Nestechiometrie, tepelná kapacita a krystalochemické modely fází

Pavel Holba NTC ZČU Plzeň

23. duben 2013

Systém chemických látek na počátku XX. století LÁTKA

ČISTÁ LÁTKAchemické individuum

SMĚSNÁ LÁTKAsměs čistých látek

PRVEK SLOUČENINA HOMOGENNÍ pravý ROZTOK

HETEROGENNÍsměs fází

daltonid berthollid

HOMOGENNÍ SOUSTAVA

HETEROGENNÍSOUSTAVA

DISPERSNÍ SOUSTAVA

KOLOIDNÍSOUSTAVA

KOLOIDNÍnepravý roztok

Selmi (1845): pseudosolutionGraham (1860) : colloid

Chemické pojmy

1590 - slovo SOLUTION ve významu ROZTOK poprvé použito v angličtině 1610 – Beguin: První (nealchymická) učebnice chemie1661 – 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: MOLEKULA1834 – Faraday: ION1878 – Gibbs: SLOŽKA & FÁZE1887 - S. Arrhenius, “Über die Dissociation der in Wasser gelösten Stoffe,”

Z. Phys. Chem., 1887, 1, 631-648.1890 – van´t Hoff: TUHÝ ROZTOK1891 - Stoney + 1911 – Millikan: : ELEKTRON 1912 – Kurnakov: BERTHOLLID & DALTONID1926 – Frenkel : VAKANCE + INTERSTICIÁL1926 – LEWIS : FOTON1930 – Schottky & Wagner: KRYSTALOVÝ DEFEKT1931 - Heisenberg: ELEKTRONOVÁ DÍRA1954 – Huggins: STRUKTON1962 - 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 Schottky1886-1976

Carl Wihelm Wagner1901-1977

Jakov Iljič Frenkel1894-1952

Defekty jsou výhodné, neboť zvyšují entropii ,

a tím snižují volnou energii

Intrinsic (niterné) defekty

Frenkelovy poruchy1926

Schottkyho poruchy1930?

Linus Pauling, The principles determining the structure of complex ionic crystals.

J. Am. Chem. Soc. 51 (1929) 1010-26

Uspořádání Počet Poloměr Příkladkoulí (R=1) vrcholů dutiny strukturyKrychle (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 látekMagnetické ferity

(Mg,Zn,Mn)xFe3-xO4+γ

Tuhé elektrolyty (CaF2-type)

(Ca,Y)(Zr, Hf, Th)1-xO2-x

Hydridy PdH0,7,, TiHxNbHx, GdHx

Supravodivé oxidyYBa2Cu3O7- δ

Bi2Sr2Ca2Cu3O10+δTl2Ba2Ca2Cu3O10+δTlBa2Ca2Cu3O9+δ

Hg2Ba2Ca2Cu3O8+δ

Oxidy aktinoidů(U, Pu,Cm, Am) O2-x

CaUO4+x

Oxidy lantanoidůCeO2-x

PrO1+x

Hydráty metanuCH4.(6x)H2O

IntermetalickésloučeninyCoSn0,69-0,72

Wolframové a molybdenové bronze

(W, Mo)O3-x

ChalkogenidyPyrhotit Fe1,0-0,8S

ZrSx, CrSx

Tuhé elektrolyty(CaTiO3-type)

La(Sr,Ca)MnO3-δ

Sr2(Sc1+xNb1-xO6-δ

Sr3CaZr1-xTa1+xO9-δThermoelektrika(CuFeO2-type)CuCr1−xMgxO2+δ

2

Protonové elektrolyty(pyrochlore type)

La2-xCaxZr2O7-δ

Složení a nestechiometrieHomogenní látka

Prvek Sloučenina Roztok

Daltonid Bertollid

Fe2O3Fe3O4

Fe:O = n:m → FenOm → FenOmo+ → Feno-Om

FeO

Molární zlomek: XO=m/(n+m)FeO XO=1/2 =0,500FeO1,056 XO=1,056/2,056 =0,514FeO1,158 XO=1,158/2,158 =0,537Fe3O4: XO=4/7 =0,571Fe3O4,112 XO=4,112/7,112 =0,587Fe2O3 XO=3/5 =0,600

Molální zlomek: YO=m/nFeO: YO=1/1 = 1,000FeO1,056 YO=1,056/1 =1,056 FeO1,158 YO=1,158/1 =1,158Fe3O4: YO=4/3 =1,333Fe3O4,112 YO=4,112/3 =1,371Fe2O3 YO=3/2 =1,500

YO →1,10 1,20 1,40

Stechiometrický (daltonský)

poměr no:mo

Odchylka od stechiometrie: = m-mo ; = no-n

[(3-)/3] Fe3O4+ = Fe3-O4

= 3/(4+ ) ; = 4/(3+)

1.531.52

1.51

0

-2

-4

-6900

1000

1100

1200

log

a X

YX

T [°C]

log aX versus T

[Y X =const]

log aX versus YX [T=const]

YX versus T [log a

X =const]

Molální zlomek YX (volné složky X) ve fázi AXY

ve vztahu k aktivitě volné složky aX a teplotě T

P = const.

