Nestechiometrie, tepelná kapacita a krystalochemické modely fází Pavel Holba NTC ZČU Plzeň 23. duben 2013

Systém chemických látek na počátku XX. století LÁTKA ČISTÁ LÁTKA chemické individuum SMĚSNÁ LÁTKA směs čistých látek PRVEK SLOUČENINA HOMOGENNÍ pravý ROZTOK KOLOIDNÍ nepravý roztok HETEROGENNÍ směs fází DISPERSNÍ SOUSTAVA daltonid berthollid HOMOGENNÍ SOUSTAVA KOLOIDNÍ SOUSTAVA HETEROGENNÍ SOUSTAVA Selmi (1845): pseudosolution Graham (1860) : colloid

Chemické pojmy 1590 - slovo SOLUTION ve významu ROZTOK poprvé použito v angličtině 1610 – Beguin: První (nealchymická) učebnice chemie 1661 – Boyle : CHEMICKÝ PRVEK (ELEMENT) 1788 – Lavoisier: ELEMENTÁRNí LÁTKY: elastická těla (plyny), kovy, zeminy, nekovy, zásady, kyseliny, 1649 - Gassendi + 1808 – Dalton: ATOM 1811 – Avogadro + 1858 - Cannizaro: MOLEKULA 1834 – Faraday: ION 1878 – Gibbs: SLOŽKA & FÁZE 1887 - S. Arrhenius, “Über die Dissociation der in Wasser gelösten Stoffe,” Z. Phys. Chem., 1887, 1, 631-648. 1890 – van´t Hoff: TUHÝ ROZTOK 1891 - Stoney + 1911 – Millikan: : ELEKTRON 1912 – Kurnakov: BERTHOLLID & DALTONID 1926 – Frenkel : VAKANCE + INTERSTICIÁL 1926 – LEWIS : FOTON 1930 – Schottky & Wagner: KRYSTALOVÝ DEFEKT 1931 - Heisenberg: ELEKTRONOVÁ DÍRA 1954 – Huggins: STRUKTON 1962 - Cotton: KLASTR (CLUSTER)

Krystalové poruchy (defekty) 1926 – J. Frenkel - předpokládá existenci kationtových vakancí a intersticiálů 1930 – Schottky a Wagner v publikaci „Theorie der geordneten Mischphasen“ vytvářejí základ pro popis krystalických látek, které nejsou tvořeny molekulami Walter Hermann Schottky 1886-1976 Carl Wihelm Wagner 1901-1977 Jakov Iljič Frenkel 1894-1952 Defekty jsou výhodné, neboť zvyšují entropii , a tím snižují volnou energii

Intrinsic (niterné) defekty Frenkelovy poruchy 1926 Schottkyho poruchy 1930?

Linus Pauling, The principles determining the structure of complex ionic crystals. J. Am. Chem. Soc. 51 (1929) 1010-26 Uspořádání Počet Poloměr Příklad koulí (R=1) vrcholů dutiny struktury Krychle (cube) 8 R =0.732 CaF2 Oktaedr 6 R ≥ 0.414 NaCl Tetraedr 4 R ≥0.225 ZnS

Barevná (F) centra v NaCl {AX} Na(g)  Na+|A| + □-|X|

Okruhy nestechiometrických látek Intermetalické sloučeniny CoSn0,69-0,72 Tuhé elektrolyty (CaF2-type) (Ca,Y)(Zr, Hf, Th)1-xO2-x Magnetické ferity (Mg,Zn,Mn)xFe3-xO4+γ Supravodivé oxidy YBa2Cu3O7- δ Bi2Sr2Ca2Cu3O10+δ Tl2Ba2Ca2Cu3O10+δ TlBa2Ca2Cu3O9+δ Hg2Ba2Ca2Cu3O8+δ Oxidy aktinoidů (U, Pu,Cm, Am) O2-x CaUO4+x Hydridy PdH0,7,, TiHx NbHx, GdHx Oxidy lantanoidů CeO2-x PrO1+x Hydráty metanu CH4.(6x)H2O Wolframové a molybdenové bronze (W, Mo)O3-x Chalkogenidy Pyrhotit Fe1,0-0,8S ZrSx, CrSx Tuhé elektrolyty (CaTiO3-type) La(Sr,Ca)MnO3-δ Sr2(Sc1+xNb1-xO6-δ Sr3CaZr1-xTa1+xO9-δ Thermoelektrika (CuFeO2-type) CuCr1−xMgxO2+δ Protonové elektrolyty (pyrochlore type) La2-xCaxZr2O7-δ 2

