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

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Prezentace na téma: "Nestechiometrie, tepelná kapacita a krystalochemické modely fází Pavel Holba NTC ZČU Plzeň 23. duben 2013."— Transkript prezentace:

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 PRVEKSLOUČENINAHOMOGENNÍ pravý ROZTOK HETEROGENNÍ směs fází daltonidberthollid HOMOGENNÍ SOUSTAVA HETEROGENNÍ SOUSTAVA DISPERSNÍ SOUSTAVA KOLOIDNÍ SOUSTAVA KOLOIDNÍ nepravý roztok Selmi (1845): pseudosolution Graham (1860) : colloid

3 Chemické pojmy 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, Gassendi – Dalton: ATOM 1811 – Avogadro Cannizaro: MOLEKULA 1834 – Faraday: ION 1878 – Gibbs: SLOŽKA & FÁZE S. Arrhenius, “Über die Dissociation der in Wasser gelösten Stoffe,” Z. Phys. Chem., 1887, 1, – van´t Hoff: TUHÝ ROZTOK Stoney – Millikan: : ELEKTRON 1912 – Kurnakov: BERTHOLLID & DALTONID 1926 – Frenkel : VAKANCE + INTERSTICIÁL 1926 – LEWIS : FOTON 1930 – Schottky & Wagner: KRYSTALOVÝ DEFEKT Heisenberg: ELEKTRONOVÁ DÍRA 1954 – Huggins: STRUKTON Cotton: KLASTR (CLUSTER)

4 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 Carl Wihelm Wagner Jakov Iljič Frenkel Defekty jsou výhodné, neboť zvyšují entropii, a tím snižují volnou energii

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

6 Linus Pauling, The principles determining the structure of complex ionic crystals. J. Am. Chem. Soc. 51 (1929) Uspořádání PočetPoloměr Příklad koulí (R=1) vrcholů dutiny struktury Krychle (cube) 8R =0.732 CaF 2 Oktaedr 6R ≥ NaCl Tetraedr 4R ≥0.225 ZnS

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

8 Okruhy nestechiometrických látek Magnetické ferity (Mg, Zn, Mn )x Fe 3-x O 4+ γ Tuhé elektrolyty (CaF 2 -type) (Ca,Y)(Zr, Hf, Th) 1-x O 2-x Hydridy PdH 0,7,, TiHx NbH x, GdH x Supravodivé oxidy YBa 2 Cu 3 O 7- δ Bi 2 Sr 2 Ca 2 Cu 3 O 10+ δ Tl 2 Ba 2 Ca 2 Cu 3 O 10+ δ TlBa 2 Ca 2 Cu 3 O 9+ δ Hg 2 Ba 2 Ca 2 Cu 3 O 8+ δ Oxidy aktinoidů (U, Pu,Cm, Am) O 2-x CaUO 4+x Oxidy lantanoidů CeO 2-x PrO 1+x Hydráty metanu CH 4.(6  x)H 2 O Intermetalické sloučeniny CoSn 0,69-0,72 Wolframové a molybdenové bronze (W, Mo)O 3-x Chalkogenidy Pyrhotit Fe 1,0-0,8 S ZrSx, CrSx Tuhé elektrolyty (CaTiO 3 -type) La(Sr,Ca)MnO 3-δ Sr 2 (Sc 1+x Nb 1-x O 6-δ Sr 3 CaZr 1-x Ta 1+x O 9-δ Thermoelektrika (CuFeO 2 -type) CuCr 1−x Mg x O 2+δ 2 Protonové elektrolyty (pyrochlore type) La 2-x Ca x Zr 2 O 7-δ

