1 / 39

Nestechiometrie , tepelná kapacita a krystalochemické modely fází

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. KOLOIDNÍ nepravý roztok. PRVEK. SLOUČENINA.

baylee
Télécharger la présentation

Nestechiometrie , tepelná kapacita a krystalochemické modely fází

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

  3. Chemické pojmy 1590 - slovo SOLUTION ve významu ROZTOK poprvé použito v angličtině 1610 – Beguin: První (nealchymická) učebnice chemie 1661 – Boyle : CHEMICKÝPRVEK (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, “ÜberdieDissociationder in WassergelöstenStoffe,” 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)

  4. Krystalové poruchy (defekty) 1926 – J. Frenkel - předpokládáexistenci kationtovýchvakancí a intersticiálů 1930 – Schottky a Wagner v publikaci „Theorie der geordnetenMischphasen“ vytvářejí základ pro popis krystalických látek, které nejsou tvořeny molekulami Walter Hermann Schottky 1886-1976 CarlWihelmWagner 1901-1977 JakovIljičFrenkel 1894-1952 Defekty jsou výhodné, neboť zvyšují entropii , a tím snižují volnou energii

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

  6. LinusPauling, Theprinciplesdeterminingthestructureofcomplexioniccrystals.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

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

  8. 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 PyrhotitFe1,0-0,8S ZrSx, CrSx Tuhé elektrolyty (CaTiO3-type) La(Sr,Ca)MnO3-δ Sr2(Sc1+xNb1-xO6-δ Sr3CaZr1-xTa1+xO9-δ Protonové elektrolyty (pyrochlore type) La2-xCaxZr2O7-δ Thermoelektrika (CuFeO2-type) CuCr1−xMgxO2+δ 2

  9. 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 Stechiometrický (daltonský) poměr no:mo 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 YO → FeO 1,10 1,20 Fe3O4 1,40 Fe2O3

  10. 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] 0 YXversus T [log aX =const] log aXversus T [YX =const] -2 1200 -4 1100 1000 -6 1.53 900 1.52 YX log aXversus YX [T=const] 1.51

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

  12. 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

  13. Tepelná kapacita - Historie 1760JosephBlack(in Glasgow) distinguishedlatentheat(transitionenthalpy) from „sensibleheat“ – specificheat (heatabsorbedatrisingtemperatureof a gram of substance by onedegree) 1819Dulong & Petit pointedouttheatomicheats(productsofspecificheatandatomic weight) ofseveralmetallic (solid) elementsequalapproximately to 3 R (R = universal gasconstant = 1,987 cal/K/mol = 8,3145 J/K/mol) as itisshown in following table: 3 R = 5,96 cal/(at.K) = 24,9 J/(at.K) 1831Franz Ernst Neumann : “Untersuchungenüber die specifischeWärmederMineralien” 1864Hermann Kopp(1817-1892): „molecular“ heatofcompoundequalsapproximately to the sum ofatomicheatsofcontainedelements. „Molecular“ heatofcompoundAB2C4 : CAB2O4= CA + 2 CB + 4 CC 1871 James C. Maxwell: distinguishedisochoric CVandisobaricCPheatcapacity 1907Albert Einstein : heatcapacityofidealcrystal : 1912Peter Debye : heatcapacityofidealcrystal : 1922 Walter Schottky: anomalyheatcapacityoftwolevelsystemsatlowtemperatures 1928 Arnold Sommerfeld(1886-1951): heatcapacityofelectrons in metals ΔelCV= γT 1952 K. Kobayashi:heatcapacitydue to FrenkeldefectsformationΔFrC

  14. Temperature dependence ofheatcapacityCandsomeofitscontributions CP ΔCdil CV Al CP= CV+ ΔdilC vibrational ΔdilC=VT.a2/b CP = Cvib + ΔelC + ΔmgC + ΔCdil Cvib = Char +Canh dilation electronic magnetic

  15. 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 Electronic Magnetic Dilation Crystal defect formations Δcdf C = ΔSchC + ΔFrC + ΔehC + … Δdil C=VT.a2/b Electron- positron pairs Schottky defects Frenkel defects ΔFrC = AFr [(ΔHFr)2/2RT2]exp(-ΔHFr/2RT)

  16. IsochoricCvandisobaricCP heatcapacities Isobaric conditions Constant pressure P: Isochoric conditions Constant volume V: 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