Složení a nestechiometrie Homogenní látka Odchylka od stechiometrie:  = m-mo ;  = no-n [(3-)/3] Fe3O4+ = Fe3-O4  = 3/(4+ ) ;  = 4/(3+) Prvek Sloučenina Roztok Daltonid Bertollid Fe:O = n:m → FenOm → FenOmo+ → Feno-Om Molální zlomek: YO=m/n FeO: YO=1/1 = 1,000 FeO1,056 YO=1,056/1 =1,056 FeO1,158 YO=1,158/1 =1,158 Fe3O4: YO=4/3 =1,333 Fe3O4,112 YO=4,112/3 =1,371 Fe2O3 YO=3/2 =1,500 Molární zlomek: XO=m/(n+m) FeO XO=1/2 =0,500 FeO1,056 XO=1,056/2,056 =0,514 FeO1,158 XO=1,158/2,158 =0,537 Fe3O4: XO=4/7 =0,571 Fe3O4,112 XO=4,112/7,112 =0,587 Fe2O3 XO=3/5 =0,600 Stechiometrický (daltonský) poměr no:mo YO → FeO 1,10 1,20 Fe3O4 1,40 Fe2O3

Molální zlomek YX (volné složky X) ve fázi AXY ve vztahu k aktivitě volné složky aX a teplotě T P = const. log aX T [°C] YX versus T [log aX =const] log aX versus T [YX =const] -2 -4 1200 1100 1000 -6 1.53 900 1.52 log aX versus YX [T=const] 1.51 YX

Vztah mezi teplotou T, obsahem Yf , a aktivitou af volné složky vyjadřuje implicitní funkce F (Yf , af , T) = 0, [P=const] kterou lze charakterizovat následující trojicí veličin: relativní parciální molární entalpií Δhf Δhf = R (∂ ln af /∂(1/T))Yf tepelnou ftochabilitou (φτωχός = chudý) κfT κfT = – (∂Yf /∂T)af vlastní (proper) plutabilitou (πλούτος = bohatý) κff κff = (∂Yf /∂ log af)T ≥ 0 mezi nimiž platí vztah: κfT = - κff (Δhf /RT2)

U0.8Pu0.2O2+δ Heat capacity of „mixed oxide fuel (MOX)“ U0.8Pu0.2O2+δ Chaleur spécifique a haute temperature des oxydes d'uranium et de plutonium (1970) Affortit C. & Marcon J-.P.; Rev. Int. Hautes Temp. Refract. 7, 236-41 δ =0.08 δ =0.045 δ =0.000 U0.8Pu0.2O2+δ δ = – 0.020 CP [cal/mol/K] T [K] Figure from Inaba H. & Naito K. (1977): Heat Capacity of Nonstoichiometric Compounds, NETSU 4 [1] 10-18

