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Thermodynamics and Kinetics of Solids
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
q T = WDT
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
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
Exothermic reactions: Electrical heating
q = RI 2 t
Determination of Heat Capacities in a Dropping
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
In the case of phase changes, undefined final states may
occur because of the fast cooling.
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
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:
Ê DG 0 ˆ
ac aD
= exp Á˜
aA aB
Ë kT ¯
As = A g
K = pA
As + Bg = Abs
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
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
Thermodynamics and Kinetics of Solids
The pressure is given by
m 2p RT
m T
= 0, 02256
tA M
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.
Mass loss mL = 44 t A a pK
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
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
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
(a) and molecular species (b)
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
ii) Pulsed acceleration potentials with separation into
groups with constant time of flight (TOF).
(advantage of i: high resolution, ii) nearly simultaneous
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:
KI + T
Ka =
I +MN I +M
I +M 2 I +N
; Kb =
I +MN I +N
I +N 2 I +M
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).
Fig. 3.5.: shows the resulting activities of Cu - Ge alloys.
(K: Geometric constant, I+: Measured ion current, T:
Absolute temperature., s: Detector efficiency, D E :
Electron beam energy).
Mg + Ng Æ Mng
results for the vapor pressures from the ionic currents
( T1 )
=IM+ IN +
In case that dimeric species M2 or N2 are being observed,
the constants of the reactions.
M2 + N Æ MN + M
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
Thermodynamics and Kinetics of Solids
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
Pt , M, MO  Electrolyte (O- -)  O2 (1 atm), Pt
M + O- - Æ MO + 2eTotal cell reaction
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)
Br2 + e- Æ Ag+ Æ Ag Br
Total reaction:
Ag +
Br2 Æ Ag Br
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
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.
The energy of the chemical reaction generates an EMF.
Problems: Suitable electrolytes, reversible electrode
Fig. 3.7. Galvanic cell for EMF measurements using molten
Thermodynamics and Kinetics of Solids
of the transfered metal. (Eutectic mixtures of LiCl und
KCl: m. p. 359 °C).
 Na2S  Na - b - Al2O3  Na2S , P H2 S  PH 2
PSO 2 , Ps2  ZrO2  PO 2
C , M  M Cl2  Cl2,g,C
Cu , Cu2S  CaS CaF2  CaS  Fe, FeS
Problems: Hydrolysis of the molten salts by atmospheric
moisture. Dispersion of the molten metal in the
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
Fig. 3.9. Sample holder for EMF measurements using solid
Abb. 3.8. EMF-measurement using a glass electrolyte
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)
Thermodynamics and Kinetics of Solids
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
Cv =
= 25.1 J/K ⋅ mol
Cp - Cv ª 0.84…2.09 J / K · mol at room temperature.
Cp ª 25.9…31.5 J / K · mol
Empirically observed temperature dependence:
Cp = a + b x 10-3 T + c x 105 T-2
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
Tm 10 -3
(Â q + 1.125 n) - 0.298 n 10
Tm 10
For Al2 (SO4)3 holds
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 ·
Tab. 3.1. Cationic contributions to the heat capacity at 298 K
Tm -2 - 2.16 n
- 0.298
Cp (298 K) = 2q (Al+++) + 3q (SO4--) = 269.03 J / K · mol
(measured value: 259.41 J / K · mol)
6.125 n + 105 n Tm -2 -3
Tm 10 - 0.298
c = -4.12 n
(n: number of atoms of the molecule, Tm: melting point in
If no more precise data are known
about the heat capacity of a
compound, one may assume
D Cp ª 0
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
Thermodynamics and Kinetics of Solids
Fig.. 4.11. shows that the Trauton constant increases
however with the boiling point:
= 0, 01037 Te + 75, 96 kJ / K ⋅ mol
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.
More precise investigations have shown, however, that
the melting entropy increases slightly with the
temperature. For fcc-metals it is:
DSm = 7,41 + 1,55 x 10-3 Tm J / K · mol
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)
A(l) = A(g)
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)
A(s) + B2(g) = AB2(g)
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
DSe = e ª 92,1 J / K ⋅ mol
Thermodynamics and Kinetics of Solids
For bcc-metals holds:
DSm = 6,78 + 0,71 x 10-3 Tm
(Table 3.3.).
J / K · mol
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
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
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
Tab. 3.5. "Latimer" Entropiebeiträge n{X} als Funktion der Ladungszahl n der
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)
and in the following contribution of the
cation mixture is observed:
Thermodynamics and Kinetics of Solids
S = -R [x lnx + (1 - x) ln (1 - x) + (1 - x) ln
+ (1 + x) ln
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
A 2Y
, aB2 Y = N
B2 Y
For ideal random mixtures RT ln aA 2 Y = -T D s A 2 Y
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
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
Analogously holds for the reduction of Cr2O3 by Al:
2 Al + Cr2O3 Æ 2 Cr + Al2O3
Êa ˆ
K = Á Cr ˜
Ë a Al ¯
Ê a Al 2 O 3 ˆ Ê a Cr ˆ 2 Ê N Al 2 O 3
˜˜ @ Á
˜ ÁÁ
Ë a Cr 2 O 3 ¯ Ë a Al ¯ Ë N Cr 2 O 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
Thermodynamics and Kinetics of Solids
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,
Volume Change and Formation
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”.
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 =
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,
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
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
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
Cr + CO (g) Æ
Cr2O3 + C
K =
O2 (g) = 2 Odissolved, Fe
DG = -233676 + 50.84 T + 38.28 T log N0 J
a Cr3 2 O3
a Cr3 PCO
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-%.
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
In equilibrium we have with DG = 0
2 log N0 + log Nsi = - 30700 T-1 + 7.20
At 1600 °C holds
2 log N0 + log NSi = -9.19
On the basis of equilibrium process (CI, CSi) follows
C 2O CSi = 2.7 · 10- 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
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.