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Thermodynamics and Kinetics of Solids 21 ________________________________________________________________________________________________________________________ II. Determination of Thermodynamic Data 3. Experimental Methods 3.1. Calorimetric Methods Determination of the enthalpy of formation or reaction by applying an apparatus with known heat capacity (calorimeter). Measurements of the temperature change. The heat capacity of the calorimeter is given by the water equivalent: q T = WDT (3.1) Classification of calorimeters: i) Isothermal Calorimeter Calorimeter temperature Tc = surrounding temperature Ts = const. Best known example: Ice calorimeter ii) Adiabatic Calorimeter Tc = Ts ≠ const. Applicable for comparatively slow reactions (e.g. the solution of metals in acids) iii) Heat Flux Calorimeter Ts - Tc = const. Simpler construction than the adiabatic calorimeter. Suitable for determining transition enthalpies but not reaction enthalpies (since the temperature may be commonly not kept constant). iv) Isoperibolic Calorimeter Ts = const. Tc is measured during and after the reaction. Most commonly applied calorimeter. Important: Exact knowledge of the reaction products is necessary. No parasitic reactions should occur. Temperature Measurement i) Mercury-in-Glass Thermometer Precision: up to ± 0.0005 °C. Calibration above 6 °C with an overlap in the range from 9-33 °C. ii) Platinum Resistance Thermometer Application of Pt wire coils. Because of precise electrical measurement 8 x higher resolution. Wheatstone’s bridge. iii) Thermocouples Sensitivity of a single thermocouple is too low; accordingly series application of thermocouples (up to 1000 elements; 10-7 °C temperature difference is measurable). In the case of 10 copper-constantan couples 1mV corresponds to 2.34 °C temperature difference. iv) Thermistors Resistance element is a semi-conductor, e.g. SiC. The (negative) temperature coefficient is much larger than in the case of metals. ª 10-6 °C temperature difference is measurable. Difficulty: Reproducibility v) Optical Pyrometers Determination of the Water Equivalent Endothermic reactions: Application of a metal (e.g. Cu, Ag or Hg in glass) with known heat capacity at approximately the same temperature as in the case of the later measurement. Al2O3 may be used at high temperatures. Exothermic reactions: Electrical heating q = RI 2 t J (3.2) Determination of Heat Capacities in a Dropping Calorimeter Determination of the heat equivalent between room temperature and various higher temperatures. Application of isoperibolic or isothermal calorimeters. The substance is being heated to the desired temperature and dropped into the calorimeter. The hot sample is either directly dropped into the liquid within the calorimeter (water, paraffin,...) or into a beaker that is surrounded by water. In the case of phase changes, undefined final states may occur because of the fast cooling. Levitation-Calorimetry Pt-resistance furnaces: T < 1800 K. For higher temperatures electromagnetic levitation. Adiabatic Calorimeter Determination of the generated heat from electrical data. Determination of Melting and Transformation Enthalpies DTA: Sample and a reference body are nearly identically treated thermally. The temperature difference between both is measured. DSC: The necessary energy for heating the sample is compared to a reference sample (within the same temperature interval). Instead o f thermometric measurement (as in the case of DTA), the electrical energy is measured differentially. Precision ª ± 0.2%. Determination of Enthalpies of Reaction and Formation, Reaction Bomb Calorimetry 15.10.01 22 Thermodynamics and Kinetics of Solids ________________________________________________________________________________________________________________________ In the calorimeter a combustion with a gas (up to 25 atm) as one of the reactants is performed. The bomb has to be closed gas tight during the reaction which may be explosion-like. The reaction is initiated by an electrical current. As gases are mostly used: O2, F2, Cl2, N2. Impurities such as C, H or N may contribute largely to the enthalpy of combustion. 