Tepelná kapacita - Historie 1760 Joseph Black (in Glasgow) distinguished latent heat (transition enthalpy) from „sensible heat“ – specific heat (heat absorbed at rising temperature of a gram of substance by one degree) 1819 Dulong & Petit pointed out the atomic heats (products of specific heat and atomic weight) of several metallic (solid) elements equal approximately to 3 R (R = universal gas constant = 1,987 cal/K/mol = 8,3145 J/K/mol) as it is shown in following table: 3 R = 5,96 cal/(at.K) = 24,9 J/(at.K) 1831 Franz Ernst Neumann : “Untersuchungen über die specifische Wärme der Mineralien” 1864 Hermann Kopp (1817-1892): „molecular“ heat of compound equals approximately to the sum of atomic heats of contained elements. „Molecular“ heat of compound AB2C4 : CAB2O4 = CA + 2 CB + 4 CC 1871 James C. Maxwell: distinguished isochoric CV and isobaric CP heat capacity 1907 Albert Einstein : heat capacity of ideal crystal : 1912 Peter Debye : heat capacity of ideal crystal : 1922 Walter Schottky: anomaly heat capacity of two level systems at low temperatures 1928 Arnold Sommerfeld (1886-1951): heat capacity of electrons in metals ΔelCV = γT 1952 K. Kobayashi: heat capacity due to Frenkel defects formation ΔFrC

CP = Cvib + ΔelC + ΔmgC + ΔCdil Temperature dependence of heat capacity C and some of its contributions CP ΔCdil CV Al CP = CV+ ΔdilC vibrational ΔdilC =VT.a2/b CP = Cvib + ΔelC + ΔmgC + ΔCdil Cvib = Char + Canh electronic magnetic dilation

Contributions to heat capacity CP According to thermodynamic tables (empirical polynomial model): CP = A1 + A2 .T + A3/T2 + A4/√T + A5.T2 According to thermodynamic and physical (theoretical) models: Cp = Char + Canh + ΔdilC + ΔelC + ΔmgC + Δcdf C + ΔothersC Non- stoichio- metry ? Neumann- Kopp & Einstein & Debye = CV Ideal crystal vibrations Dilation Electronic Magnetic Crystal defect formations Δcdf C = ΔSchC + ΔFrC + ΔehC + … Δdil C =VT.a2/b Schottky defects Frenkel defects Electron- positron pairs ΔFrC = AFr [(ΔHFr)2/2RT2]exp(-ΔHFr/2RT)

Isochoric Cv and isobaric CP heat capacities Isochoric conditions Constant volume V: Isobaric conditions Constant pressure P: Increasing temperature T Increasing pressure P Increasing temperature T Increasing volume V Cv = (∂U/∂T)V CP = (∂H/∂T)P Internal energy U Enthalpy H = U + P.V ΔdilC = CP - Cv

Difference between isobaric and isochoric heat capacity (∂U/∂T)V = T (∂S/∂T)V = CV ; (∂H/∂T)P =T (∂S/∂T)P = CP ; α = (∂V/∂T)P /V → (∂V/∂T)P = αV ; β = - (∂V/∂P)T /V → (∂V/∂P)T = - βV H = U + P.V → dH = TdS + VdP ; F = U – TS → dF = - SdT - PdV CP≡(∂H/∂T)P = (∂U/∂T)P + P .(∂V/∂T)P dU = TdS  P.dV (∂U/∂T)P = (∂U/∂T)V + (∂U/∂V)T .(∂V/∂T)P (∂U/∂V)T = T .(∂S/∂V)T - P (∂U/∂T)P = Cv + (T .(∂S/∂V)T - P).(∂V/∂T)P (∂S/∂V)T = ∂2F/∂T∂V=∂2F/∂V∂T=(∂P/∂T)V (∂U/∂T)P = Cv + (T .(∂P/∂T)V - P).(∂V/∂T)P CP = [Cv + (T .(∂P/∂T)V - P).(∂V/∂T)P] + P .(∂V/∂T)P CP = CV + T .(∂P/∂T)v.(∂V/∂T)P + {- P (∂V/∂T)P + P.(∂V/∂T)P} CP = CV +T .(∂P/∂T)v .(∂V/∂T)P (∂P/∂T)v = -(∂V/∂T)P /(∂V/∂P)T = (-V.α)/(-V.β) = α/β CP = CV +T .(∂V/∂T)P α/β = CV - T V α2/β Maxwell´s relation 1860 (Clairaut 1743): CP = CV + T.V. α2/β ; CP - CV ≡ Δdil C = T.V. α2/β