9 Složení a nestechiometrie Homogenní látka PrvekSloučeninaRoztok Daltonid Bertollid Fe 2 O 3 Fe 3 O 4 Fe:O = n:m → Fe n O m → Fe n O mo+  → Fe no-  O m FeO Molární zlomek: X O =m/(n+m) FeO X O =1/2 =0,500 FeO 1,056 X O =1,056/2,056 =0,514 FeO 1,158 X O =1,158/2,158 =0,537 Fe 3 O 4: X O =4/7 =0,571 Fe 3 O 4,112 X O =4,112/7,112 =0,587 Fe 2 O 3 X O =3/5 =0,600 Molální zlomek: Y O =m/n FeO: Y O =1/1 = 1,000 FeO 1,056 Y O =1,056/1 =1,056 FeO 1,158 Y O =1,158/1 =1,158 Fe 3 O 4: Y O =4/3 =1,333 Fe 3 O 4,112 Y O =4,112/3 =1,371 Fe 2 O 3 Y O =3/2 =1,500 Y O → 1,101,201,40 Stechiometrický (daltonský) poměr n o :m o Odchylka od stechiometrie :  = m-m o ;  = n o -n [(3-  )/3] Fe 3 O 4+  = Fe 3-  O 4  = 3  /(4+  ) ;  = 4  /(3+  )

10 log a X YXYX T [°C] log a X versus T [Y X =const] log a X versus Y X [T=const] Y X versus T [log a X =const] Molální zlomek Y X (volné složky X) ve fázi AX Y ve vztahu k aktivitě volné složky a X a teplotě T P = const.

11 Vztah mezi teplotou T, obsahem Y f, a aktivitou a f volné složky vyjadřuje implicitní funkce F (Y f, a f, T) = 0, [P=const] kterou lze charakterizovat následující trojicí veličin: 1.relativní parciální molární entalpií Δh f Δh f = R (∂ ln a f /∂(1/T)) Y f 2.tepelnou ftochabilitou (φτωχός = chudý) κ fT κ fT = – (∂Y f /∂T) a f 3.vlastní (proper) plutabilitou (πλούτος = bohatý) κ ff κ ff = (∂Y f /∂ log a f ) T ≥ 0 mezi nimiž platí vztah: κ fT = - κ ff ( Δ h f /RT 2 )

12 Figure from Inaba H. & Naito K. (1977): Heat Capacity of Nonstoichiometric Compounds, NETSU 4 [1] U 0.8 Pu 0.2 O 2+δ δ =0.08 δ =0.045 δ =0.000 δ = – C P [cal/mol/K] T [K] Chaleur spécifique a haute temperature des oxydes d'uranium et de plutonium (1970) Affortit C. & Marcon J-.P.; Rev. Int. Hautes Temp. Refract. 7, Heat capacity of „mixed oxide fuel (MOX)“ U 0.8 Pu 0.2 O 2+δ

13 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 ( ): „molecular“ heat of compound equals approximately to the sum of atomic heats of contained elements. „Molecular“ heat of compound AB 2 C 4 : C AB 2 O 4 = C A + 2 C B + 4 C C 1871 James C. Maxwell: distinguished isochoric C V and isobaric C P 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 ( ): heat capacity of electrons in metals Δ el C V = γT 1952 K. Kobayashi: heat capacity due to Frenkel defects formation Δ Fr C Tepelná kapacita - Historie

14 magnetic Temperature dependence of heat capacity C and some of its contributions Al CPCP CVCV ΔC dil C P = C vib + Δ el C + Δ mg C + ΔC dil vibrational electronic dilation C vib = C har + C anh C P = C V + Δ dil C Δ dil C =VT.  2 / 

15 Contributions to heat capacity C P C p = C har + C anh + Δ dil C + Δ el C + Δ mg C + Δ cdf C + Δ others C Schottky defects Frenkel defects Electron- positron pairs Dilation = C V Ideal crystal vibrations ElectronicMagnetic Non- stoichio- metry ? Δ cdf C = Δ Sch C + Δ Fr C + Δ eh C + … Crystal defect formations Δ dil C =VT.  2 /  Δ Fr C = A Fr [(ΔH Fr ) 2 /2RT 2 ]exp(-ΔH Fr /2RT) C P = A 1 + A 2.T + A 3 /T 2 + A 4 / √T + A 5.T 2 According to thermodynamic tables ( empirical polynomial model): According to thermodynamic and physical ( theoretical ) models: Neumann- Kopp & Einstein & Debye