Vztah mezi teplotou T, obsahem Yf , a aktivitou af volné složkyvyjadřuje implicitní funkce

F (Yf , af , T) = 0, [P=const]kterou lze charakterizovat následující trojicí veličin:1. relativní parciální molární entalpií Δhf

Δhf = R (∂ ln af /∂(1/T))Yf

2. tepelnou ftochabilitou (φτωχός = chudý) κfT

κfT = – (∂Yf /∂T)af

3. vlastní (proper) plutabilitou (πλούτος = bohatý) κff

κff = (∂Yf /∂ log af)T ≥ 0mezi nimiž platí vztah:

κfT = - κff (Δhf /RT2)

Figure from Inaba H. & Naito K. (1977): Heat Capacity of Nonstoichiometric Compounds, NETSU 4 [1] 10-18

U0.8Pu0.2O2+δ

δ =0

.08

δ =0.

045

δ =0.0

00δ =

– 0.0

20

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, 236-41

Heat capacity of „mixed oxide fuel (MOX)“ U0.8Pu0.2O2+δ

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 capacity1907 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 temperatures1928 Arnold Sommerfeld (1886-1951): heat capacity of electrons in metals ΔelCV = γT1952 K. Kobayashi: heat capacity due to Frenkel defects formation ΔFrC

Tepelná kapacita - Historie

magnetic

Temperature dependence of heat capacity Cand some of its contributions

Al

CP

CV

ΔCdil

CP = Cvib + ΔelC + ΔmgC + ΔCdil

vibrational

electronic dilationCvib = Char + Canh

CP = CV+ ΔdilCΔdilC =VT.a2/b

Contributions to heat capacity CP

Cp = Char + Canh + ΔdilC + ΔelC + ΔmgC + Δcdf C + ΔothersC

Schottkydefects

Frenkeldefects

Electron-positron pairs

Dilation= CVIdeal crystalvibrations

Electronic Magnetic Non-stoichio-metry ?

Δcdf C = ΔSchC + ΔFrC + ΔehC + …

Crystal defect

formations

Δdil C =VT.a2/b

ΔFrC = AFr [(ΔHFr)2/2RT2]exp(-ΔHFr/2RT)

CP = A1 + A2 .T + A3/T2 + A4/√T + A5.T2

According to thermodynamic tables (empirical polynomial model):

According to thermodynamic and physical (theoretical) models:

Neumann-Kopp &

Einstein & Debye

Isochoric Cv and isobaric CP heat capacitiesIsochoric conditions

Constant volume V:

Increasingtemperature T Increasing

pressure P

Isobaric conditionsConstant pressure P:

Increasingtemperature T

Increasingvolume V

Cv = (∂U/∂T)V CP = (∂H/∂T)P

Internal energy U Enthalpy H = U + P.VΔdilC = 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 = - βVH = U + P.V → dH = TdS + VdP ; F = U – TS → dF = - SdT - PdVCP≡(∂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/βCP = CV +T .(∂V/∂T)P α/β = CV - T V α2/β

Maxwell´s relation1860

(Clairaut 1743):

CP = CV + T.V. α2/β ; CP - CV ≡ Δdil C = T.V. α2/β

Heat capacity and nonstoichiometry

Measured in sealed ampoule

Heat capacity of sample with free component f (element X)

Measured under controlled atmosphere

Constant fractionof 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, YfCP, af

ΔsatCP = CP, af – CP, Yf

Isoplethal conditions Isodynamical conditions

Molal fraction Yf = Nf/∑iNi≠fYf = Xf/(1-Xf)

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

(∂H/∂Nf)T = (∂(H/n)/∂(Nf /n)T =(∂(H/n)/∂Yf)T ≡ hf = Hfo + Δhf

availablefrom

TG data

Partial Molal Enthalpy : (∂H/∂Nf)T,P ≡ hf

CP,af = CP,Yf + hf. (∂Yf /∂T)af ; ΔsatCP = hf (∂Yf /∂T)af

Isoplethal CP : CP,Yf = (∂H/∂T)P,Nf

Isodynamical CP: CP,af = (∂H/∂T)P,af

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

Relative partial molal enthalpy Δhf

Partial molalenthalpy = Molar enthalpy of

pure component f + Relative partialmolal enthalpy

(∂H/∂Nf)T,P,Ni≠f ≡ hf (Yf ,T) = Ho

f (T) + Δhf (af ,T, Yf)Partial molal Gibbs energy gf = chemical potential μf

(∂G/∂Nf)T,P,Ni≠f ≡ gf (Yf ,T) = Go

f (T) + Δgf (af ,T, Yf)