Heat capacity and nonstoichiometry Heat capacity of sample with free component f (element X) Measured in sealed ampoule Molal fraction Yf = Nf/∑iNi≠f Yf = Xf/(1-Xf) Measured under controlled atmosphere Constant fraction of volatile component Yf AnXm+γ = n A1Xy Yf = (m+Δ)/n = = y = m/n + δ = yo+δ af = pf /pof Constant activity of volatile component af CP, Yf CP, af Isoplethal conditions Isodynamical conditions ΔsatCP = CP, af – CP, Yf

Saturation contribution ΔsatCP Internal energy as a compound function U = U(T, V(T)) dU = T.dS – P.dV + ∑ μi.dNi ; (∂U/∂T)V, Ni = T.(∂S/∂T)V,Ni = CV,Ni (∂U/∂T)P = (∂U/∂T)V + (∂U/∂V)T .(∂V/∂T)P Enthalpy as a compound function H = H(T, Nf(T)) Additional variable Nf = amount of free (volatile component) Molal fraction of free component Yf = Nf/∑iNi≠f = Xf/(1-Xf) = Nf /n (n in AnXNf) dH = T.dS + V.dP + ∑ μi.dNi ; (∂H/∂T)P, Ni = T.(∂S/∂T)P,Ni = CP,Ni (∂H/∂T)P,af = (∂H/∂T)P,Nf + (∂H/∂Nf)T .(∂Yf /∂T)af available from TG data Isoplethal CP : CP,Yf = (∂H/∂T)P,Nf Isodynamical CP: CP,af = (∂H/∂T)P,af Partial Molal Enthalpy : (∂H/∂Nf)T,P ≡ hf (∂H/∂Nf)T = (∂(H/n)/∂(Nf /n)T =(∂(H/n)/∂Yf)T ≡ hf = Hfo + Δhf CP,af = CP,Yf + hf. (∂Yf /∂T)af ; ΔsatCP = hf (∂Yf /∂T)af

Relative partial molal enthalpy Δhf Molar enthalpy of pure component f Relative partial molal enthalpy = + (∂H/∂Nf)T,P,Ni≠f ≡ hf (Yf ,T) = Hof (T) + Δhf (af ,T, Yf) Partial molal Gibbs energy gf = chemical potential μf (∂G/∂Nf)T,P,Ni≠f ≡ gf (Yf ,T) = Gof (T) + Δgf (af ,T, Yf) (∂G/∂Nf)T,P,Ni≠f ≡ μf (Yf ,T) = μof (T) + RT ln af (Yf) Gibbs-Helmholtz equation (∂(G/T)/∂(1/T))P,Ni = H → (∂(μf /T)/∂(1/T)) P,Ni= hf (∂(μf /T)/∂(1/T))P,Ni = (∂(Gof/T)/∂(1/T)) + R(∂ ln af /∂(1/T)) = hf Hof + Δhf = hf R (∂ ln af /∂(1/T))P,Nf= Δhf ΔsatCP = hf (∂Yf/∂T)af = [Hof + R (∂ ln af/∂(1/T))P,Nf](∂Yf /∂T)af

Thermal phtochability κfT and proper plutability κff ΔsatCP = hf (∂Yf /∂T)af = – hf κfT YBa2Cu3O6+z ; YO =(6+z)/6 → δ = z/6 „thermal phtochability“ (φτωχός = poor) κfT = – (∂Yf /∂T)af ≥ 0 κfT is obtainable from TG measurements (from the slope in Yf vs. T dependence) Thermogravimetry of YBa2Cu3O6+z „proper plutability“ (πλούτος = rich) κff = (∂Yf /∂ log af)T ≥ 0 κff = 1/(∂ log af /∂Yf) T κff is obtainable from coulometric titrations (from the slope in log af vs Yf dependence) Transforming κfT = – (∂Yf /∂T)af = + (∂Yf /∂(1/T))af /T2 and considering af = const +(∂Yf /∂(1/T)af (1/T2) = -(1/T2 )[∂ log af /∂(1/T)]/[∂ log af /∂Yf] κfT = - (Δhf /RT2)/(1/κff) = - κff (Δhf /RT2) ΔsatCP = - (Hof +Δhf) κfT = + (Hof +Δhf) (Δhf /RT2) κff