16 Isochoric C v and isobaric C P heat capacities Isochoric conditions Constant volume V: Increasing temperature T Increasing pressure P Isobaric conditions Constant pressure P: Increasing temperature T Increasing volume V C v = (∂U/∂T) V C P = (∂H/∂T) P Internal energy UEnthalpy H = U + P.V Δ dil C = C P - C v

17 Difference between isobaric and isochoric heat capacity (∂U/∂T) V = T (∂S/∂T) V = C V ; (∂H/∂T) P =T (∂S/∂T) P = C P ; 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 C P ≡ (∂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 = C v + (T.(∂S/∂V) T - P).(∂V/∂T) P (∂S/∂V) T = ∂ 2 F/∂T∂V=∂ 2 F/∂V∂T = (∂P/∂T) V (∂U/∂T) P = C v + (T.(∂P/∂T) V - P).(∂V/∂T) P C P = [C v + (T.(∂P/∂T) V - P).(∂V/∂T) P ] + P.(∂V/∂T) P C P = C V + T.(∂P/∂T) v.(∂V/∂T) P + {- P (∂V/∂T) P + P.(∂V/∂T) P } C P = C V +T.(∂P/∂T) v.(∂V/∂T) P (∂P/∂T) v = -(∂V/∂T) P /(∂V/∂P) T = (-V.α)/(-V.β) = α/β C P = C V +T.(∂V/∂T) P α/β = C V - T V α 2 /β Maxwell´s relation 1860 (Clairaut 1743): C P = C V + T.V. α 2 /β ; C P - C V ≡ Δ dil C = T.V. α 2 /β

18 Heat capacity and nonstoichiometry Measured in sealed ampoule Heat capacity of sample with free component f (element X) Measured under controlled atmosphere Constant fraction of volatile component Y f Constant activity of volatile component a f A n X m+ γ = n A 1 X y Y f = (m+ Δ )/n = = y = m/n + δ = y o +δ a f = p f /p o f C P, Y f C P, a f Δ sat C P = C P, a f – C P, Y f Isoplethal conditions Isodynamical conditions Molal fraction Y f = N f / ∑ i N i≠f Y f = X f /(1-X f )

19 Saturation contribution Δ sat C P Internal energy as a compound function U = U(T, V(T)) dU = T.dS – P.dV + ∑ μ i.dN i ;(∂U/∂T) V, N i = T.(∂S/∂T) V,Ni = C V,Ni (∂U/∂T) P = (∂U/∂T) V + (∂U/∂V) T.(∂V/∂T) P (∂H/∂N f ) T = (∂(H/n)/∂(N f /n) T =(∂(H/n)/∂Y f ) T ≡ h f = H f o + Δh f available from TG data Partial Molal Enthalpy : (∂H/∂N f ) T,P ≡ h f C P,af = C P,Yf + h f. (∂Y f /∂T) a f ; Δ sat C P = h f (∂Y f /∂T) a f Isoplethal C P : C P,Yf = (∂H/∂T) P,N f Isodynamical C P : C P,af = (∂H/∂T) P,a f Enthalpy as a compound function H = H(T, N f (T)) Additional variable N f = amount of free (volatile component) Molal fraction of free component Y f = N f /∑ i N i≠f = X f /(1-X f ) = N f /n (n in A n X N f ) dH = T.dS + V.dP + ∑ μ i.dN i ; (∂H/∂T) P, N i = T.(∂S/∂T) P,Ni = C P,Ni (∂H/∂T) P,a f = (∂H/∂T) P,N f + (∂H/∂N f ) T.(∂Y f /∂T) a f