(∂G/∂Nf)T,P,Ni≠f ≡ μf (Yf ,T) = μo

f (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 = (∂(Go

f/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 = [Ho

f + R (∂ ln af/∂(1/T))P,Nf](∂Yf /∂T)af

Thermal phtochability κfT and proper plutability κff

Thermogravimetryof YBa2Cu3O6+z

YBa2Cu3O6+z ; YO =(6+z)/6 → δ = z/6

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)

„thermal phtochability“ (φτωχός = poor) κfT = – (∂Yf /∂T)af

≥ 0

„proper plutability“ (πλούτος = rich)

κff = (∂Yf /∂ log af)T ≥ 0κff = 1/(∂ log af /∂Yf) T

ΔsatCP = hf (∂Yf /∂T)af = – hf κfT

κfT is obtainable from TG measurements(from the slope in Yf vs. T dependence)

κff is obtainable from coulometric titrations (from the slope in log af vs Yf dependence)

ΔsatCP = - (Hof +Δhf) κfT = + (Ho

f +Δhf) (Δhf /RT2) κff

30

20

10

13000

1400 1500 1600 1700 1800 1900

MagnetiteHaematite

Liqu

id

pO2=1

pO2=0.01

pO2=0.0001

Metastablemagnetite

Stablemagnetite

Δ satC

P (J.

mol

-1.K

-1)

FeO4/3+δ

T (K)

Difference of heat capacity ΔdevCP(δ) = CP(δ≠0)-CP(δ=0) due to deviation from stoichiometry

CP,b(Yf≠Yof) = CP,b(Y

of) + ∫(∂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,Pδ ≡ Yf –Yo

f → dδ = dYf ; hf = Ho

f + Δhf → (∂hf/∂T)Yf = CP

of + (∂Δhf/∂T)Yf

Cpo

f (T) = (∂Hof /∂T) (= heat capacity of pure component f)

CP,b(δ≠0) = CP,b(δ=0) + δ.Cpo

f +∫(∂Δhf/∂T)δ,P dδ

ΔdevCP(δ) = δ.Cpo

f +∫(∂Δhf/∂T)δ,P dδ

Temperature dependence of relative partial molal enthalpy Δhf

(∂Δhf/∂T)δ,P = R (∂[∂ ln af/∂(1/T)]δ/∂T)

δ =

= -(R/T2)(∂2 ln af/ ∂(1/T)2)δ

ΔdevCP(δ) = δ.Cpo

f +∫0

δ(∂Δhf/∂T)δ,P dδ

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.Cp

of

(it means validity of Neumann-Kopp rule)

Quantities required for determination of nonstoichiometric contributions to heat capacity

Δhf = R.[∂ ln af/∂(1/T)]Yf

ΔdevCP(δ) = δ.Cpo

f +∫(∂Δhf/∂T)δ,P dδ

(∂Δhf/∂T) = -(R/T2)(∂2 ln af/ ∂(1/T)2)δ

κfT = – (∂Yf /∂T)af 1/ κff =

=(∂ log af /∂Yf)T

ΔsatCP = [Hof + R (∂ ln af/∂(1/T))P,Nf

](∂Yf /∂T)af =

ΔsatCP = - κfT (Hof +Δhf) = +(Ho

f +Δhf) (Δhf /RT2) κff

Thermodynamictables

Experiment resultsImplicit function

F (Yf, log af, T) = 0

How can it be fitted by crystal defect models ?

Mikroskopické složky:Atomy, Molekuly

Ionty, Elektrony/DíryVakance/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.,

Thermodynamic model of defect crystalThermodynamic 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 – Swhere 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.λr

eq +νjf.δwhere λr

eq are equilibrium degrees of conversion determined from conditions of minimum Gibbs free energy G

(∂G/∂λr)δ,T,P =∑jνjr (μo

j +RT ln aj) = ΔGor + RT ln Kr = 0

λreq = Ar exp (-ΔHo

r/RT); (∂λreq/∂T) = Ar (ΔHo

r/RT2) exp (-ΔHor/RT)

Gibbs free energy G of the crystal with equilibrium amounts of defects is then:

G = Gid + ∑r λr

eq (ΔGor + RT ln Kr )

and enthalpy H of crystal is given using Gibbs-Helmholtz equation: H = (∂(G/T)/∂(1/T)) = Hid + ∑

r λr

eq (ΔHor + R [∂ ln Kr /∂(1/T)])

Heat capacity CP of crystals with equilibrium defects is then (approximately):