FeO4/3+δ ΔsatCP (J.mol-1.K-1) 30 pO2=1 pO2=0.01 pO2=0.0001 20 Metastable magnetite ΔsatCP (J.mol-1.K-1) Liquid 10 Haematite Stable magnetite Magnetite 1300 1400 1500 1600 1700 1800 1900 T (K)

(∂CP/∂Nf)T,P = ∂2H/∂T∂Nf=∂2H/∂Nf∂T= (∂hf/∂T)Yf,P Difference of heat capacity ΔdevCP(δ) = CP(δ≠0)-CP(δ=0) due to deviation from stoichiometry CP,b(Yf≠Yof) = CP,b(Yof) + ∫(∂CP,b/∂Yf)T,PdYf Partial molal heat capacity: (∂CP,b/∂Yf)T,P=∂(CP/n)/∂(Nf/n) =∂CP/∂Nf Maxwell-like relation (use of Clairaut's theorem published in 1743): (∂CP/∂Nf)T,P = ∂2H/∂T∂Nf=∂2H/∂Nf∂T= (∂hf/∂T)Yf,P δ ≡ Yf –Yof → dδ = dYf ; hf = Hof + Δhf → (∂hf/∂T)Yf = CPof + (∂Δhf/∂T)Yf Cpof (T) = (∂Hof /∂T) (= heat capacity of pure component f) CP,b(δ≠0) = CP,b(δ=0) + δ.Cpof +∫(∂Δhf/∂T)δ,P dδ ΔdevCP(δ) = δ.Cpof +∫(∂Δhf/∂T)δ,P dδ

Temperature dependence of relative partial molal enthalpy Δhf ΔdevCP(δ) = δ.Cpof +∫0δ(∂Δhf/∂T)δ,P dδ (∂Δhf/∂T)δ,P = R (∂[∂ ln af/∂(1/T)]δ/∂T)δ = = -(R/T2)(∂2 ln af/ ∂(1/T)2)δ if the dependence ln af vs (1/T) is linear for any δ (inside a given integration interval), then : (∂Δhf/∂T)δ,P = 0 so that : ΔdevCP(δ) ≅ δ.Cpof (it means validity of Neumann-Kopp rule)

ΔsatCP = [Hof + R (∂ ln af/∂(1/T))P,Nf](∂Yf /∂T)af = Quantities required for determination of nonstoichiometric contributions to heat capacity ΔsatCP = [Hof + R (∂ ln af/∂(1/T))P,Nf](∂Yf /∂T)af = ΔsatCP = - κfT (Hof +Δhf) = +(Hof +Δhf) (Δhf /RT2) κff 1/ κff = =(∂ log af /∂Yf)T κfT = – (∂Yf /∂T)af Thermodynamic tables Δhf = R.[∂ ln af/∂(1/T)]Yf Experiment results Implicit function F (Yf, log af, T) = 0 (∂Δhf/∂T) = -(R/T2)(∂2 ln af/ ∂(1/T)2)δ ΔdevCP(δ) = δ.Cpof +∫(∂Δhf/∂T)δ,P dδ How can it be fitted by crystal defect models ?

