2O-Nb2O5 based
on Reisman et al. [113]. The vertical dashed line indicates the
phases missing in Reisman‘s diagram within 80% of Nb2O5
34
3-8 Schematic binary phase diagram of the system Na2O-Nb2O5 based
on Appendino’s diagram [115]
34
4-1 X-ray diffractogram of the starting material NBA (Asea Brown Boveri
AG)
45
4-2 XRD patterns for commercially obtained yttria-stabilized zirconia
(Friatec)
46
4-3 Schematic sketch of the galvanic cell using Na-ß/ß"-Al2O3 as solid
electrolyte
50
4-4 Schematic sketch of the galvanic cell employed to determine the
thermodynamic stability of the Na-Me-O phase equilibria (Me = Mo,
Nb, Ti) with respect to determination of sodium oxide activity.
51
4-5 Schematic sketch of the galvanic cell employed to determine the
thermodynamic stability of the Na-Me-O phase equilibria (Me = Mo,
Nb, Ti) with respect to determination of sodium oxide activity of Na-
Me-O system.
51
4-6 Schematic set-up of the galvanic cell employed to determine the
thermo-dynamic stability of the Na-Me-O (Me = Mo, Nb, Ti) phase
equilibria with respect to the determination of sodium activity.
52
4-7 Gold sputtering on a NBA pellet and crucible 52
4-8 Galvanic cell set-up in the gas tight quartz tube 53
4-9 Experimental set-up 55
5-1 DTA traces of pure Na2MoO4 and of the eutectic mixture Na2MoO4 + 56
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viii
Na2Mo2O7 prepared from Na2MoO4 and MoO3 with different molar
percentage as shown in the legend. 1 and 2 are the endothermic
and 3, 4 and 5 are the exothermic peaks
5-2 Optical micrograph of the powder compact of the eutectic mixture.
Dark phase is Na2Mo2O7 (a) and bright phase is Na2MoO4 (b)
57
5-3 XRD patterns of the eutectic mixture Na2MoO4 + Na2Mo2O7, before
and after the cell voltage measurement, prepared from 76 mol%
Na2MoO4 + 24 mol% MoO3
58
5-4 Sodium oxide activity of the eutectic mixture Na2MoO4 + Na2Mo2O7
as a function of the sodium activity at various temperatures
(measured by the cell configuration (I))
59
5-5 Logarithm of the activity of sodium oxide dissolved in the phase
mixture Na2MoO4 + Na2Mo2O7 as a function of the inverse
temperature (according to the ordinates height of the Fig. 5-4)
60
5-6 Sudden change of the activity plateau towards higher ordinate values
at 400 °C obtained from cell (I) for the eutectic mixture Na2MoO4 +
Na2Mo2O7
61
5-7 Time dependence of the sodium oxide activity while stepwise
changing the sodium activity from one extreme to another (logarithm
of the sodium activity from =Naalog -17.96 to -14.79 and vice versa
at 475 °C)
62
5-8 Sodium oxide activity of the eutectic mixture Na2MoO4 + Na2Mo2O7
as a function of the sodium activity at various temperatures mea-
sured by cell configuration (II)
63
5-9 Logarithm of the activity of sodium oxide dissolved in the phase mix-
ture Na2MoO4 + Na2Mo2O7 as a function of the inverse temperature
(according to the ordinates height of the Fig. 5-4 and Fig. 5-8)
64
5-10 Voltage of the cell (III) as a function of the sodium activity of the
carbonate electrode at various temperatures
65
5-11 Cell voltage plotted as a function of time at 525 °C for different
sodium activities regime from the one extreme to the other. The
66
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ix
different values of the logarithm of the sodium activity established by
the experimental conditions are displayed
5-12 Temperature dependence of the fitting parameter resulting from the
nonlinear regression of the data of Fig. 5-10 according to Eq. 4-13
(section 4.1.4) 1: [124], 2: [91], 3: [75] and circle represents this
work
67
5-13 Difference of the standard Gibbs energies of formation of Na2MoO4
and Na2Mo2O7 as a function of temperature. 1, 2 and 3 refer to the
results obtained with cell (I), (II) and (III) respectively. Dashed line 4:
[91], solid line 5: [75]
68
5-14 The standard Gibbs free energy of formation of Na2Mo2O7 as a
function temperature. 1, 2 and 3 from the obtained results with the
cell configurations (I), (II) and (III), respectively, (cf. section 4.1.1 and
Table 4-1), (4: [75], 5: [91])
68
5-15 Voltage of the cell (IV) as a function of the CO2 partial pressure at
various temperatures
70
5-16 Comparison of the experimental and theoretical (RT/2F) slopes of
the voltage vs. 2COpln plot (Fig. 5-15)
70
5-17 Voltage of cell (IV) as a function of the CO2 partial pressure at 525
°C. The solid and dotted lines correspond to the experimental and
calculated voltage, respectively
71
5-18 Voltage of cell (IV) as a function of the CO2 partial pressure at 450
°C. The solid is experimental and the dotted line is calculated
71
5-19 The result of p-electronic conduction parameter on sodium-beta-
alumina obtained from (IV) as a function of sodium activity in the
measuring electrode
72
5-20 Time-dependent change of the logarithm of the sodium oxide activity
as a function of the logarithm of the sodium activity after heating up
the cell and, for the first time exposing the as-prepared cell
components to the measuring conditions at 550 °C
74
5-21 Sodium oxide activity of the phase mixture Na2Ti3O7 + Na2Ti6O13 as
a function of sodium activity at various temperatures obtained from
75
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x
cell (I)
5-22 Logarithm of the activity of sodium oxide dissolved in Na2Ti3O7 +
Na2Ti6O13 plotted as a function of the inverse temperature. 3:[66] and
4: [103]
76
5-23 Sodium oxide activity of the phase mixture Na2Ti3O7 + Na2Ti6O13 as
a function of sodium activity at various temperatures obtained from
cell (II)
77
5-24 XRD pattern of the phase mixture Na2Ti3O7 + Na2Ti6O13 before and
after cell voltage measurement
78
5-25 The standard Gibbs energy of formation of the phase Na2Ti6O13 as a
function of temperature. 1, 2: [66], 3: [2] and 4: [103]
79
5-26 Voltage of cell (III) for the phase mixture Na2Ti3O7 + Na2Ti6O13 as a
function of sodium activity of the carbonate electrode at various
temperatures
80
5-27 Logarithm of sodium oxide activity of the phase mixture Na2Ti7O13 +
TiO2 as a function of sodium activity obtained from cell (I)
82
5-28 Logarithm of sodium oxide activity of the phase mixture Na2Ti7O13 +
TiO2 as a function of sodium activity obtained from cell (II)
82
5-29 Voltage of cell (I) for the phase mixture Na2Ti7O13 + TiO2 as a
function of CO2 partial pressure
83
5-30 XRD patterns of the synthesized eutectic phase mixture NaNbO3 +
Na3NbO4
85
5-31 DTA trace of the synthesized eutectic mixture NaNbO3 + Na3NbO4
[113, 114]
86
5-32 Phase distribution of the eutectic mixture NaNbO3 + Na3NbO4 86
5-33 EDX spectra for the eutectic mixture NaNbO3 + Na3NbO4 87
5-34 Voltage of the cell configuration (I) for the mixture Na3NbO4 +
NaNbO3 is plotted as a function of CO2 partial pressure
88
5-35 Sodium oxide activity of the eutectic mixture obtained from cell
voltage measurement is plotted as function of the sodium activity
established at the interface between the carbonate pellet and the gas
mixture
88
5-36 Voltage of cell (III) for the eutectic mixture NaNbO3 + Na3NbO4 as a 90
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xi
function of the sodium activity of the measuring electrode at various
temperatures
5-37 Sodium activities of the eutectic phase mixture Na3NbO4 + NaNbO3,
determined from the zero voltage in Fig. 5-36, are plotted against
inverse temperature
91
5-38 Difference of the Gibbs energy of formation of the eutectic mixture
NaNbO3 + Na3NbO4 determined from the measured value of
equilibrium sodium activity and the corresponding oxygen partial
pressure, plotted against temperature. Solid line: computed values
from Lindemer et al. [97] estimated entropy and enthalpy values of
these phases
92
5-39 Time-dependent change of the logarithm of the sodium oxide activity
as a function of the logarithm of the sodium activity of the phase
mixture NaNbO3 + Na2Nb4O11 after heating the cell and, for the first
time exposing the as-prepared cell components to the measuring
conditions at 550°C
94
5-40 Sodium oxide activity of the phase mixture NaNbO3 + Na2Nb4O11 as a
function of sodium activity established at the interface between the
carbonate pellet and gas phase at various temperatures
95
5-41 Sodium oxide activity of the phase mixture Na2Nb4O11 + NaNb3O8 as
a function of sodium activity established at the interface between the
carbonate pellet and gas phase at various temperatures
96
5-42 Sodium oxide activity of the phase mixture NaNb3O8 + Na2Na8O21 as