3.2. Equilibria with a Gas Phase. Determination of the change in Gibbs energy from the equilibrium constants: A+B=C+D K= Ê DG 0 ˆ ac aD = exp Á˜ aA aB Ë kT ¯ (3.3) e.g., As = A g K = pA As + Bg = Abs K= 1 pB As = Adessolved In the case that the gas phase is a complex mixture of species (for example, MoO3(s) Æ Mo3O9(g), Mo4O12(g), Mo5O15(g)) it is necessary to measure the individual gaseous components, which is commonly very difficult. Static Methods for the Determination of Vapor Pressures i) Application of manometers, e.g. a quartz spiralmanometer with a mirror or membrane-zeromanometer. Determination of the vapor concentration by optical absorption or emission (T up to 1000 °C) ii) Gas-Condensed Phase Equilibria in Closed Systems. Dewpoint method: Electrically heated furnace with two independently heated regions. Temperature increase of the entire furnace, afterwards cooling of the part of the furnace without sample until dew occurs. Precision ± 1 °C. Example: sample = brass; condensation of zinc. The vapor pressure of zinc at the temperature of the formation of dew corresponds to the zinc pressure of brass at the temperature of the sample. Isopiestic Method: Formation of an equilibrium vapor pressure over an alloy at high temperature and the pure volatile component at lower temperature. The temperatures at the hot and cold end are fixed. The equilibrium composition of the alloy is being determined. Sievert’s Method: Determination of the solubility of gases in metals. Heating of the metal in a closed cylinder combined with a burette which is connected via a three-wayvalve with a pump and a gas supply. A known gas volume is given and the decrease in volume is observed. Dynamic Vapor Pressure Methods Boiling Point Method: Determination of the boiling point (vapor pressure = atmospheric pressure) from the discontinuity of the weight-temperature curve or the pressure change at constant temperature. Transport Method: For the determination of the vapor pressure of a metal or the volatile component of an alloy, a constant flux of inert gas is passed over the sample. The gas takes up the vapor at a rate which depends on the relative pressure and flow rate. The vapor is condensed at a lower temperature and the mass is determined. Other Heterogeneous Equilibria Systems which contain more than 1 gas. Consideration of reactions between one gas and one condensed phase with the formation of at least one volatile product. H2 - CH4 - equilibria: H2, metal, its carbides and methane. H2 - NH3 - equilibria: Nitridation of iron 2 Fe 4 N + 3 H 2 (g) = 2( NH 3 ) g + 8 Fe H2 - H2O – equilibria: Reduction of metal oxide (e.g. 'FeO' + H2 = Fe+ H2O) H2 - H2S - equilibria: e.g. Ag2S + H2 (g) = 2 Ag + H2S (g) at 600 - 1280 °C. Other equilibria: CO, CO2, SO2 - O2 - SO3. Methods on the Basis of Evaporation Rates Determination of the vapor pressure of a substance from the evaporation rate into a vacuum: i) Knudsen ii) Langmuir Knudsen: Effusion 15.10.01 Thermodynamics and Kinetics of Solids 23 ________________________________________________________________________________________________________________________ The pressure is given by p= m 2p RT m T = 0, 02256 atm tA M tA M (3.4) m: Mass of the vapor with the molecular weight M, which evaporates from an area A during the period of time t. Langmuir: Sample is exposed to vacuum (no equilibrium as in the case of the Knudsen method) The mass is mostly much lower. M Mass loss mL = 44 t A a pK T a: Evaporation coefficient (0< a £ 1). pK: Vapor pressure as determined by the Knudsen method. The Knudsen cell is often used in combination with a mass spectrometer (Identification of the gaseous species). Knudsen-Effusion: Determination of the mass at room temperature before and after the experiment or in combination with a vacuum microbalance (25 g, 1 mg resolution) with continuous monitoring of the mass (example: Determination of the activity of Si in transition metal-silicides; by mixing with SiO2, SiO vapor instead of Si vapor is generated and the measuring temperature is reduced from > 2000 K to 700 K). For highest resolution: Condensation of a known fraction of a gas with a radioactive isotope onto a target and Fig. 3.2. Combination of a Knudsen cell with a mass spectrometer radiochemcial analysis. Example (Fig. 3.1.): Determination of the chromium activity in chromium alloys (1400 °C) with condensation