Makroskopické složky: (fenomenologické složky) Strukturní popis fáze (kontinua) Mikroskopické složky: Atomy, Molekuly Ionty, Elektrony/Díry Vakance/Intersticiály Příměsové atomy/ionty Termodynamický model fáze Tekuté: Plyny, Kapaliny Tuhé::Nekrystalické, Krystalické Makroskopické složky: (fenomenologické složky) Prvky, Sloučeniny Fenomenologický popis soustavy Chování a chemické složení fáze Waals, J. van der and Kohnstamm, P. (1927) Lehrbuch der Thermostatik I.,II.,

Thermodynamic model of defect crystal Thermodynamic model of crystalline phase: G = ∑μj nj = ∑(μoj + RT ln aj) nj nj = amounts of crystal defects obtained from balances of elements, electric charge and crystallographical sites in a form: nj = νjo+∑rνjr.λr +νjf.δ aj = activities of crystal defects are assumed as proportional to their amounts aj = kj.nj = kj.(νjo+∑rνjr.λr +νjf.δ) λr = conversion degree of r-th independent reaction between crystal defects r ∊ (1, R) ; R = M – N – S where M = number of defect species; N =number of chemical elements; S = number of crystal sublattices

Equilibrium amount of defects and its contribution to CP njeq = equilibrium amounts of crystal defects at δ = const njeq = νjo+∑νjr.λreq +νjf.δ where λreq are equilibrium degrees of conversion determined from conditions of minimum Gibbs free energy G (∂G/∂λr)δ,T,P =∑jνjr (μoj +RT ln aj) = ΔGor + RT ln Kr = 0 λreq = Ar exp (-ΔHor/RT); (∂λreq/∂T) = Ar (ΔHor/RT2) exp (-ΔHor/RT) Gibbs free energy G of the crystal with equilibrium amounts of defects is then: G = Gid + ∑r λreq (ΔGor + RT ln Kr ) and enthalpy H of crystal is given using Gibbs-Helmholtz equation: H = (∂(G/T)/∂(1/T)) = Hid + ∑r λreq (ΔHor + R [∂ ln Kr /∂(1/T)]) Heat capacity CP of crystals with equilibrium defects is then (approximately): CP = (∂H/∂T) ≅ CP,id + ∑r ΔHor (∂λreq/∂T) ΔcdfCP = CP – CPid = ∑r Ar (ΔHor2/RT2) exp (-ΔHor/RT)

Equilibrium nonstoichiometry (simple model) Thermodynamic model of crystalline phase: G = ∑μj nj = ∑(μoj + RT ln aj) nj nj = amounts of crystal defects obtained from balances of elements, electric charge and crystallographical sites in a form: nj = νjo+∑νjr.λr +νjf.δ aj = activities of crystal defects are assumed as proportional to their amounts aj = kj.nj = kj.(νjo+∑rνjr.λr +νjf.δ) at constant λr (conversion degrees of reactions between crystal defects) Equilibrium nonstoichiometry δeq at constant conversion degrees λr is determined from: (∂G/∂δ) T,P,λr, =∑jνjf (μoj +RT ln aj) = μof +RT ln af ≠ 0 ΔGoIf = ∑jνjf μoj - μof ; ln Kif = ∑jνjf ln aj - ln af Incorporation reaction: ΔGIf = ΔGoIf - RT ln Kif = 0

Δhf /R ≈ ΔHoIf /R → Δhf ≈ ΔHoIf Relative partial molal enthalpy Δhf and phtochability κfT and parameters of incorporation reaction (If) ΔGIf = [∑ νjf.Gjo – Gfo ] + RT [∑jνjf ln aj – ln af] = 0 ΔHoIf - TΔSoIf = ΔGoIf = – RT ln Kif ΔHoIf /T – ΔSoIf = - R ∑jνjf ln aj νjf + R ln af ln af = ΔHoIf /RT – ΔSoIf /R + ∑jνjf ln (kj.(νjo+∑rνjr.λr +νjf.δ) (∂ ln af/∂(1/T))δ = Δhf /R = ΔHoIf /R + ∑νjf (∂ ln (kj.nj)/ ∂(1/T)) if ∂ (∑j νjf ln kj nj)/∂(1/T) ≪ ΔHoIf /R then Δhf /R ≈ ΔHoIf /R → Δhf ≈ ΔHoIf κfT = – (∂Yf /∂T)af ≈ – (ΔGoIf /RT2)(∑jνjf /nj)