a function of sodium activity established at the interface between the
carbonate pellet and gas phase at various temperatures
96
5-43 Logarithm of the activity of sodium oxide dissolved in the phase
mixtures as a function of the inverse temperature. A = NaNbO3 +
Na2Nb4O11, B = Na2Nb4O11 + NaNb3O8 and C = NaNb3O8 +
Na2Nb8O21
97
5-44 Temperature dependence of the difference between the standard
Gibbs energy of formation of NaNbO3 +Na2Nb4O11, Na2Nb4O11 +
NaNb3O8 and NaNb3O8 + Na2Na8O21 determined from the ordinate
height of Fig. 5-40, Fig. 5-41 and Fig. 5-42 according to the
98
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xii
equations in Table 4-3 (section 4.1.1)
5-45 The potential differences of the cell (III) for the phase mixture
NaNbO3 + Na2Nb4O11 as a function of sodium activity established at
the carbonate electrode
100
5-46 The difference of the Gibbs free energy of formation of the phase
mixture NaNbO3 + Na2Nb4O11. The circles indicate the values
obtained from cell (I) and the rectangles those obtained from cell (III)
through nonlinear regression procedure
100
5-47 Sodium oxide activity of the phase mixture NaNb3O8 + Na2Nb8O21
obtained from the cell configuration (II) as a function sodium activity
at various temperatures
101
5-48 Sodium oxide activity of the phase mixture NaNb3O8 + Na2Nb8O21
obtained from the cell configuration (I) and (II) as a function of
inverse temperature
103
5-49 Difference of Gibbs free energy of formation of the phase mixture
NaNb3O8 + Na2Nb8O21 obtained from the cell configuration (I) and (II)
104
5-50 The standard Gibbs energy of formation of difference phases of Na-
Nb-O system obtained from the individual cell configuration is plotted
as a function of temperature. NaNbO3: Lindemer et al. [97]
estimation, Na3NbO4: cell (III), Na2Nb4O11: cell (I) and (II), NaNb3O8:
cell (I) and Na2Nb8O21: cell (I) and (II)
105
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xiii
List of Tables
Table Contents Page
2-1 Defect formation reactions along with their mass action law 9
2-2 Sodium chemical potential dependence of the concentration of charge
species in different regions of Brouwer diagram for undoped Na-beta-Al2O3
12
2-3 Defect formation reactions along with their mass action law [25] 12
2-4 Oxygen pressure dependence of the concentration of charge species in
different regions of the Brouwer diagram for yttria-stabilized zirconia [25]
13
3-1 e. m. f. of two phase sodium alloy systems at 120 °C (vs Na). M (Na)
represents M saturated with sodium [62]
26
3-2 Heterogeneous phase equilibria of ternary systems which are used as
reference electrode in alkali concentration cell in potentiometric measure-
ments
28
4-1 Cell configurations for characterization of the Na-Me-O systems (Me = Mo,
Ti, Nb)
36
4-2 Equilibrium reactions and thermodynamic expressions in terms of sodium
activity (assuming that the activity of the constituents phases is unity
except sodium)
36
4-3 Equilibrium reactions and thermodynamic expressions in terms of sodium
oxide activity (assuming that the activity of the constituents phases is unity
except sodium oxide)
37
4-4 Solid electrolytes for characterizing the phase mixtures 45
4-5 Eutectic chemical composition used for the preparation of phase mixtures
Na2MoO4 + Na2Mo2O7
46
4-6 Details of the preparation of single phases of the Na-Nb-O system 47
4-7 Preparation of Na2Ti3O7 and Na2Ti6O13 48
4-8 Conditions for fabrication of the bi-phasic solids 48
4-9 Sputtering conditions 52
5-1 Sodium activity of the eutectic mixture determined from the zero line
crossing of the Fig. 5-36 and corresponding oxygen partial pressures at
respective temperatures
91
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1. Introduction
One of the major applications of sodium-metal-oxygen systems is as electrode
material in combination with suitable solid electrolytes for potentiometric measure-
ments. The electrode material is an important component of a galvanic cell mea-
surement. As far as sodium ion conducting solid electrolytes are concerned the
level of sodium activity in the electrode systems plays an important role for
accurate data characterization with such electrolytes.
A cell can promise the desired properties if composed of an electrode with thermo-
dynamically well defined phases stable under the operational conditions and free
from any impact of electronic transference through the electrolyte.
The level of activity of elemental sodium lies almost within the ionic domain of
sodium ion-conductors and electronic conduction parameter may have no pro-
nounced effect but a lot of other problems arise to use it as reference electrode.
An alternative to the application of elemental sodium is the use of binary sodium
intermetallic compounds which were studied as electrode materials for batteries.
The level of sodium activity in the alloy systems is favourable in combination with a
Na-ion-conductor. But the major complication of using these binary alloys as
reference electrodes is the narrow temperature range over which they are solid.
Possible alternative materials are equilibria of phases in ternary systems. An im-
portant group of materials consists of compounds of alkaline metals with transition
metal oxides to form compounds of the type AXMeYOZ (A = Li, Na, K; Me =Ti, Nb,
Mo, V, Cr, Fe etc.). Thermodynamics characterization is essential to popularize
them to be used as attractive reference materials and other applications.
The electrochemical characterization of the thermodynamic stability of hetero-
geneous phase equilibria comprising sodium-containing compounds is usually
accomplished by the determination of the sodium activity of these phase mixtures,
for instance, by means of potentiometric measurements on galvanic cells using a
sodium ion conductor as the solid electrolyte e. g. Na-beta-Al2O3, NASICON. The
level of the sodium activity to be determined is extremely low. It comes close to the
lower limit of the ionic conduction domain of the electrolytes employed for such
measurements. If Na-(ß+ß")-Al2O3 [1, 2, 3] and NASICON [4] exhibit a non-
negligible extent of electronic conduction under oxidizing conditions, e.g. if exposed
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2
to sodium carbonate and CO2/O2 gas atmosphere, this would have serious con-
sequence for the performance of CO2 sensors or Na/S battery. Therefore, the
contradiction arises.
The p-electronic conduction parameter of these ion conductors is a function of the
chemical potential of the species in the surroundings that is the neutral counterpart
of the mobile ion in the electrolyte [5, 6 7]. As a consequence, the effect of
electronic transference appears to be less pronounced or even not present. This
phenomenon creates confusion in the literature about the role of electronic
conduction.
As a contribution to overcome this confusion and in order to get reliable data on not
yet characterized phase equilibria, an approach has to be applied which definitely
eliminates the electronic impact if there is any and simultaneously allows to in-situ
check the co-existence of the phases under investigation which is the important
criterion in potentiometric measurements on galvanic cell for reliable thermo-
dynamic data evaluation.
The objective of the present work is to electrochemically characterize the thermo-
dynamic properties of sodium containing phase equilibria with extremely low activity
at elevated temperatures. These findings open a pathway to the characterization
not only of sodium containing compounds but also of other phase equilibria.
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3
2. Theoretical background
2.1 Electrochemical phenomena
A galvanic cell usually consists of two electrodes in contact with an ionic conductor
which generates a difference of the electrical potential between the surfaces of an
electrolyte as a result of the spontaneous reaction occurring in the electrodes. The
potential difference directly permits the determination of Gibbs energy or chemical
potential differences.
Ideal solid electrolytes should have the following characteristics:
i. Ionic crystal bonding;
ii. Principal charge carriers are ions which means the ionic transference number
(tion) is nearly unity.
Point defects are primarily responsible for the electrical conduction in solid electro-
lytes. Ionic solids contain these defects at all temperatures above 0 K [8].