20 Relative partial molal enthalpy Δh f Partial molal enthalpy = Molar enthalpy of pure component f + Relative partial molal enthalpy (∂H/∂N f ) T,P,N i≠f ≡ h f (Y f,T) = H o f (T) + Δh f (a f,T, Y f ) Partial molal Gibbs energy g f = chemical potential μ f (∂G/∂N f ) T,P,N i≠f ≡ g f (Y f,T) = G o f (T) + Δg f (a f,T, Y f ) (∂G/∂N f ) T,P,N i≠f ≡ μ f (Y f,T) = μ o f (T) + RT ln a f (Y f ) Gibbs-Helmholtz equation (∂(G/T)/∂(1/T)) P,N i = H → (∂(μ f /T)/∂(1/T)) P,N i = h f (∂(μ f /T)/∂(1/T)) P,N i = (∂(G o f /T)/∂(1/T)) + R(∂ ln a f /∂(1/T)) = h f H o f + Δh f = h f R (∂ ln a f /∂(1/T)) P,N f = Δh f Δ sat C P = h f (∂Y f /∂T) a f = [H o f + R (∂ ln a f /∂(1/T)) P,N f ](∂Y f /∂T) a f

21 Thermal phtochability κ fT and proper plutability κ ff Thermogravimetry of YBa 2 Cu 3 O 6+z YBa 2 Cu 3 O 6+z ; Y O =(6+z)/6 → δ = z/6 Transforming κ fT = – (∂Y f /∂T) a f = + (∂Y f /∂(1/T)) a f /T 2 and considering a f = const +(∂Y f /∂(1/T) a f (1/T 2 ) = -(1/T 2 )[∂ log a f /∂(1/T)]/[ ∂ log a f / ∂ Y f ] κ fT = - ( Δ h f /RT 2 )/(1/ κ ff ) = - κ ff ( Δ h f /RT 2 ) „thermal phtochability“ (φτωχός = poor) κ fT = – (∂Y f /∂T) a f ≥ 0 „proper plutability“ (πλούτος = rich) κ ff = (∂Y f /∂ log a f ) T ≥ 0 κ ff = 1/ ( ∂ log a f / ∂ Y f ) T Δ sat C P = h f (∂Y f /∂T) a f = – h f κ fT κ fT is obtainable from TG measurements (from the slope in Y f vs. T dependence) κ ff is obtainable from coulometric titrations (from the slope in log a f vs Y f dependence) Δ sat C P = - (H o f +Δ h f ) κ fT = + (H o f +Δ h f ) ( Δ h f /RT 2 ) κ ff

22 Magnetite Haematite Liquid p O 2 =1 p O 2 =0.01 p O 2 = Metastable magnetite Stable magnetite Δ sat C P (J.mol -1.K -1 ) FeO 4/3+ δ T (K)

23 Difference of heat capacity Δ dev C P (δ) = C P (δ≠0)-C P (δ=0) due to deviation from stoichiometry C P,b (Y f ≠Y o f ) = C P,b (Y o f ) + ∫(∂C P,b /∂Y f ) T,P dY f Partial molal heat capacity: (∂C P,b /∂Y f ) T,P =∂(C P /n)/∂(N f /n) =∂C P /∂N f Maxwell-like relation (use of Clairaut's theorem published in 1743) :Clairaut's theorem (∂C P /∂N f ) T,P = ∂ 2 H/∂T∂N f =∂ 2 H/∂N f ∂T= (∂h f /∂T) Y f,P δ ≡ Y f –Y o f → dδ = dY f ; h f = H o f + Δh f → (∂h f /∂T) Y f = C P o f + (∂ Δ h f /∂T) Y f C p o f (T) = (∂H o f /∂T) (= heat capacity of pure component f) C P,b (δ≠0) = C P,b (δ=0) + δ.C p o f +∫(∂ Δ h f /∂T) δ,P dδ Δ dev C P (δ) = δ.C p o f +∫(∂ Δ h f /∂T) δ,P dδ

24 Temperature dependence of relative partial molal enthalpy Δh f (∂ Δ h f /∂T) δ,P = R ( ∂[∂ ln a f /∂(1/T)] δ /∂T) δ = = -(R/T 2 )(∂ 2 ln a f / ∂(1/T) 2 ) δ Δ dev C P (δ) = δ.C p o f +∫ 0 δ (∂ Δ h f /∂T) δ,P dδ if the dependence ln a f vs (1/T) is linear for any δ (inside a given integration interval), then : (∂ Δ h f /∂T) δ,P = 0 so that : Δ dev C P (δ) ≅ δ. C p o f (it means validity of Neumann-Kopp rule)