Dominating incorporation reaction (If) and proper plutability κff ln af = ΔGoIf /RT + ∑jνjf ln (kj.nj) = = ΔGoIf /RT +∑jνjf ln (nj) + ∑jνjf ln (kj.) If some defect (d) is dominating in the sum ∑jνjf ln (nj) : νdf ln (nd) ≫ ∑j≠dνjf ln (nj) and nd ∝ δ then ln af = ΔGoIf /RT + ∑jνjf ln (kj.) + ∑j≠dνjf ln (nj) + νdf ln (δ) log af ≈ B + νdf log δ log δ ≈ - B/νdf + (1/νdf ) log af κff = (∂Yf /∂ log af)T =(∂δ /∂ log af)T ; d ln δ = dδ/δ κff = δ (∂ ln δ /∂ log af)T = 2.303 δ. (∂ log δ /∂ log af)T κff ≈ 2.303 δ/νdf

KIO2 = ([Fe3+M]2.[O2-X]1.[□M]3/4/ ([Fe2+M]2.aO21/2) Incorporation reaction of oxygen (f = O2 ) in magnetite Fe3O4+γ (the simplest model) 2 Fe2+|M| + ½ O2(g) = 2 Fe3+|M| + O2-|X| + ¾ □ |M| (IO2) KIO2 = ([Fe3+M]2.[O2-X]1.[□M]3/4/ ([Fe2+M]2.aO21/2) KIO2 =[(2+2g )/(1-2g )]2 [3g /4]¾ aO2–½ δ = γ/3 ↔ γ = 3δ [□ M] = nd = (3 γ / 4) = (9 δ / 4) ; ΔGIO2 = ΔGoIO2 + RT ln KIO2 = 0 → ½ ln aO2 ~ ¾ ln γ → log γ ~ ²/₃ log aO2 d log aO2 /d log γ = 3/2 ↔ d log γ/d log aO2 =2/3 log Δ = log 3 + log γ – log (4+γ) log af ≈ B + νdf log δ ↔ νdf = 3/2; (1/νdf )= 2/3

log pO2 vs log Δ in Fe 3-ΔO4 log pO2 log Δ +2 dlog pO2/dlog Δ = 3/2 γ-Fe2O3 =(3/4)Fe8/3O4 +2 dlog pO2/dlog Δ = 3/2 log pO2 - 2 Models - 4 Frenkel defects Sockel H-G. „Coulometrische Titration an Übergangsmetalloxiden“, Dissertation, Technische Hochschule Clausthal (1968); H.G. Sockel, H. Schmalzried :Ber. Bunsen- ges. phys. Chem. 72 [1968] 745-754 - 6 without Fr. def. Inverse spinel experiment - 8 σ log Δ =σΔ /Δ - 4 - 3 - 2 - 1 log(1/3) ≅-0.48 log Δ

Range of partial molal enthalpy of oxygen in magnetite Flood & Hill (1957) : [kJ] 142 - 87 T = HI0-TSI0= RT ln{[(2+2g)/(1-2g)]2 [3g/4 ]¾}/aO2½} ln aO2 = 2 ln{[(2+2g)/(1-2g)]2 [3g/4 ]¾} - (2H0/R)(1/T) +2S0/R CP, aO2 = CP, YO + ½ hO2 (∂g/∂T) P,aO2,NFe (∂H/∂g) = ½ (∂H/∂NO2) = ½ (HoO2 + ΔhO2); ΔhO2 = f (g, T) ΔhO2 = R(∂ ln aO2 /∂(1/T))YO = - 2HI0 = -284 kJ Flood & Hill (1957) : ΔhO2 = f (g >0, T) = const = -284 kJ/mol O2 Gordeev (1966): ΔhO2 = f (g = 0, T) = -620 kJ/mol O2 !!!