Aliovalent impurities also introduce excess defects whose concentrations are fixed
mainly by the compositions of the impurities.
2.2 Structure of solid electrolyte
Yttria-stabilized zirconia (YSZ) and Na-beta-alumina (NBA) have been selected as
solid electrolytes in the present investigation. To understand the properties of
these solid electrolytes their structure is discussed here.
2.2.1 Structure of beta alumina
Beta-aluminas are ceramic oxides composed of Na2O and Al2O3, often with small
amounts of MgO and/or Li2O as dopants. The most of the information concerning
the structure of alkali beta-alumina has been obtained from X-ray diffraction,
although in recent years several other experimental techniques have been applied
to determine details of the structure and the properties resulting from the con-
duction properties. The basic crystal structure was revealed by Beevers and Ross
[9] in 1937. The structure of Na-beta-alumina is shown in Fig. 2-1. Al3+ and O2- are
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4
packed in the same fashion as in MgAl2O4 spinel of the oxygen sub-lattice usually
called as “spinel block”. Al3+ ions occupy the octahedral sites; the tetrahedral sites
are occupied by Mg2+ ions if present. The spinel blocks are separated from each
other by loosely packed planes containing Na+ and O2-.
Fig. 2-1 Structure of Na-β -Al2O3 (left) and Na- ''β -Al2O3 (right)
Due to the loose packing, space is available for movement of the alkali ions
leading to a high ionic conductivity. The conductivity is limited to this plane since a
movement of ions along the c axis is exceedingly difficult. The material, therefore,
is highly anisotropic.
The conduction plane of Na-β -Al2O3 is a mirror plane, with a face centred cubic
packing arrangement of oxide ions in the ambience shown in Fig. 2-2. This
-
5
packing arrangement is slightly different in Na- ''β -Al2O3 since in this phase the
conduction plane is not a mirror plane. As can be seen in Fig. 2-2, it takes three
spinel-type blocks before the stacking arrangement is repeated, and for this
reason, Na- ''β -Al2O3 is called a “3-block” while the Na-ß -Al2O3 is called a “2-block”
material. Other modifications of the spinel block stacking arrangement have been
reported [10] and given the names '''β and ''''β -Al2O3.
Although spinel is cubic, the conductive alkali inter-layers lead to a hexagonal
crystal structure for Na-ß -Al2O3 and a rhombohedral structure for Na- ''β -Al2O3.
.
Fig. 2-2 Ion packing arrangement in ß-Al2O3 (left) and 'ß' -Al2O3 (right) (letters refer
to stacking arrangements where ABC represent face-centred cubic packing while
ABAB represents hexagonal packing)
The lattice constants of Na- ''β -Al2O3 are a = 0.559 nm and c = 2.253 nm and of
Na-ß -Al2O3 are a = 0.559 nm and c = 3.423 nm. The most probable position of
ACBA
ABCA
ACBA
Mirror plane
Mirror plane
O2- Na+
ß-Al2O3
ACBA
ACBA
BACB
CBAC
Single cell
ß"-Al2O3
CAxis
-
6
sodium ions for the Na-ß -Al2O3 have been determined by Beevers and Ross [9] is
shown in Fig. 2-3 and is called BR (”Beevers Ross”) position.
Fig. 2-3 Ideal structure of the conducting plane of beta alumina. Solid circles are
column oxygen ions; open circles are mobile cations on BR sites; unoccupied
hexagon vertices are aBR sites; and sites between neighbouring BR and aBR are
mO sites. A mobile cation in ideal structure is in a deep potential well indicated by
dotted lines
The ideal composition of Na-ß-Al2O3 is NaAl11O17. In this stoichiometric com-
position all the sites should be filled. However, Felsche [11] found that the sodium
sites are only partially occupied. Even though Peters et al. [12] studied typical
crystals containing 29% excess sodium and concluded that the sodium is smeared
out from the conduction plane into the spinel blocks. They postulated that excess
sodium is charge compensated by aluminium vacancies. Therefore, the formula
can be written more accurately as Na1+xAl11-x/3O17 where x is usually 0.15-0.30.
Two possible positions for the excess sodium are shown in Fig. 2-3. The sites
-
7
labelled aBR refer to “anti Beevers-Ross” positions. The other positions lie
between the oxide ions and are labelled mO for “mid-oxygen”.
Peters et al. [12] measured the electron density due to Na+ and found that the BR
sites were only 75% occupied. The remaining Na+ electron density was found in a
diffuse fashion around the mO sites. No Na+ was found at aBR sites.
2.2.2 Structure of yttria-stabilized zirconia (YSZ)
Among the crystal structures of oxide solid electrolytes, the fluorite crystal
structure is attractive for useful electrolytes since it exhibits very high oxygen-ion
conductivity [13]. The fluorite structure is a face-centered-cubic arrangement of the
cations with the anions occupying all the tetrahedral sites. In this structure each
metal cation is surrounded by eight oxygen ions, and each oxygen ion is
tetrahedrally coordinated with four metal cations. In this fluorite structure all the
octahedral sites are empty. Thus it is a rather “open” one, and rapid interstitial
diffusion might be expected along the octahedral sites [13]. Among the fluorite-
type materials, ZrO2- and ThO2-based electrolytes have been studied most
extensively and found suitable for a wide range of applications like fuel cells,
oxygen monitors, oxygen pumps, and for various thermodynamic and kinetic
measuring devices. ZrO2-based electrolytes have the advantage of higher
conductivity at a given temperature and are used over a wide range of oxygen
partial pressure [8].
Pure ZrO2 has three well-defined polymorphs, i.e., the monoclinic, tetragonal and
cubic structure [14] (Fig. 2-4). The monoclinic phase is stable up to about 1100 °C,
where it transforms over a 100 K temperature range to the tetragonal phase [15];
at 2370 °C, the compound adopts the cubic fluorite structure (Fig. 2-4), in which
oxygen ions are located on a primitive cubic structure inside the face-centred cubic
structure of zirconium ions. When lower-valent cations (such as Y3+, Ca2+, Mg2+)
are incorporated into ZrO2, the cubic fluorite structure is stabilized to lower
temperatures, referred to as stabilized zirconia. If the content is high enough the
cubic structure is stable down to room temperature. The substitution of such lower
valent cations on the Zr4+ sublattice sites creates oxygen vacancies. As one anion
vacancy is produced for every pair of trivalent cations, the corresponding con-
-
8
centration of anion vacancies in the ZrO2 lattice can be as high as 4-8 % [14]. The
oxygen deficiency leads to the high ionic conductivity.
(a) Cubic (b) Tetragonal (c) Monoclinic
Fig. 2-4 Structure of the cubic, tetragonal and monoclinic ZrO2 phase [16]
The solid solutions formed by doping ZrO2 with Y3+ can be written as Zr1-xYxO2-x/2
[17]. The use of yttria-stabilized zirconia as a solid oxide electrolyte goes back to
Nernst, who in 1899 invented the “Nernst light” [18]. This electrolyte was also used
in the first solid oxide fuel cell constructed by Bauer and Preis in 1937 [19].
2.3 Defect Chemistry
In 1956 Kröger and Vink [20, 21] proposed the commonly used nomenclature for
the description of defects. The point defects are considered as dilute species and
the solid as the solvent.
There are essentially three ways of establishment of equilibrium of defects in ionic
crystals:
1. Intrinsic defect equilibria. This includes Frenkel and Schottky defect equili-
bria.
2. Doping; i.e. the intentional manipulation of defect types and concentration
by the incorporation of specific dopant into the bulk of a crystal.
3. Defect reactions at interfaces, e.g. the incorporation of species from the
“outside” into the crystal via defects or the opposite, the loss of atoms to the
ambience generating defects in crystal.
-
9
In cation conductors, like beta-alumina and NASICON lattice disorders occur pre-
dominantly in the cation sub-lattice. Frenkel defects [22, 23], i.e. pairs of metal
interstitials ( •iM ) and metal vacancies ('MV ), are highly mobile while the anions are
immobile. In the following, the discussion will be focused on Na-ß-Al2O3 as solid
electrolyte which is also valid for other monovalent cation conductors. The charged
sodium defects may be compensated by electronic carriers, such as electrons (e')
and holes (h•) or by charged ionic defects. It is assumed here that the con-
centration of interstitial sodium ions is much larger than that of electrons and
defect electrons.