of chromium onto molybdenum as target disc; dissolution of Cr in acid and determination of the radioactivity. Alternatively, MoO3 was formed by oxidation, which could be pressed into pellets. Problems may be the interaction with the sample holder and temperature gradients. Therefore, resistance furnaces are being used. Complex gas phases: Application of a mass spectrometer. Ionization of the effusion molecular beam by bombardment with monoenergetic electrons. The Fig. 3.1. Effusion cell for the determination of the vapor pressures of metalls (1200 - 1400 ° C) Fig. 3.3. Ion current vs. electron energy for monoatomic species gis (a) and molecular species (b) 15.10.01 24 Thermodynamics and Kinetics of Solids ________________________________________________________________________________________________________________________ ionization source is as close to the Knudsen cell as possible (Fig. 3.2.). 2 methods for the separation of the ions: i) Continuous extraction by fixed acceleration potentials, ii) Pulsed acceleration potentials with separation into groups with constant time of flight (TOF). (advantage of i: high resolution, ii) nearly simultaneous detection) Fig. 3.3. shows a typical ionization efficiency curve for a simple monoatomic gas and the fragmentation of complex molecules (e.g., M2 + e- Æ M+ + M + 2e-). The ionic current is commonly measured by a photomultiplier. Relationship between the peak intensity of the species in the mass spectrometer and the pressure: p= KI + T sDDE Ka = I +MN I +M I +M 2 I +N ; Kb = I +MN I +N I +N 2 I +M (3.7) Instrumental and geometrical factors are eliminated in this case. Practical difficulties often: Pressure of the dimer M2+ is commonly one order of magnitude lower than that of the monomer. While the lower vapor pressure limit is ª 10- 4 mm Hg in the case of the Knudsen method, measurements according to the Langmuir method may be performed at considerably lower pressures. The Langmuir Method is often applied in order to increase the rate of the weight loss (especially suitable for substances with high sublimation energies). Examples: Fig. 3.5.: shows the resulting activities of Cu - Ge alloys. (3.5) (K: Geometric constant, I+: Measured ion current, T: Absolute temperature., s: Detector efficiency, D E : Electron beam energy). From Mg + Ng Æ Mng results for the vapor pressures from the ionic currents ∂ ∂ ( T1 ) + log I MN T DHo =IM+ IN + R (3.6) In case that dimeric species M2 or N2 are being observed, the constants of the reactions. M2 + N Æ MN + M and N2 + M Æ MN + N may be described in a good approach by the relative amounts of the ionic currents: Fig. 3.4. : Cu - Ge (l) : Ion current ratios at 1400 °C Fig. 3.4. shows the ratio of the ion currents of copper and gemenium in the case of liquid Cu – Ge alloys 15.10.01 Thermodynamics and Kinetics of Solids 25 ________________________________________________________________________________________________________________________ Fig. 3.6. shows induction heated metals cooling of the processes, electronic conduction of the electrodes. Determination of the dissociation pressure of an M/MO System: Pt , M, MO Electrolyte (O- -) O2 (1 atm), Pt M + O- - Æ MO + 2eTotal cell reaction 1 2 1 2 O2 + 2e- Æ O- - O2 + M Æ MO Dissociation pressure of Ag / Ag Br: Ag Ag Br Br2 , C Ag Æ Ag+ + e- Fig. 3.5. Activities in the system Cu - Ge (l) 1 2 Br2 + e- Æ Ag+ Æ Ag Br Total reaction: Ag + 1 2 Br2 Æ Ag Br DG=-nFE Compared to the application of solid electrolytes, molten salts have the general disadvantage that several ions are commonly mobile. Moltoen salts: Often alcali chlorides with dissolved salt Fig. 3.6. Langmuir apparatus for the determination of reaction pressures surface (e.g. Cu) in a molydeum brat. Oxides dissociate in the case of evaporation; the large evaporation enthalpies results in a is observed because of. EMF-Measurements The energy of the chemical reaction generates an EMF. Problems: Suitable electrolytes, reversible electrode Fig. 3.7. Galvanic cell for EMF measurements using molten chlorides 15.10.01 26 Thermodynamics and Kinetics of Solids ________________________________________________________________________________________________________________________ of the transfered metal. (Eutectic mixtures of LiCl und KCl: m. p. 359 °C). PH 2 