Equilibrium nonstoichimetry – more general model for Δhf (∂G/∂δ) T,P,λr =∑jνjf (μoj +RT ln aj) = μof +RT ln af (∂G(δ,{λr(δ)})/∂δ) T,P = (∂G/∂δ) T,P,λr,+∑(∂G/∂λr)(∂λr/∂δ) (∂G/∂λr)δ,T,P = ΔGr = ΔGor + RT ln Kr (∂G/∂δ) T,P,= ΔGIf + ∑r ΔGr (∂λr/∂δ) = μof +RT ln af ∂(ΔGr /T)/∂(1/T) = ΔHr Δhf = R ∂ln af /∂(1/T) ≅ ΔHIf + ∑rΔHr (∂λr/∂δ) Interdependence coefficients κrf = (∂λr/∂δ)

Flood-Hill model of nonstoichiometric magnetite FeO3/4+δ (spinel structure) 2 Fe2+|M| + ½ O2(g) = 2 Fe3+|M| + O2-|X| + ¾ □ |M| (R1) Fey+|M| = Fey+|I| + □ |M| (R2) ((3y-1)/4)Fe2+|M| + ¾Fey+|I|+ ½O2(g) = ((2y+2)/4)Fe3+|M|+ O2-|X| (R3) Δhf = R ∂ln af /∂(1/T) ≅ ΔHIf + ∑rΔHr (∂λr/∂δ) Δhf = ΔHR1 + ΔHR2 (∂λr/∂δ) limδ→0 Δhf = ΔHR1 + (-4/3) ΔHR2 = ΔHR3

Relative partial molal entalpy of oxygen in nonstoichiometric magnetita Fe3O4+γ according to model with Frenkel defects -620 ≤ ΔhO2 = ΔHIO2 + ΔHFr (∂λFr/∂γ) ≤ - 284 kJ/mol O2 (∂λFr/∂γ) ≡ κFrf (γ) ∊ (-3/4);0) -284 kJ/mol κFrf =(∂λFr/∂γ) = 0 Dominating incorporation R1 -300 kJ/mol O2 -400 1350°C 1200°C 1500°C ΔhO2 1050°C -500 -600 -620 kJ/mol Dominating incorporation R3 κFrf =(∂λFr/∂γ) = -4/3 -5 -4 -3 -2 -1 log γ

T dΔhO/dT [k J/K/g-at. O] ΔhO [kJ/g-at O] 1000 δ -100 1 18.78 Flood & Hill 1957 1 18.78 4.15 = 1000.δ dΔhO/dT [k J/K/g-at. O] -200 .5 3.43 -300 Gordeev 1966 5 10 2 1.97 ΔhO [kJ/g-at O] 1.25 -400 ∫0δ (dΔhO/dT) dδ [J/K] 0.52 1 1300°C 1400°C 1500°C 5 10 T 1000 δ Spencer & Kubashewski 1978

Temperature dependence of relative partial molal enthalpy Δhf ΔdevCP(δ) = δ.Cpof +∫0δ(∂Δhf/∂T)δ,P dδ (∂Δhf/∂T)δ,P = R (∂[∂ ln af/∂(1/T)]δ/∂T)δ = = -(R/T2)(∂2 ln af/ ∂(1/T)2)δ if the dependence ln af vs (1/T) is linear for any δ (inside a given integration interval), then : (∂Δhf/∂T)δ,P = 0 so that : ΔdevCP(δ) ≅ δ.Cpof if δ = 0.004 then (0.004/2)xCPoO2 = 0.002x37 J/K  0.074 J/K (it means validity of Neumann-Kopp rule) However, in the case of magnetite FeO4/3+δ the contribution due to ∫0δ(∂Δhf/∂T)δ,P dδ in interval from δ=0 to δ=0.004 (at T=1400°C) is equal to about 2.0 J/K (in FeO4/3+δ), it is about 1,5% of the tabulated value (ca. 68 J/K).