The relevant defect formation reactions in Na-ß-Al2O3, along with their mass action
relations, assuming dilute solution, are given in Table 2-1.
Table 2-1 Defect formation reactions along with their mass action law
Type of reaction Reaction Law of mass action Eq.
Intrinsic defect formation 'NaiiNa VNaVNa +↔+•
]V].[Na[
]V].[Na[K
iNa
'Nai
F
•
= 2-1
•++↔ hVNaNa 'NaNa NaNa
'k
p a]Na[
]h].[V[K
•
= 2-2a
'eNaVNa Na'Na +↔+
1Na'
Na
Nan a
]V[
]'e].[Na[K −= 2-2b
Electron-hole generation •+↔ h'e0 ]h].['e[Ke•= 2-3
recombination
Where Naa is the sodium activity and Vi and 'NaV are the interstitial and sodium
vacancies, respectively. [ ] denotes the concentration of ion or electron defects
and KF, Ke, Kp and Kn are the constants of mass action equilibria, having the
general form of
∆−= ο
RT
HexpK)T(K , Eq. 2-4
where οK includes an entropy term, H∆ is the reaction enthalpy, T is the absolute
Interaction with the
surroundings
-
10
temperature and R is the gas constant. In addition, the electro-neutrality condition
has to be taken into account, i.e.
]Na[]h[]'e[]V[ i'Na
•• +=+ . Eq. 2-5
Eqs. 2-1, 2-2, 2-3 and 2-5 allow to calculate the defect concentration as a function
of the sodium chemical potential and temperature. The defect concentration [D] in
a solid electrolyte as a function of the chemical potential of the neutral species in
the ambience of solid electrolyte is represented by the Brouwer diagram [24]. Fig.
2-5 represents the Brouwer diagram for Na-ß-Al2O3.
Fig. 2-5 Brouwer diagram for undoped Na-β -Al2O3
The Brouwer diagram can be divided into three different regions. In each of them,
one of the defects on either side of Eq. 2-5 controls the neutrality equation, and
thus, the sodium activity dependence of the defects involved.
Due to the interaction of the solid with the surroundings the charge neutrality is
maintained by decreasing concentration of the negatively charged sodium
vacancies (Eq. 2-2b) and increasing the number of the positively charged sodium
interstitials (Eq. 2-2a). In the extreme case
]Na[]'e[ i•=]Na[]V[ i
'Na
•=]V[]h[ 'Na=•
]V[ 'Na
]h[ •
]Na[ i•
]'e[
]Na[ i•
]'e[
]V[ 'Na
]h[ •
Naalg
lg[D
]
]Na[]'e[ i•=]Na[]V[ i
'Na
•=]V[]h[ 'Na=•
]V[ 'Na
]h[ •
]Na[ i•
]'e[
]Na[ i•
]'e[
]V[ 'Na
]h[ •
Naalg Naalg
lg[D
]lg
[D]
-
11
]V[]h[ 'Na=• Eq. 2-6
is considered for charge neutrality. Inserting the new neutrality condition into the
mass action Eqs. 2-1, 2-2 and 2-3, the defect concentration in the solid electrolyte
can be calculated.
In the vicinity of the stoichiometric point, the concentration of sodium ions on
interstitial sites and vacancies are much larger than the concentrations of
electrons and defect electrons, the relative changes in the concentrations of
electronic charge carriers are much larger than that of the interstitial ions ( •iNa )
and the vacancies (middle of Fig. 2-5). Hence the concentrations of sodium ion
vacancies and sodium ions in the interstitial site may be considered as virtually
being constant
.const]Na[]V[ i'Na ==
• Eq. 2-7
Applying the law of mass action to Eq. 2-2a and taking Eq. 2-7 into consideration
one obtains the relation
1Naa]h[−• ∝ . Eq. 2-8
In a similar fashion the expression for the concentration of the electrons in this
range can be obtained by incorporating Eq. 2-2b:
Naa]'e[ ∝ . Eq. 2-9
The situation is different for large deviations from ideal stoichiometry. If the
chemical potential of sodium is very high, the concentration of the holes and
vacancies may be neglected, hence the electro-neutrality condition Eq. 2-5
reduces to
]Na[]'e[ i•= . Eq. 2-10
Applying the law of mass action to Eq. 2-2b and taking into account the above
relation one obtains
[e’] 21
Naa∝ . Eq. 2-11
These results based on the considerations stated above are summarized in Table
2-2 and shown schematically in Fig. 2-5.
In case of doped material (like yttria-stabilized zirconia or magnesium stabilized
Na-beta-Al2O3), the relevant defect formation reactions along with their mass
-
12
action relationship, concentration of charge species in different regions of the
Brouwer diagram can be established as well.
Table 2-2 Sodium chemical potential dependence of the concentration of charge
species in different regions of Brouwer diagram for undoped Na-beta-Al2O3
Electro-neutrality condition
(limit case)Naa ]V[
'Na ]Na[ i
• ]h[ • ]'e[
]V[]h[ 'Na=• low 21
Naa−
∝ 21
Naa∝2
1
Naa−
∝ 21
Naa∝
]Na[]V[ i'Na
•= middle .const≅ .const≅Naa
1∝ Naa∝
]Na[]'e[ i•= high 21
Naa−
∝ 21
Naa∝2
1
Naa−
∝ 21
Naa∝
In a similar way defect formation reactions along with mass action law for yttria-
stabilized zirconia is given in Table 2-3 and oxygen pressure dependence of the
concentration of charge species in different regions of the Brouwer diagram is
given in Table 2-4 and plotted in Fig. 2-6.
Table 2-3 Defect formation reactions along with their mass action law [25]
Type of reaction Reaction Law of mass action Eq
Intrinsic defect formation ''iOiO OVVO +↔+••
]V].[O[
]O].[V[K
iO
''iO
F
••
= 2-12
'e2VO21
O O2O ++↔•• 2
1
OO
2O
p 2p
]O[
]'e].[V[K
••
= 2-13a
••• +↔+ h2OVO21
OO2 21
OO
2O
n 2p
]V[
]h].[O[K
−
••
•
= 2-13b
Electron-hole generation •+↔ h'e0 ]h].['e[Ke•= 2-14
recombinationThe dissolution of yttria into the
fluorite phase of zirconia •••• ++→++ OO'ZrZr5.1 V2
1O5.1YVVYO O
'''' 2
const]Y[ 'Zr =
Interaction with thesurroundings
-
13
Table 2-4 Oxygen pressure dependence of the concentration of charge species in
different regions of the Brouwer diagram for yttria-stabilized zirconia [25]
Electro-neutrality condition
(limit case)2Op ]V[ O
•• ]O[ ''i ]h[ • ]'e[
]V[2]'e[ O••= low 61
O2p
−∝ 6
1p
2O∝ 6
1p
2O∝ 6
1p
−∝
2O
]O[]V[ ''iO =•• medium .const≅ .const≅ 41
O2p∝ 4
1
O2p
−∝
]'Y[]V[2 ZrO =•• medium .const≅ .const≅ 41
O2p∝ 4
1p
−∝
2O
.const]'Y[]h[ Zr ==• high 21p
−∝
2O2
1
O2p∝
.const≅ .const≅
]O[2]h[ ''i =• high 61p
−∝
2O6
1p
2O ∝6
1p
2O∝ 6
1p
−∝
2O
Fig. 2-6 Brouwer diagram for yttria-stabilized zirconia [25]
-
14
2.4 Conductivity
The materials under consideration exhibit two types of conductivity: electronic and
ionic. As conduction mechanisms for ionic motion three types are regarded as
important in ionic conductors, these are vacancy diffusion, interstitial mobility and
interstitialcy motion. The vacancy mechanism involves movement of atoms
through the crystal in which lattice atoms jump to neighbouring unoccupied lattice
sites (Fig. 2-7). The interstitial mechanism involves jumps of particles directly from
one interstitial site to another (Fig. 2-8). The interstitialcy mechanism involves the
displacement of a particle located on a regular lattice site into the interstitial lattice
by an interstitial particle, which itself then occupies the regular site (Fig. 2-9).