Na2S Na - b - Al2O3 Na2S , P H2 S PH 2 or PSO 2 , Ps2 ZrO2 PO 2 C , M M Cl2 Cl2,g,C or Cu , Cu2S CaS CaF2 CaS Fe, FeS Problems: Hydrolysis of the molten salts by atmospheric moisture. Dispersion of the molten metal in the electrolyte. Determination of the EMF by extrapolation of the current-voltage curve to I = 0: Ag Ag Cl M Pb Cl2 Pb Glass electrolytes: Determination of the Na – activity in molten Na - Hg- and Na - Cd - systems (300 - 400 °C), or the Ag-activity in Ag-Au. Ceramic solid electrolytes Experimental arrangement Fig. 3.9. Sample holder for EMF measurements using solid Abb. 3.8. EMF-measurement using a glass electrolyte electrolytes. Kiukkola + Wagner (1957): Pt , Ni , NiO ZrO2 Fe , FeO , Pt 3.3. Estimation of Thermodynamic Data Because of the lack of available thermodynamic data it is Cell reaction: NiO + Fe = FeO + Ni Electronic Conductivity of the electrolyte dependent on P O 2 . Application of ZrO2 and ThO2 in series. Reference electrode. Inert Gas / Vacuum. Gas electrode: H2 - H2O , CO - CO2 , ... Secondary equilibria: Pt, MnO, MnS (SO2 = 1 atm) ZrO2 O2 , Pt Left hand electrode reaction: MnS + 30-- Æ SO2 + MnO + 6eElectrolyte with dispersed second phase, e.g. Ni Ni F2 Sr F2 Sr F2 - La F3 Co , Co F2 Electrolyte with gas sensitive electrode: Fig. 3.10.: Coulometric titration of Cu, Cu2O ThO2-Y2O3 (O) Pb (l) 15.10.01 Thermodynamics and Kinetics of Solids 27 ________________________________________________________________________________________________________________________ of large interest to estimate data with sufficient precision. Heat Capacities. Dulong-Petit’s law: atomic heat of the elements ª 6.2 cal / K at room temperature. Since the atoms of solids are fixed in the lattice there are no degrees of freedom by rotation or translation. However, there exist 3 degrees of freedom of vibration (which have to be counted twice). Accordingly, we have above Debye’s temperature Cv = 6 = 25.1 J/K ⋅ mol 2R (3.8) Cp - Cv ª 0.84…2.09 J / K · mol at room temperature. Accordingly Cp ª 25.9…31.5 J / K · mol Âq Empirically observed temperature dependence: Cp = a + b x 10-3 T + c x 105 T-2 (3.9) Kellogg (1967): Estimation of heat capacities of predominant ionic compounds at 298 K by adding the contributions of a cationic and anionic groups (q (cat) , q (an)). Average values were determined from these experimental data (Tables 3.1. and 3.2.) C p (298 K) = Tab. 3.2. Anionic contributions to the heat capacity at 298 K (The T-2 term reflects the bending at lower temperatures above 298 K and at the Debye temperature). The result of analyzing 200 inorganic compunds is (3.10) a= Tm 10 -3 ( q + 1.125 n) - 0.298 n 10 Tm 10 For Al2 (SO4)3 holds b= The heat capacities increase with temperature and are approximately the same for all compounds per ion or atom at the melting point. Ünal (1977): 30.3 ± 2.1 J / K · mol. Tab. 3.1. Cationic contributions to the heat capacity at 298 K -3 5 Tm -2 - 2.16 n - 0.298 (3.12) Cp (298 K) = 2q (Al+++) + 3q (SO4--) = 269.03 J / K · mol (measured value: 259.41 J / K · mol) (3.11) 6.125 n + 105 n Tm -2 -3 Tm 10 - 0.298 c = -4.12 n Âq (3.13) (3.14) (n: number of atoms of the molecule, Tm: melting point in K) If no more precise data are known about the heat capacity of a compound, one may assume D Cp ª 0 (3.15) for reactions in the condensed state (postulate of the additivity of the heat capacities of the elements or reactants = Neumann-Kopp’s Rule). This holds well for alloys but also in a first approach for compounds with 15.10.01 28 Thermodynamics and Kinetics of Solids ________________________________________________________________________________________________________________________ Fig.. 4.11. shows that the Trauton constant increases however with the boiling point: Le = 0, 01037 Te + 75, 96 kJ / K ⋅ mol Te (3.18) Melting. The melting entropy is not a constant as in the case of evaporation). The change of the ordering by melting is smaller than by evaporation. The variation of the ordering of a solid material by the various chemical binding forces results in a proportionally large effect on the melting entropy. Crompton (1895), Richards (1897), Tammann (1913): Pure metals: Fig. 3.11. Le / Te vs Te for pure elements and inorganic DSm ª 9,2 J / K = const. compounds More precise investigations have