  17. Differencebetweenisobaricandisochoricheatcapacity (∂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 ≡ΔdilC = T.V. α2/β

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

  19. 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 variableNf= 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

  20. RelativepartialmolalenthalpyΔhf Partial molal enthalpy 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

  21. 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 κfTisobtainable 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 κffis obtainable from coulometric titrations (from the slope in log af vs Yf dependence) TransformingκfT = – (∂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

  22. 30 pO2=1 pO2=0.01 pO2=0.0001 FeO4/3+δ 20 Metastable magnetite ΔsatCP (J.mol-1.K-1) Liquid 10 Stable magnetite Haematite Magnetite 0 1300 1400 1500 1600 1700 1800 1900 T (K)

  23. 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

  24. 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)

  25. 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 ?

  26. 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 ThermostatikI.,II.,

  27. 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

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

  29. 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 nonstoichiometryδeq at constant conversion degrees λris determined from: (∂G/∂δ)T,P,λr,=∑jνjf(μoj+RT ln aj) = μof+RT ln af≠ 0 ΔGoIf = ∑jνjfμoj- μof; ln Kif = ∑jνjfln aj - ln af ΔGIf = ΔGoIf - RT ln Kif = 0 Incorporation reaction:

  30. Relative partial molal enthalpy Δhf and phtochability κfTand parameters of incorporation reaction (If) ΔGIf = [∑ νjf.Gjo– Gfo] + RT[∑jνjf ln aj– ln af] = 0 ΔHoIf - TΔSoIf= ΔGoIf = – RTln Kif ΔHoIf /T – ΔSoIf= - R ∑jνjf ln ajνjf + 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/Rthen Δhf /R ≈ ΔHoIf /R → Δhf ≈ ΔHoIf κfT = – (∂Yf/∂T)af ≈ – (ΔGoIf /RT2)(∑jνjf /nj)

  31. 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∝δthen 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 lnδ = dδ/δ κff= δ(∂ ln δ/∂ log af)T = 2.303 δ. (∂ log δ/∂ log af)T κff≈ 2.303 δ/νdf

  32. 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]¾aO2–½ δ = γ/3 ↔ γ = 3δ [□ M] = nd= (3 γ / 4) = (9 δ / 4) ; ΔGIO2= ΔGoIO2+ RT lnKIO2 = 0 → ½ lnaO2~ ¾ lnγ → 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 + νdflog δ ↔ νdf= 3/2; (1/νdf)= 2/3

  33. logpO2vs logΔ in Fe3-ΔO4 γ-Fe2O3 =(3/4)Fe8/3O4 +2 0 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 σlog Δ=σΔ/Δ - 4 - 3 - 2 - 1 log(1/3) ≅-0.48 log Δ

  34. Rangeofpartialmolalenthalpyof oxygen in magnetite Flood & Hill (1957) : [kJ] 142 - 87 T = HI0-TSI0= RT ln{[(2+2g)/(1-2g)]2 [3g/4 ]¾}/aO2½} lnaO2 = 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!!!

  35. Equilibriumnonstoichimetry –more general model forΔhf (∂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 + ∑rΔGr (∂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)

  36. Flood-Hill model ofnonstoichiometric 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

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

  38. -100 Flood & Hill 1957 1 18.78 dΔhO/dT [k J/K/g-at. O] 4.15 = 1000.δ -200 .5 3.43 0 -300 Gordeev 1966 5 10 2 1.97 ΔhO[kJ/g-at O] 1.25 ∫0δ (dΔhO/dT) dδ[J/K] -400 1 0.52 0 1300°C 1400°C 1500°C 5 10 T 1000 δ Spencer & Kubashewski 1978

  39. 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 ifthe dependence lnafvs (1/T) islinearforanyδ (inside a givenintegration interval), then : (∂Δhf/∂T)δ,P = 0 sothat : ΔdevCP(δ) ≅δ.Cpof ifδ = 0.004 then(0.004/2)xCPoO2 = 0.002x37 J/K  0.074J/K (itmeans validity ofNeumann-Kopp rule) However, in the case of magnetite FeO4/3+δthecontributiondue to ∫0δ(∂Δhf/∂T)δ,P dδin intervalfromδ=0 to δ=0.004 (at T=1400°C) isequal to about2.0 J/K (in FeO4/3+δ), itisabout 1,5% ofthe tabulatedvalue (ca. 68 J/K).

More Related