Fig. 2-9 Interstitialcy mechanism showing two
possible locations of ions after movement
Fig. 2-7 Vacancy mechanism for transport of ions
Fig. 2-8 Interstitial mechanism for transport of ions
-
15
The electrical conductivity ( s ) of a solid is related to its defect concentration by
[26]:
hheeii ii pqnqqc µ+µ+µ∑=σ Eq. 2-15
where c is the ionic defect concentration, q is the charge, µ is the mobility (the
mean particle velocity per unit potential gradient), and the subscripts i, e and h
denote ions, electrons and holes, respectively.
Electronic defects can be formed by thermal excitation of electrons from the
valence band to the conduction band. Equilibrium between free electrons (e') in
the conduction band and electron holes in the valence band can be expressed by
Eq. 2-3. The expression for thermal equilibrium of electrons and holes leads to the
equation [13]:
−∝= •
kT
E exp]h]['e[K g1 Eq. 2-16
where [e'] and ]h[ • denote the concentration of electrons and electron holes,
respectively, K1 is the equilibrium constant and Eg the energy difference between
the valence and conduction band (band gap energy) and k the Boltzmann
constant.
2.5 Electrolytic domain
The region with respect to temperature and activity or partial pressure of the
potential determining species of the electrolyte within which the electronic
contribution to the total conductivity is less than 1 % is called the electrolytic
domain [27]. In contrast, the ionic domain is much wider than the electrolytic
domain in an alkali ion conductor by two orders of magnitude and in an oxygen ion
conductor by eight orders of magnitude [23].
2.5.1 Ionic domain of Na-ß/ß"-Al2O3 (NBA)
The ionic domain of NBA is depicted in Fig. 2-10. The parameters /a and Va
represent those sodium-activities at which the electron conductivities ps and ns ,
respectively, are equal to the ionic conductivity [28]. Thus, they are the limits of
-
16
the sodium-activity range of prevailing ionic conduction, i.e. the limits of the ionic
domain of the electrolyte [27].
The sodium chemical potential of ternary oxides is usually extremely low and thus
it is close to the lower limits of the ionic domain of the electrolyte. Therefore, any
attempt to measure it by means of a galvanic cell using NBA as solid electrolyte
may provide erroneous result due to partial electronic short circuit within the cell.
8 10 12 14 16 18 20 22 24 26-50
-40
-30
-20
-10
0
10
T [°C]
800 600 400 200 150300
1/ T [10-4 K -1]
log
aN
a
31
24
5
6
7
11
8
10
9
12
a/
aV
Fig. 2-10 Limits of the ionic domain of Na-beta-alumina indicated by dotted lines.
The upper part of the sodium activity scale is defined by the n-electronic con-
duction parameter a0 and the lower by the p-electronic conduction parameter ⊕a
(1: [29], 2: [30], 3 :[31], 4: [32], 5: [33], 6,7: [5], 8: [34], 9: [35], 10,11: [2], 12: [36])
-
17
-50
-40
-30
-20
-10
0
10
20
30
5 6 7 8 9 10 11 12
log
(p O
[P
a])
2
1/T [10-4 K-1]
YSZ
p0
p/
Fig. 2-11 Limits of the ionic domain of YSZ. The upper part of the oxygen partial
pressure scale is defined by the p-electronic conduction parameter /p and lower
part by the n-electronic conduction parameter p0 [37]
2.5.2. Ionic domain of yttria-stabilized zirconia (YSZ)
The ionic domain of YSZ is shown in Fig 2-11 [37]. The parameters /p and p0
represent those oxygen partial pressure at which the electron conductivities ps
and ns , respectively, are equal to the ionic conductivity [28]. The hatched areas
represent the scattering regions of these parameters [37].
-
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-
19
αϕ the electrical potential of phase a.
The electrochemical potentials of two phases I and II according to equation Eq. 2-
21 is:
Phase I ⇔ Phase II
IkI
kI
k Fz η=ϕ+µ II
kII
kII
k Fz ϕ+µ=η .
When electrochemical equilibrium is established, IIkI
k η=η .
Therefore, )(Fz IIIkII
kI
k ϕ−ϕ=µ−µ .
The most general formulation of the condition for electrochemical equilibrium is:
0dv......dvdvk
Nk
N
kk
IIk
IIk
k
Ik
Ik =ξ∑ ∑ η++ξη+ξ∑ η . Eq. 2-22
2.7 Galvani voltage
According to the condition of electrochemical equilibrium between two phases I
and II:
( ) ( ) 0FzFz ''k''kk
''k
k
'k
'kK
' =ϕ+µ∑ ν+∑ ϕ+µν . Eq. 2-23
Therefore,
( ) ( )∑ µν+µν−=∑ ϕν+ϕνk
''k
''k
'k
'k
k
'k
'k
''k
''k FzFz . Eq. 2-24
The sum of charges disappearing and emerging in phase I are equal to the sum of
charges disappearing and emerging in phase II. From this statement rz , the
charge number of the electrode reaction r can be defined as [38]:
∑ ν=∑ ν=k
k''k
kK
'kr zzz - . Eq. 2-25
From the equation Eq. 2-24 and Eq. 2-25 the electrical potential defference can be
written as:
( )FzFz r
kkk
r
K
''k
''k
'k
'k
'''∑ µν
=∑ µν+µν
=ϕ−ϕ and Eq. 2-26
( )''''','eqU ϕ−ϕ= . Eq. 2-27'','
eqU is equilibrium Galvani voltage.
-
20
Chemical potential of the species k can be given by:
kkk alnRT+µ=µο .
Where οµk is the standard chemical potential of k (reference state) and ka is the
activity referred to okµ .
From the equation Eq. 2-26, Eq. 2-27 and the chemical potential of k, equilibrium
galvani voltage can be written as:
∏+= νοk
kr
'','eq
k'','
aln.Fz
RTUU Eq. 2-28
( οο µ∑ ν= kk
krFz
1U
'',' standard Galvani voltage).
ϕ
'','eqU '','
eqU
2.8 Cell voltage with electronic transference
According to Wagner [39, 40] the voltage, U, of a solid electrolyte (SE) galvanic
cell in the most general form is calculated from the balance of the ion and electron
charge carrier current densities. The generalized version of the Wagner cell
voltage equation for an arbitrary electrolyte with one sort of mobile ion reads as
follows.
∫ µ
σσ
+ξ=
ξ
ξ
ξ
µ
µ
''X
'X
X
i
eid
1
1Fz
1U - . Eq. 2-29
ξX , ξ and ξµX stand for the neutral particle corresponding to the ion i, the number
I II III
Fig. 2-12 Generation of
Galvani voltage between two
phases
-
21
of X-atoms associated in the standard state and the chemical potential of ξX ,
respectively. The superscripts ' and " denote the positions of the electrolyte
surfaces which are identical with the reference and measuring electrode,
respectively, of the galvanic cell.
Within a mixed ionic electronic conductor a local equilibrium can be assumed to
exist between the ionic charge carriers, the respective neutral particles and the
electrons, e' [39]:
ξξ⇔+ X
1'ezi i . Eq. 2-30
Hence, the chemical potentials of these species are interrelated:
ξµ
ξ=µ+µ Xeii grad
1gradzgrad . Eq. 2-31
As solid electrolytes are heavily doped materials with a high concentration of
mobile ionic charge carriers, the chemical potential of the ions may be assumed to
be practically constant with respect to changes of the chemical potential of ?X
throughout the electrolyte [41, 42]:
0grad i ≈µ . Eq. 2-32
The electronic conductivity may be due to electrons and/or holes:
pne s+s=s . Eq. 2-33
The chemical potentials of the electrons and holes are interrelated by the intrinsic
electronic defect equilibrium:
pn -gradgrad µ=µ . Eq. 2-34
Usually in solid electrolytes the concentration of electronic charge carriers, ce, is
very small, thus the activity can approximately be replaced by the concentration:
ee clnRTgradgrad ≈µ . Eq. 2-35
With Eqs. 2-32, 2-34 and 2-35, the relationship for the chemical potential de-
pendence of the concentration of the electronic charge carriers can be derived by
integrating Eq. 2-31. Assuming that the mobility of the electronic charge carriers is
independent of the activity-Xξ , this relationship reads in terms of the partial
conductivities:
-
22
ξ
σ=σ
ξ
⊕ iX
z
1
aa
ip . Eq. 2-36a
and
ξ
σ=σ i?X z
1
a
ain .