shown, however, that the melting entropy increases slightly with the temperature. For fcc-metals it is: (3.16) DSm = 7,41 + 1,55 x 10-3 Tm J / K · mol (3.20) coordination lattices. In other words, it is DG (T) = DH (298 K) - TDS (298 K) The enthalpies of transitions, melting and evaporation have to be taken into consideration, however. Estimated average values for changes of the heat capacity for different reactions with gases: Cp [J / K · mol] A(s) = A(g) -7.5 A(l) = A(g) -9.6 AxBy(s) = AxBy(g) -9.6 (x+y) AxBy(l) = AxBy(g) -11.3 (x+y) A(g) + x B2(g) = AB2x(s) +12.5x A(s) + B2(g) = AB2x(l) +14.2x A(s) + B2(g) = AB2(g) -9.2 (3.19) Tab. 3.3. Melting entropies for inorganic compounds Enthalpies and entropies of transitions, melting and evaporation Methods for the estimation of these data are rather reliable if the molecular structure of the substance is known. Evaporation: Pictet (1876), Trouton (1884): The entropy of evaporation (i.e. enthalpy / absolute temperature of the evaporation is approximately the same for all compounds L DSe = e ª 92,1 J / K ⋅ mol Te (3.17) 15.10.01 Thermodynamics and Kinetics of Solids 29 ________________________________________________________________________________________________________________________ For bcc-metals holds: DSm = 6,78 + 0,71 x 10-3 Tm (Table 3.3.). J / K · mol (3.21) For covalent metals, DS m is substantially higher than 9.2 J / K. Compounds: Predictions are difficult since the melting entropy depends on the nature of the atomic ordering and the type of chemical bond. To a certain degree, the crystalline structure provides an indication of the type of binding; however, AgCl and NaCl oder CaCl2 and MgF2 have the same structure but different melting entropies Entropies and Entropy Changes. Standard entropies: Nearly all elements have been measured. For inorganic compounds, Latimer (1951) found that the standard entropies may be added up from empirically observed values for the anionic and cationic constituents (308 mainly ionic compounds). (Tables 3.4., 3.5.) In order to obtain the standard entropy of a solid compound, the value for the cation has to be multiplied by the number of cations in the molecule and added to the value obtained for the anion. Example: S (Al2(SO4)3 , 298 K) = (2 x 23.4) + (3 x 64.2) = 239.4 J / K · mol. Tab. 3.4. "Latimer" Entropiebeiträge {M} Entropies of Mixing (Non-metallic Solutions): Example: Mixing of cations in double oxides (spinels, MX2O4, with 1/3 of the cations on tetrahedral sites and 2/3 on octahedral sites; no mixing effect at “correct” occupation of sites; if, however, X occupies partially tetrahedral sites and M partially occupies octahedral sites, (Mx X1-x) [M1-x X1+x] O4 an effect of mixing occurs: x = 0: normal spinel; x = 1: inverse spinel. Tab. 3.5. "Latimer" Entropiebeiträge n{X} als Funktion der Ladungszahl n der Kationen The value x may be determined from the equilibrium constant of the exchange reaction (M) + [X] = [M] + (X). The result is: DH (exchange) = - RT ln (1- x )2 x(1+ x) (3.22) and in the following contribution of the cation mixture is observed: 15.10.01 30 Thermodynamics and Kinetics of Solids ________________________________________________________________________________________________________________________ S = -R [x lnx + (1 - x) ln (1 - x) + (1 - x) ln + (1 + x) ln 1+x ] 2 1-x 2 (3.23) Temkin’s-Rule for the calculation of the activities in mixtures of non-metallic compounds (which provides a relationship between the activities and numbers of atoms in each molecular species). For mixtures of A2Y - B2Yholds: aA 2 Y = N 2 A 2Y , aB2 Y = N 2 B2 Y (3.24) ( For ideal random mixtures RT ln aA 2 Y = -T D s A 2 Y ) the are often not very precise and are restricted to a relatively small number of compounds. The enthalpies of the elements in their standard state at 298.15 K are set to 0. The temperature dependence of the formation enthalpies is generally small. In order to obtain a consistent basis for comparison, the formation of 1 mol AxBy with x + y = 1 is being considered. It is expected that the compound with the highest melting point has the highest formation enthalpy. If the melting points of the other compounds (of the same system) are considerably smaller, straight lines to the result is D sA2 Y = -2R ln N A 2Y = -R ln N 2A 2 Y (3.25) Analogously holds for the reduction of Cr2O3 by Al: 2 Al + Cr2O3 Æ 2 Cr + Al2O3 Êa ˆ K = Á Cr ˜ Ë a Al ¯ 2 Ê a Al 2 O 3 ˆ Ê a Cr ˆ 2 Ê N Al 2 O 3 ÁÁ ˜˜ @ Á ˜ ÁÁ Ë a Cr 2 O 3 ¯ Ë a Al ¯ Ë N Cr 2 O 3 ˆ ˜˜ ¯ 2 (3.26) 3 (Fe, Mn)3 C : a Fe 3 C = N Fe 3 C Another contribution to the entropy besides the configuration entropy is the thermal entropy (by the change of the vibration of the cations and their surrounding oxygen ions when mixed oxides are formed). For spinels such as Fe3O4, FeAl2O4, FeV2O4 und FeCr2O4 holds DS = -7.32 + DSm J / K · mol. Abb. 3.12. Lithium-tin phase diagram pure elements may be drawn in the DH - x – presentation in a first approach. Formation Enthalpies The determination of the Gibbs energy requires information about the formation enthalpies. The methods Tab. 3.6. Thermodynamics and cation distribution in spinels Abb. 3.13. Formation enthalpies in the lithium-tin system 15.10.01 Thermodynamics and Kinetics of Solids 31 ________________________________________________________________________________________________________________________ Homologeous Series There is a certain relationship between the formation enthalpies of metal compounds and the order number of the metal in the periodic system in the case of the same stoichiometry and same radical. Pettifor (1986): Reorganisation of the periodic table according to "Mendeleev-numbers” (Figs. 3.14, 3.15) Volume Change and Formation Enthalpy Fig. 3.15. The line shows the sequence of the elements through the modified periodic Originally it has been assumed system according to “Mendeleev’s number”. specifically for intermetallic compounds that the deformation or polarization of the atoms of both metal atoms by the 100 eMV AV formation of the alloy depends on the affinity. The DV = AV relationship holds, however, also for simple inorganic saltlike compounds, though the change intensity is mainly caused by the formation of ions. Percentage of volume AV : Sum MV: Molecular volume of the compound, change: of the atomic volumina of both components, e = 0.95 (CsCl-structure), 0.825 (NaCl-structure) (Fig.. 3.16). (  )   Fig. 3.16. Heat generation and degrees in volume at the formation of compounds with simple structures The deviation from the curve is £ 25 kJ/g-atom. 4. Examples of Thermochemical Treatment of Materials Problems Fig. 3.14. Formation enthalpies of carbides and nitrides with a cubic NaCl-structure. The plot is made in the Pettifor’s 4.1. Iron and Steel Production arrangement of the periodic system 15.10.01 32 Thermodynamics and Kinetics of Solids ________________________________________________________________________________________________________________________ Removal of dissolved oxygen in molten steel by the addition of an element which may form an oxide of higher stability than that of iron. i) Si: SiO2 (s) =Si (l) + O2 (g) DG0 = 952697 - 203.8 T J 2 3 Cr + CO (g) Æ 1 3 Cr2O3 + C (4.7) 1 K = O2 (g) = 2 Odissolved, Fe DG = -233676 + 50.84 T + 38.28 T log N0 J a Cr3 2 O3 2 a Cr3 PCO (4.8) This allows to calculate Pco for the equilibrium of an FeCr-C-alloy with pure Cr2O3. Equilibrium constant: Si (l) = Sidissolved, Fe DG = -131378 + 15.02 T + 19.14 T log NSi J log K = 12580 T- 1 - 9.10 For the reaction SiO2 (s) = Sidissolved, Fe + 2 Odissolved, Fe Reduction of the C-content to ª 0.01 weight-%. Reaction: (4.1) (4.9) aCr = 0.1 fi aC = 3 x 10-4 (for pure Cr2O3 and pCO = 1 atm at 2000 K). this results in the following Gibbs reaction energy DG = 587643 - 137.94 T + 38.28 T log N0 + 19.14 T log NSi (4.2) In equilibrium we have with DG = 0 2 log N0 + log Nsi = - 30700 T-1 + 7.20 (4.3) At 1600 °C holds 2 log N0 + log NSi = -9.19 (4.4) On the basis of equilibrium process (CI, CSi) follows C 2O CSi = 2.7 · 10- 5 (4.5) The concentrations are rather small and a small amount of Si has to be added. ii) Al Analogously holds for Al2O3 (s) = 2 Aldissolved, Fe + 3 Odissolved, Fe (4.6) C 3O C 2Al = 10-13 This value does not correspond, however, to the experimental result. Possible reasons: Reaction of Al is faster than the dissolution of Al in the melt; formation of the spinels FeAl2O4 from FeO and Al2O3. Removal of carbon from Fe-Cr-C- and Fe-Si-C-alloys (l): Liquid Fe-Cr-alloys are being formed by the reduction of oxides in spark-arc furnaces at T ª 1700 °C. 15.10.01