25 Quantities required for determination of nonstoichiometric contributions to heat capacity Δ h f = R. [∂ ln a f /∂(1/T)] Yf Δ dev C P (δ) = δ.C p o f +∫(∂ Δ h f /∂T) δ,P dδ (∂ Δ h f /∂T) = -(R/T 2 )(∂ 2 ln a f / ∂(1/T) 2 ) δ κ fT = – (∂Y f /∂T) a f 1/ κ ff = = ( ∂ log a f / ∂ Y f ) T Δ sat C P = [H o f + R (∂ ln a f /∂(1/T)) P,N f ](∂Y f /∂T) a f = Δ sat C P = - κ fT ( H o f +Δ h f ) = +( H o f +Δ h f ) ( Δ h f /RT 2 ) κ ff Thermodynamic tables Experiment results Implicit function F (Y f, log a f, T) = 0 How can it be fitted by crystal defect models ?

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

27 Thermodynamic model of defect crystal Thermodynamic model of crystalline phase: G = ∑μ j n j = ∑(μ o j + RT ln a j ) n j n j = amounts of crystal defects obtained from balances of elements, electric charge and crystallographical sites in a form: n j = ν jo +∑ r ν jr.λ r +ν jf.δ a j = activities of crystal defects are assumed as proportional to their amounts a j = k j.n j = k j.(ν jo +∑ r ν jr.λ r +ν jf.δ) λ r = 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 C P n j eq = equilibrium amounts of crystal defects at δ = const n j eq = ν jo +∑ν jr.λ r eq +ν jf.δ where λ r eq are equilibrium degrees of conversion determined from conditions of minimum Gibbs free energy G (∂G/∂λ r ) δ,T,P =∑ j ν jr (μ o j +RT ln a j ) = ΔG o r + RT ln K r = 0 λ r eq = A r exp (-ΔH o r /RT); (∂λ r eq /∂T) = A r (ΔH o r /RT 2 ) exp (-ΔH o r /RT) Gibbs free energy G of the crystal with equilibrium amounts of defects is then: G = G id + ∑ r λ r eq (ΔG o r + RT ln K r ) and enthalpy H of crystal is given using Gibbs-Helmholtz equation : H = (∂(G/T)/∂(1/T)) = H id + ∑ r λ r eq (ΔH o r + R [∂ ln K r /∂(1/T)]) Heat capacity C P of crystals with equilibrium defects is then (approximately): C P = (∂H/∂T) ≅ C P,id + ∑ r ΔH o r (∂λ r eq /∂T) Δ cdf C P = C P – C P id = ∑ r A r (ΔH o r 2 /RT 2 ) exp (-ΔH o r /RT)

29 Equilibrium nonstoichiometry (simple model) Thermodynamic model of crystalline phase: G = ∑μ j n j = ∑(μ o j + RT ln a j ) n j n j = amounts of crystal defects obtained from balances of elements, electric charge and crystallographical sites in a form: n j = ν jo +∑ν jr.λ r +ν jf.δ a j = activities of crystal defects are assumed as proportional to their amounts a j = k j.n j = k j.(ν jo +∑ r ν jr.λ r +ν jf.δ) at constant λ r ( conversion degrees of reactions between crystal defects) Equilibrium nonstoichiometry δ eq at constant conversion degrees λ r is determined from: (∂G/∂δ) T,P,λr, =∑ j ν jf (μ o j +RT ln a j ) = μ o f +RT ln a f ≠ 0 ΔG o If = ∑ j ν jf μ o j - μ o f ; ln K if = ∑ j ν jf ln a j - ln a f ΔG If = ΔG o If - RT ln K if = 0 Incorporation reaction:

30 Relative partial molal enthalpy Δh f and phtochability κ fT and parameters of incorporation reaction (If) ΔG If = [∑ ν jf.G j o – G f o ] + RT [∑ j ν jf ln a j – ln a f ] = 0 ΔH o If - TΔS o If = ΔG o If = – RT ln K if ΔH o If /T – ΔS o If = - R ∑ j ν jf ln a j ν jf + R ln a f ln a f = ΔH o If /RT – ΔS o If /R + ∑ j ν jf ln (k j.(ν jo +∑ r ν jr.λ r +ν jf.δ) (∂ ln a f /∂(1/T)) δ = Δh f /R = ΔH o If /R + ∑ν jf (∂ ln (k j.n j )/ ∂(1/T)) if ∂ (∑ j ν jf ln k j n j )/∂(1/T) ≪ ΔH o If /R then Δh f /R ≈ ΔH o If /R → Δh f ≈ ΔH o If κ fT = – (∂Y f /∂T) a f ≈ – ( ΔG o If /RT 2 )(∑ j ν jf /n j )

31 Dominating incorporation reaction (If) and proper plutability κ ff ln a f = ΔG o If /RT + ∑ j ν jf ln (k j.n j ) = = ΔG o If /RT +∑ j ν jf ln (n j ) + ∑ j ν jf ln (k j.) If some defect (d) is dominating in the sum ∑ j ν jf ln (n j ) : ν df ln (n d ) ≫ ∑ j≠d ν jf ln (n j ) and n d ∝ δ then ln a f = ΔG o If /RT + ∑ j ν jf ln (k j.) + ∑ j≠d ν jf ln (n j ) + ν df ln (δ) log a f ≈ B + ν df log δ log δ ≈ - B/ ν df + (1/ ν df ) log a f κ ff = (∂Y f /∂ log a f ) T =(∂δ /∂ log a f ) T ; d ln δ = dδ/δ κ ff = δ (∂ ln δ /∂ log a f ) T = δ. (∂ log δ /∂ log a f ) T κ ff ≈ δ/ ν df

32 Incorporation reaction of oxygen (f = O 2 ) in magnetite Fe 3 O 4+γ (the simplest model) 2 Fe 2+ |M| + ½ O 2 (g) = 2 Fe 3+ |M| + O 2- |X| + ¾ □ |M| ( I O 2 ) K I O 2 = ([Fe 3+ M ] 2.[O 2- X ] 1.[ □ M ] 3/4 / ( [Fe 2+ M ] 2.a O 2 1/2 ) K I O 2 =[(2+2  )/(1-2  )] 2 [3  /4] ¾ a O2 –½ δ = γ/3 ↔ γ = 3 δ [ □ M ] = n d = (3 γ / 4) = (9 δ / 4) ; ΔG IO 2 = ΔG o IO 2 + RT ln K I O 2 = 0 → ½ ln a O2 ~ ¾ ln γ → log γ ~ ²/₃ log a O2 d log a O2 /d log γ = 3/2 ↔ d log γ/d log a O2 =2/3 log Δ = log 3 + log γ – log (4+γ) log a f ≈ B + ν df log δ↔ ν df = 3/2; (1/ν df )= 2/3

33 log p O2 vs log Δ in Fe 3-Δ O log Δ log p O2 experiment Frenkel defects without Fr. def. Inverse spinel Models σ log Δ =σ Δ /Δ dlog p O2 /dlog Δ = 3/2 γ-Fe 2 O 3 =(3/4)Fe 8/3 O 4 Sockel H-G. „Coulometrische Titration an Übergangsmetalloxiden“, Dissertation, Technische Hochschule Clausthal (1968); H.G. Sockel, H. Schmalzried :Ber. Bunsen- ges. phys. Chem. 72 [1968] log(1/3) ≅ -0.48

34 Range of partial molal enthalpy of oxygen in magnetite Flood & Hill (1957) : [kJ] T =  H I 0 -T  S I 0 = RT ln{[(2+2  )/(1-2  )] 2 [3  ¾ }/a O2 ½ } C P, a O 2 = C P, Y O   ½ h O2 (∂  ∂T) P,a O 2,N Fe (∂H/∂  ) = ½ (∂H/∂N O2 ) = ½ (H o O2 + Δ h O2 ) ; Δ h O2 = f ( , T) Δ h O2 = R(∂ ln a O2 /∂(1/T)) Y O = - 2  H I 0 = -284 kJ ln a O2 = 2 ln{[(2+2  )/(1-2  )] 2 [3  ¾ } - (2  H 0 /R)(1/T) +2  S 0 /R Gordeev (1966): Δ h O2 = f ( , T) = -620 kJ/mol O 2 !!! Flood & Hill (1957) : Δ h O2 = f ( , T) = const = -284 kJ/mol O 2