V. Eq. 2-36b
Substituting Eqs. 2-36a and 2-36b into Eq. 2-31 enables the Wagner equation to
be integrated [43, 44] under the boundary condition:
( )4
1iz
1
aa
-
23
( ) ( )( ) ( )/V
V/
aaaa
aaaaln
F
RTU
'Me
''Me
'Me
''Me
++
++−= Eq. 2-41
where ''Mea is the metal activity of the measuring electrode and 'Mea is the metal
activity of the reference electrode.
The n-type conductivity prevails at high Me-activities and the p-type conductivity at
low activities. The consequence is that Va is larger than /a by several orders of
magnitude and larger than the Me activities of the electrode used in the present
work:
'Me
''Me a,aa >>V . Eq. 2-42
(Eq. 2-41) can be further simplified:
/
/ aa
aaln
F
RTU
'Me
''Me
+
+−= . Eq. 2-43
-
24
3. Literature survey
3.1 General description
Thermodynamics provides a useful tool for predicting chemical stability of mater-
ials and compatibility with other materials, particularly at high temperatures. Thus
there is a continuous need for accurate thermodynamic data for existing and future
ceramic materials. In the production practice and scientific research they play an
important role.
Among the thermodynamic functions, Gibbs free energy is the most informative
function. There are different methods of measuring Gibbs free energies of sub-
stances and reactions:
1. Computation from heats of formation, entropies and specific heats, resulting
from calorimetric techniques.
2. Measurement of the equilibrium constant of reactions, using spectroscopic
techniques.
3. Potentiometric (e. m. f.) measurements.
Among these methods, potentiometric technique is the most promising, reliable,
versatile and widely used method.
Solid state electrochemical measurements can be carried out by employing both:
(a) cation conductors and
(b) anion conductors.
Kiukkola and Wagner [45] demonstrated the use of calcia stabilized zirconia (CSZ)
as a solid electrolyte in equilibrium e. m. f. measurements for the determination of
thermodynamic properties of oxides at high temperatures. Subsequent to their
pioneering work, a large number of galvanic cell studies were reported in the
literature making use of Daniel type cells [46-58].
The present work has been focused on the characterization of sodium transition
metal oxide system (Na-Me-O; Me = Mo, Ti, Nb) by potentiometric technique. In
this technique mainly two criteria must be fulfilled to get accurate thermodynamic
stability data:
-
25
1. The cell voltage must be free from any impact of electronic transference
through the solid electrolyte and
2. The phase mixture to be characterized must be stable and the constituent
phases must co-exist under the operating conditions.
Sodium ion conductors are usually used as solid electrolyte for the characteriza-
tion of Na-metal-oxide system and Na2CO3/CO2, O2 is often used as measuring
electrode. The level of sodium activity either at measuring electrode or at counter
electrode plays an important role for the electronic conductivity properties of the
solid electrolyte.
3.2 Elemental sodium
The level of sodium activity in elemental sodium lies within the ionic domain of
sodium ion conductors. Therefore, it can be used as reference electrode for
sensors or the evaluation of thermodynamic stability data. Early approaches for
sodium reference electrodes were employed elemental sodium in combination with
two phase Na-ß/ß"-Al2O3 and NASICON solid electrolytes for gas sensors [59, 60]
which creates a number of problems. Since, these cells must be operated at
higher temperatures (400 °C) which results in high reactivity of the liquid metal
(melting point: 98 °C) with the sealing of the reference electrode causing leakage
and making it unsuitable for both sensors or the evaluation of thermodynamic
stability data. The following galvanic cells were used to sense the NO2, O2 and
CO2 gases using elemental sodium as reference electrode:
Pt | Na | Na- ''ß/ß -Al2O3 | NaCO3 |Pt, CO2 (g), O2 (g) [59] (a)
Pt | Na | Na- ''ß/ß - Al2O3 | NaNO3 |Pt, NO2(g), O2(g) [60] (b)
However, a number of problems arise such as side reactions [59], a large
electrolyte /electrode interface resistance at temperatures below 550 K [60] which
reduces the use of elemental sodium as electrode.
It seems to be advantageous to find out solid sodium reference electrodes which
can be used in sensors or other applications.
-
26
3.3 Sodium alloys
An alternative to the application of elemental sodium is the use of two-phase
binary sodium alloys. In two phase regions the sodium activity is constant i.e.,
independent of the overall composition in the view of Gibbs phase rule and also
the level of sodium activity is within the ionic conduction domain. A number of
sodium alloy systems were studied by S. Crouch et al. [61], as for example, Na-Al,
Na-Si, Na-Zn, Na-B, Na-Pb, and Na-Sn. Among these systems only Na-Pb and
Na-Sn were found to have reasonable properties in terms of their thermodynamic
and kinetic behavior at 120 °C. Other systems were found not to have these
properties at the same temperature.
H. Schettle et al. [62] used the following galvanic cell (c):
Pt | Na | Na- ''ß/ß -Al2O3 | M (Na)-NaM| Pt where M= Sb, Bi, Pb, Sn. [62]. (c)
They reported that the open-circuit cell voltage turned out to be independent of the
composition corresponding to the existence of two phases in the alloy electrode.
But the major complication of using these binary alloys as reference electrodes is
the narrow temperature range over which they are solid and also sealing could not
be avoided. The binary alloy systems are investigated listed in the Table 3-1.
Table 3-1 e. m. f. of two phase sodium alloy systems at 120 °C (vs Na).
M (Na) represents M saturated with sodium [62]
Phases Electrode potential [mV]
Sb(Na)-NaSb 750
Bi(Na)-NaBi 710
Pb(Na)-NaPb 350
Sn(Na)-NaSn 440
-
27
3.4 Termary systems
3.4.1 Sodium-metal-oxide systems
Possible electrode materials are phase equilibria of ternary systems, important
group is AXMeYXZ: (A = Li, Na, K; Me = Ti, V, Cr, Mn, Fe, Co, Ni, Nb, Mo, La, Ta,
W, and X = O, etc.). Of these systems the sodium bronzes [62] and the ternary
oxides shown in Table 3-2 are used as electrode materials. Sodium bronzes
reference electrodes need to be encapsulated. If the reference side is encap-
sulated, as in case of CO2 sensors, the cell signal is necessarily dependent on the
oxygen pressure. Non-stoichiometric phases like NaxCoO2-y [62] yield reference
potentials depending of course on x and y.
The sodium metal oxide ternary systems are most probable materials for reference
electrodes or other applications at high temperatures. For their applications, the
characterization of the thermodynamic stability of the heterogeneous phase equili-
bria is mandatory.
The electrochemical characterization of the thermodynamic stability of the hetero-
geneous phase equilibria comprising sodium containing compounds finally deduce
to the determination of the sodium activity of these phase mixtures. This is usually
accomplished by the means of potentiometric measurements on a galvanic cells
using a sodium ion conductor, e.g. Na-beta-Al2O3 or NASICON, as solid
electrolyte. According to the information in the literature [22], the potential of the
measuring electrode of a potentiometric solid state CO2 sensor, based on NBA as
electrolyte, is so low that it is close to the lower limits of the ionic domain of the
electrolyte. There is also consideration in literature on the electronic conductivity of
Na-( ''ß/ß )-Al2O3 [1, 2, 3] and of NASICON [4]. There is no sodium ion conductor
known so far having ionic domain larger than Na-beta-Al2O3. Therefore, it is
expected that previous measurements, as long as they have been carried out in
the conventional way, might be erroneous due to the partial electronic short-circuit
in the measuring cell.
To check this electronic impact on thermodynamic properties compounds with
different levels of sodium chemical potential have to be considered. For this
reason system Na-Me-O with Me = Mo, Ti, and Nb has been considered.