CP = (∂H/∂T) ≅ CP,id

+ ∑r ΔHo

r (∂λreq/∂T)

ΔcdfCP = CP – CPid = ∑r Ar (ΔHo

r2/RT2) exp (-ΔHo

r/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.δaj = 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 (μo

j +RT ln aj) = μof +RT ln af ≠ 0

ΔGoIf = ∑

jνjf μo

j - μof ; ln Kif = ∑

jνjf ln aj - ln af

ΔGIf = ΔGoIf - RT ln Kif = 0 Incorporation reaction:

Relative partial molal enthalpy Δhf and phtochability κfT and parameters of incorporation reaction (If)

ΔGIf = [∑ νjf.Gjo – Gf

o ] + RT [∑jνjf ln aj – ln af] = 0

ΔHoIf - TΔSo

If = ΔGoIf = – RT ln Kif

ΔHoIf /T – ΔSo

If = - R ∑jνjf ln aj

νjf + R ln af

ln af = ΔHoIf /RT – ΔSo

If /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 thenΔhf /R ≈ ΔHo

If /R → Δhf ≈ ΔHoIf

κfT = – (∂Yf /∂T)af ≈ – (ΔGo

If /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≠d

νjf 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/δκff = δ (∂ ln δ /∂ log af)T = 2.303 δ. (∂ log δ /∂ log af)T

κff ≈ 2.303 δ/νdf

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.aO2

1/2)

KIO2 =[(2+2 )/(1-2 )]2 [3 /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 ~ ln ¾ γ → log γ ~ / log ² ₃ aO2d log aO2 /d log γ = 3/2 ↔ d log γ/d log aO2 =2/3log 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

+2

0

- 2

- 4

- 6

- 8

- 4 - 3 - 2 - 1log Δ

log pO2

experi

ment

Frenkel defects

without Fr. def.

Inverse spinel

Models

σ log Δ =σΔ /Δ

dlog p O2/dlog Δ = 3/2

γ-Fe2O3 =(3/4)Fe8/3O4

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

log(1/3)≅-0.48

Range of partial molal enthalpy of oxygen in magnetite

Flood & Hill (1957) :

[kJ] 142 - 87 T = HI0-TSI

0= RT ln{[(2+2)/(1-2)]2 [3/4 ]¾}/aO2½}

CP, aO2 = CP, YO + ½ hO2 (∂/∂T) P,aO2,NFe

(∂H/∂) = ½ (∂H/∂NO2) = ½ (HoO2 + ΔhO2); ΔhO2 = f (, T)

ΔhO2 = R(∂ ln aO2 /∂(1/T))YO = - 2HI

0 = -284 kJ

ln aO2 = 2 ln{[(2+2)/(1-2)]2 [3/4 ]¾} - (2H0/R)(1/T) +2S0/R

Gordeev (1966):ΔhO2 = f ( = 0, T) = -620 kJ/mol O2 !!!

Flood & Hill (1957) :ΔhO2 = f ( >0, T) = const = -284 kJ/mol O2

Equilibrium nonstoichimetry –more general model for Δhf

(∂G/∂δ) T,P,λr =∑

jνjf (μo

j +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) = μo

f +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

-300

-400

-500

-600

kJ/mol O2

-5 -4 -3 -2 -1 0log γ

-284 kJ/mol

-620 kJ/mol

1050

°C 1200

°C

1350

°C

1500

°C

-620 ≤ ΔhO2 = ΔHIO2 + ΔHFr (∂λFr/∂γ) ≤ - 284 kJ/mol O2

ΔhO

2

Dominating incorporation R1

Dominating incorporation R3

κFrf =(∂λFr/∂γ) = 0

κFrf =(∂λFr/∂γ) = -4/3

(∂λFr/∂γ) ≡ κFrf (γ) ∊ (-3/4);0)

-100

-200

-300

-400

1300°C 1400°C 1500°C

0.52

1.25

1.97

3.43

18.78

Flood & Hill 1957

Gordeev 1966

Δh O [kJ/g

-at O]

T

4.15 = 1000.δ

0

1

2

5 101000 δ

∫ 0δ (dΔh O/

dT) dδ [J/

K]dΔhO/dT

[k J/K/g

-at. O]

0

.5

1

5 10

Spencer & Kubashewski 1978

Temperature dependence of relative partial molal enthalpy Δhf

(∂Δhf/∂T)δ,P = R (∂[∂ ln af/∂(1/T)]δ/∂T)

δ =

= -(R/T2)(∂2 ln af/ ∂(1/T)2)δ

ΔdevCP(δ) = δ.Cpo

f +∫0

δ(∂Δhf/∂T)δ,P dδ

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.Cp

of

if δ = 0.004 then (0.004/2)xCPo

O2 = 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 ∫

0

δ(∂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).