35 Equilibrium nonstoichimetry – more general model for Δ h f (∂G/∂δ) T,P,λ r =∑ j ν jf (μ o j +RT ln a j ) = μ o f +RT ln a f (∂G(δ,{λ r (δ)})/∂δ) T,P = (∂G/∂δ) T,P,λ r, +∑(∂G/∂λ r )(∂λ r /∂δ) (∂G/∂λ r ) δ,T,P = ΔG r = ΔG o r + RT ln K r (∂G/∂δ) T,P, = ΔG If + ∑ r ΔG r (∂λ r /∂δ) = μ o f +RT ln a f ∂(ΔG r /T)/∂(1/T) = ΔH r Δh f = R ∂ln a f /∂(1/T) ≅ ΔH If + ∑ r ΔH r (∂λ r /∂δ) Interdependence coefficients κ rf = (∂λ r /∂δ)

36 Flood-Hill model of nonstoichiometric magnetite FeO 3/4+δ (spinel structure) 2 Fe 2+ |M| + ½ O 2 (g) = 2 Fe 3+ |M| + O 2- |X| + ¾ □ |M| (R1) Fe y+ |M| = Fe y+ |I| + □ |M| (R2) ((3y-1)/4) Fe 2+ |M| + ¾Fe y+ |I|+ ½O 2 (g) = ((2y+2)/4)Fe 3+ |M|+ O 2- |X|(R3) Δh f = R ∂ln a f /∂(1/T) ≅ ΔH If + ∑ r ΔH r (∂λ r /∂δ) Δh f = ΔH R1 + ΔH R2 (∂λ r /∂δ) lim δ→0 Δh f = ΔH R1 + (-4/3) ΔH R2 = ΔH R3

37 Relative partial molal entalpy of oxygen in nonstoichiometric magnetita Fe 3 O 4+γ according to model with Frenkel defects kJ/mol O log γ -284 kJ/mol -620 kJ/mol 1050°C 1200°C 1350°C 1500°C -620 ≤ Δ h O2 = ΔH IO2 + ΔH Fr (∂λ Fr /∂γ) ≤ kJ/mol O 2 Δh O2 Dominating incorporation R1 Dominating incorporation R3 κ Frf =(∂λ Fr /∂γ) = 0 κ Frf =(∂λ Fr /∂γ) = -4/3 (∂λ Fr /∂γ) ≡ κ Frf (γ) ∊ (-3/4);0)

38 °C1400°C1500°C Flood & Hill 1957 Gordeev 1966 Δh O [k J /g-at O] T 4.15 = δ δ ∫ 0 δ (d Δh O /dT) dδ [ J /K] d Δh O /dT [k J /K/g-at. O] Spencer & Kubashewski 1978

39 Temperature dependence of relative partial molal enthalpy Δh f (∂ Δ h f /∂T) δ,P = R ( ∂[∂ ln a f /∂(1/T)] δ /∂T) δ = = -(R/T 2 )(∂ 2 ln a f / ∂(1/T) 2 ) δ Δ dev C P (δ) = δ.C p o f +∫ 0 δ (∂ Δ h f /∂T) δ,P dδ if the dependence ln a f vs (1/T) is linear for any δ (inside a given integration interval), then : (∂ Δ h f /∂T) δ,P = 0 so that : Δ dev C P (δ) ≅ δ. C p o f if δ = then (0.004/2)xC P o O 2 = 0.002x37 J/K  J/K (it means validity of Neumann-Kopp rule) However, in the case of magnetite FeO 4/3+δ the contribution due to ∫ 0 δ (∂ Δ h f /∂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 FeO 4/3+δ ), it is about 1,5% of the tabulated value (ca. 68 J/K).


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

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