-
28
Table 3-2 Heterogeneous phase equilibria of ternary systems which are used as
reference electrode in alkali concentration cell in potentiometric measurements
Literature Phase equilibria (reference electrode) Electrolyte used
[63, 64, 65, 66] Na2Ti3O7, Na2Ti6O13 NASICON/NBA
[66, 67] Na2Ti6O13, TiO2 NBA
[68] Na2SnO3, SnO2 NBA
[69, 70] Na2ZrO3, ZrO2, NBA
[71] Na2Fe2O5, Fe2O5 NBA
[72] Na3Fe5O9, Fe2O9 NBA
[73] Sb2O4, NaSbO3 NBA
[74] Na2CrO4, Cr2O3, NBA
[75] Na2MoO4, Na2Mo2O7, NASICON
[75] NiO, Na2NiO2 NASICON
[76] Na2Si2O5, SiO2, NBA
[76] Na2Si2O5, Na2SiO3, NBA
[76] Na2Ge4O9, GeO2, NBA
[77] Na2MoO4, Mo3, NBA
[78] Na2WO4, WO3, NBA
[79] Na-a-Al2O3, Na-ß-Al2O3 NASICON
[80, 81] W, WS2, Na2S NBA
[82] Ni, NiF2, NaF NBA
[83, 36] Na-beta-Al2O3/borate glass, NiO,
FeO/FeNi
NBA
[83, 7] K-beta-Al2O3/borate glass, NiO, FeO/FeNi KBA
[84, 6] NaSiyO2y+0.5, SiO2 (y = 1.5) NBA
[84, 7] KSi1.5O3.5, SiO2 KBA
3.4.2 Previous knowledge on the thermodynamic stability of Na-Me-O sys-
tems (Me = Mo, Ti, Nb)
3.4.2.1 Na-Mo-O system
In the system (sodium + molybdenum + oxygen) the phase fields have been
identified by Gnanasekharan et al. [85, 86] in the temperature range 673-923 K as
-
29
shown in Fig. 3-1. The phases Na2MoO4 and Na2Mo2O7 constitute a eutectic
phase mixture in the system MoO3-Na2MoO4 shown in Fig. 3-2. But different
eutectic chemical compositions of the mixture are reported by different invest-
igators [87-90]. In this system attention will be paid to characterize the eutectic
phase mixture Na2MoO4 + Na2Mo2O7. For calculating the °∆ Gf values of both
phases individually one has to rely on known °∆ Gf values of the other phase. If
one value, e. g. for ο∆42MoONaf
G , is not accurate then the same error is transferred
to the calculation of the ο∆722 OMoNaf
G value. ο∆42MoONaf
G data from different sources
are plotted in Fig. 3-3 [75, 91, 92]. Fortunately the data seem to be corrected in the
sense that they do not scatter despite of different sources. Mathews et al. [75]
computed ο∆42MoONaf
G taking into account the estimated
Fig. 3-1 Phase diagram of {(1-x1-x2)Na + x1Mo + x2O} at 673-923 K [85, 86]
standard enthalpy of formation of Na2MoO4 at 298.15 K from Lindemer et al. [93],
enthalpy and entropy increments of Na2MoO4 measured calorimetrically by Iyer et
al. [94] and the enthalpy and entropy increments of Na (l), Mo (s) and O2 (g) from
-
30
[95]. Iyer et al. stated that ο∆42MoONaf
G were taken from [96]. Barin [92] optimized
the ο∆42MoONaf
G value from low temperature properties of entropy and enthalpy of
the phase [97] and his estimated heat capacity value. All of the data used for the
computation of 42MoONafG°∆ are either estimated or calorimetrically determined.
Fig. 3-2 Schematic phase diagram of the MoO3-Na2MoO4 system based on [88]
constitutes eutectic phase mixture Na2MoO4 + Na2Mo2O7
-1210
-1190
-1170
-1150
660 700 740 780 820
∆fG
°
[
kJ m
ol-1
]N
a 2M
oO
4
T [K]
1
2
3
Fig. 3-3 Gibbs energy of formation of Na2MoO4 as a function of temperature from
different sources. (1: [92], 2: [75] and 3: [91])
MoO3 Na2MoO4Mol%
T [°
C]
T [°
C]
Na 2
Mo 2
O7
1000
aß
?
d
?
MoO3 Na2MoO4Mol%
T [°
C]
T [°
C]
Na 2
Mo 2
O7
1000
aß
?
d
?
-
31
3.4.2.2 Na-Ti-O system
The equilibrium phase diagram of the Na2O-TiO2 system is depicted in Fig. 3-4
[98]. In this system attention will be focused on Na2Ti3O7 + Na2Ti6O13 and
Na2Ti6O13 + TiO2 phase mixtures due to their wide application as electrode
material [63-67]. As mentioned earlier, for the characterization of a phase mixture,
one has to know the standard Gibbs energy values, for at least, one phases of the
mixture. Fig. 3-5 shows ο∆732 OTiNaf
G as a function of temperature from different
sources [92, 99, 100]. It appears that the data are coincided and pretended same
objectivity. After checking the sources of the data confusion becomes obvious.
Fig. 3-4 Phase diagram of Na2O-TiO2 system [98]
Bennington et al. [99] determined the enthalpy of formation of Na2Ti3O7 calori-
metrically at 298.15 K. Combining it with auxiliary data from different sources
resulted in Gibbs free energies of formation as function of temperature. The high
temperature enthalpy and entropy data for Na2O and O2 were taken from the
-
32
JANAF tables [101]. The value of S°298K for rutile was from Kelley and King [102]
and the enthalpy and entropy data above 298 K from Arthur [103] and Naylor
[104]. The entropy of Na2Ti3O7 at 298 K was taken from Shomate [105] and the
enthalpy and entropy increment from Naylor [106].
All necessary enthalpy and entropy data of sodium were from Hultgren [107] and
the data for titanium from [108]. Most of the data mentioned above were either
estimated or calorimetrically determined.
Barin [92] and Eriksson [100] took the data of Na2Ti3O7 [99] and optimized them
through special algorithm. Nevertheless the original data are the same. If error
associates with the original data it will remain since no independent measurement
is done.
The standard Gibbs energy of formation of Na2Ti6O13 from different sources [2, 66,
100] is shown in Fig. 3-6. In this case data are not coincided completely. Plots 1
and 2 of Fig. 3-6 are experimentally reported data [66]. The plot 3 [100] is
optimized data and of plot 4 is assessed [2] in the light of electronic impact taking
into consideration of the data of plot 1.
-3200
-3000
-2800
400 500 600 700 800 900 1000
∆fG
°
[kJ/
mo
lN
a 2T
i 3O
7
T [K]
1
2
3
4
Fig. 3-5 The standard Gibbs free energy of formation of Na2Ti3O7 as a function of
temperature, (1: [92] and 2: [100], 3 and 4: [99])
-
33
-5450
-5350
-5250
-5150
750 800 850 900 950
∆fG
°
[ k
J/m
ol]
T [K]
Na 2
Ti 6
O13
1
2
3
4
Fig. 3-6 The standard Gibbs free energy of formation of Na2Ti6O13 from different
sources plotted against temperature. (1, 2: [66], 4: [2 ] and 3: [100])
3.4.2.3 Na-Nb-O system
The complete system Na2O-Nb2O5 has been studied by Reisman et al. [109] (Fig.
3-7) and Shafer et al. [110] but the two sets of results do not wholly agree. Both
groups of investigators found the three phases Na3NbO4, NaNbO3 and Na2Nb8O21.
They also reported one another niobium-rich phase, to which they assigned the
composition Na2Nb28O71, whereas Shafer et al. [110] reported two niobium-rich
phases, given as NaNb7O18 and Na2Nb20O51. An alkali-rich phase, Na5NbO5, was
reported by Spitzyn and Lapitskii [111] but it has not been observed by other in-
vestigators. Half of the system (niobium-rich side) has been studied by Appendino
[112] and Irle et al. [113]. Both of them reported one additional alkali-rich phase,
Na2Nb4O11 and Appendino also reported another phase, assigned as NaNb3O8.
Schematic binary phase diagram of the system Na2O-Nb2O5 based on
Appendino’s data is shown in Fig. 3-8. Andersson [114] has also found two phases
Na2Nb4O11 and NaNb3O8.
-
34
Fig. 3-7 Schematic binary phase diagram of the system Na2O-Nb2O5 based on
Reisman et al. [109]. The vertical dashed lines indicate the phase missing in
Reisman’s diagram within 80% of Nb2O5
Fig. 3-8 Schematic binary phase diagram of the system Na2O-Nb2O5 based on
Appendino’s diagram [112]
Nb2O5
Na 5
NbO
5
Na 2
Nb 4
O11
NaN
b 3O
8
10 40 50 60 70 80 903020
Na 3
NbO
4 Na 2
Nb 2
0O51N
aNb 7
O18
Na2O
Mol %
T [°
C]
T [°
C]
NaN
bO3
Na 2
Nb 8
O21
Nb2O5
Na 5
NbO
5
Na 2
Nb 4
O11
NaN
b 3O
8
10 40 50 60 70 80 903020
Na 3
NbO
4 Na 2
Nb 2
0O51N
aNb 7
O18
Na2O
Mol %
T [°
C]
T [°
C]
NaN
bO3
Na 2
Nb 8
O21
Na2O Nb2O5Mol%
Na 2
Nb 6
O16
Na 2
Nb 4
O11
NaN
b 7O
18
NaN
b 13O
33
50 60 70 80 10090
Na 2
Nb 8
O21
T [
°C]
Liquid
Na2O Nb2O5Mol%
Na 2
Nb 6
O16
Na 2
Nb 4
O11
NaN
b 7O
18
NaN
b 13O
33
50 60 70 80 10090
Na 2
Nb 8
O21
T [
°C]
Liquid
-
35
After the Na2Nb8O21 in the phase diagram (Fig. 3-7 and 3-8), a number of other
niobium-rich phases were established by different investigators [114, 115].
There is no information regarding experimental thermodynamic data of any phases
of the system except of estimated data on the change of standard enthalpy of
formation and the standard entropy for the phases Na3NbO4 and NaNbO3 [93].
-
36
4. Experimental
4.1 Measuring principle
4.1.1 Cell configurations
Three types of cell configurations ((I)-(III) have been considered to evaluate the
thermodynamic stability of the systems Na-Me-O (Me = Mo, Ti, Nb), (Table 4-1).
Cell type (IV) has been considered to evaluate the p-electronic conduction
parameter of NBA. Equilibrium reactions and thermodynamic expressions in terms
of sodium and sodium oxide activity are given in Table 4-2 and 4-3, respectively.
Table 4-1 Cell configurations for characterization of the Na-Me-O systems
(Me = Mo, Ti, Nb)
Cell configurations Cell denotation
Pt, O2, CO2 | Na2CO3 (Au) | Na-Me-O (Au) | YSZ | O2, (CO2) Pt (I)
Pt, O2, CO2 | Na2CO3 (Au) | NBA (Au) | Na-Me-O (Au) | YSZ | O2, (CO2) Pt (II)
Pt, O2, CO2 | Na2CO3 (Au) | NBA (Au) |YSZ|NBA (Au)|Na-Me-O(Au)O2,(CO2) Pt (III)
Pt, O2, CO2 | Na2CO3 (Au) | NBA | Na-Me-O (Au) | O2, (CO2), Pt (IV)
Table 4-2 Equilibrium reactions and thermodynamic expressions in terms of
sodium activity (assuming that the activity of the constituent phases is unity)
Equilibrium reactions Thermodynamic expressions
2Na2MoO4 = Na2Mo2O7
+ 2Na + 1/2O2272242 ONaOMoNafMoONaf
plnRT2
1alnRT2GG2 +=∆−∆ οο
2Na2Ti3O7 = Na2Ti6O13 +
2Na +1/2O221362732 ONaOTiNafOTiNaf
plnRT2
1alnRT2GG2 +=∆−∆ οο
Na3NbO4 = NaNbO3
+ 2Na+ 1/2O22343 ONaNaNbOfNbONaf
plnRT2
1alnRT2GG +=∆−∆ °ο
4NaNbO3 = Na2Nb4O11 +
2Na + 1/2O2211423 ONaONbNafNaNbOf
plnRT2
1alnRT2GG4 +=∆−∆ οο
-
37
Table 4-3 Equilibrium reactions and thermodynamic expressions in terms of
sodium oxide activity (assuming that the activity of the constituent phases is unity)
Systems
Na-Me-O
Equilibrium
reactions
Thermodynamic expressions
Na-Mo-O 2Na2MoO4 =
Na2Mo2O7 + Na2OONaONafOMoNafMoONaf 2272242
alnRTGGG2 +∆+∆=∆ οοο
Na-Ti-O 2Na2Ti3O7 =
Na2Ti6O13 + Na2OONaONafOTiNafOTiNaf 221362732
alnRTGGG2 +∆+∆=∆ οοο
Na-Ti-O Na2Ti6O13 = 6TiO2
+ Na2OONaONafTiOfOTiNaf 2221362
alnRTGG6G +∆+∆=∆ οοο
Na-Nb-O Na3NbO4 =
NaNbO3 + Na2OONaONafNaNbOfNbONaf 22343
alnRTGGG +∆+∆=∆ οοο
Na-Nb-O 4NaNbO3 =
Na2Nb4O11 + Na2OONaONafONbNafNaNbOf 2211423
alnRTGGG4 +∆+∆=∆ οοο
Na-Nb-O 3Na2Nb4O11 =
4NaNb3O8 + Na2OONaONafONaNbfONbNaf 22831142
alnRTGG4G3 +∆+∆=∆ οοο
Na-Nb-O 8NaNb3O8 =
3Na2Nb8O21+Na2OONaONafONbNafONaNbf 22218283
alnRTGG3G8 +∆+∆=∆ οοο
4.1.2 Sodium chemical potential of the carbonate/gas electrode
The sodium activity ( Naa ′′ ) at the interface between the (CO2, O2) gas atmosphere
and the sodium carbonate results from the equilibrium of Na2CO3 with Na and the
gas components CO2 and O2 according to the following reaction [22]:
Na2O2
1COCONa 2232 ++⇔ . Eq. 4-1
The activity can be calculated according to the following equation:
222CO3CO2Na
OCONa lnp4
1 -lnp
2
1-
2RT
Gf?-Gf?alnοο
=′′ . Eq. 4-2
Quantities ο∆2COf
G and ο∆32CONaf
G denote the standard Gibbs free energies of form-
-
38
ation of the involved substances. 2COp and 2Op are the partial pressures of CO2
and O2 gas atmosphere, respectively.
4.1.3 Determination of the activity of sodium oxide dissolved in the phase
mixture
The measurement is based on the configurations of cell (I) and (II). The solid state
galvanic cell employs an oxygen concentration chain where yttria-stabilized zir-
conia (YSZ) is used as the solid electrolyte. The measuring electrode consists of a
sintered pellet i.e. a mixture of two adjacent phases of the system Na-Me-O, in
contact with a pellet of Na2CO3 or NBA. Both of these pellets are electrically short-
circuited by randomly distributed thin gold wires. The platinized surface of YSZ is
used as the reference electrode. The Na2CO3 pellet and YSZ are exposed to the
same CO2, O2 gas atmosphere. Only one of the gas components i.e. O2 acts as
the potential determining species at the reference electrode side while, at the
measuring electrode, both CO2 and O2 are the potential determining species.
Thus, the galvanic cell used for the present investigation without or including a
NBA pellet can be represented as follows for defining situation at the interfaces:
2Op′ 2Op ′′ 2Op′
Pt, CO2, O2 | Na2CO3 (Au) | Na-Me-O (Au) | YSZ |O2, Pt (I) Naa ′′′′ Naa ′′′ Naa ′′
and
2Op′ 2Op ′′ 2Op′
Pt, CO2, O2 | Na2CO3 (Au) | NBA (Au)| Na-Me-O (Au) | YSZ |O2, Pt (II) Naa ′′′′′ Naa ′′′′ Naa ′′′ Naa ′′
respectively.
The quantities 2Op′ and 2Op ′′ are the oxygen partial pressures at the parallel surface
of the YSZ pellet with the electrodes. Naa ′′ , Naa ′′′ , Naa ′′′′ and Naa ′′′′′ denote the sodium
activities established at the respective interfaces. The electrical potential differ-
ence between the surfaces of the YSZ pellet with the electrodes generates the
voltage U of the cells:
-
39
2
2
O
O
p
pln
F4RT
U′
′′= . Eq. 4-3
The sodium oxide dissolved in the phase mixture Na-Me-O, is the connecting link
between the sodium activity and the oxygen partial pressure by the following
dissociation equilibrium at interface " [116]:
2O2
1Na2O2Na +⇔