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Measurement of Thermodynamic Properties
For equilibrium calculations we need:• Equilibrium constant K or ΔG for a reaction• Enthalpy ΔH for a reaction
Standardized values for compounds:
Enthalpy of formation at 298 K ΔfH(298)Standard-entropy S0(298)Molar heat capacity cP(T)Enthalpies of transformation ΔtrH(Ttr)
Methods:• Calorimetry ΔfH, ΔtrH, cp, S0,….• Vapor pressure measurements• Electromotive force measurements } pi(T), ai(T), K, ΔG, ..
1 Thermodynamic Data
Literature: O.Kubaschewski, C.B.Alcock and P.J.Spencer: Materials Thermochemistry, Pergamon 1993.
Calorimetry“Measurement of heat exchange connected with a change in temperature (or a change in the physical or chemical state)”
Connection of ΔT and ΔQ: TQTC
T ΔΔ
=Δ 0lim)(a
Classification of methods:
1) Tc = Ts = const.; variation of Q ⇒ Isothermal Cal. 2) Tc = Ts ≠ const.; variation of Tc, Ts with Q ⇒ Adiabatic Cal. 3) Ts = const.; Tc varies with Q ⇒ Isoperibol Cal.
Tc…temperature of the calorimeter Ts…temperature of the surrounding Q…heat produced per unit of time
2 Thermodynamic Data
Observed thermal effects
T
t, time
ΔT dtTc∫ Δ×ΔT
Adiabatic Isoperibol, near adiabatic Isoperibol
cTQ ×Δ= cTQ ×Δ=&
Constant “c” obtained from calibration!
3 Thermodynamic Data
Bomb calorimetry
Can also be used for the indirectdetermination of ΔfH(298)
e.g.:C(s)+ O2(g) = CO2(g) - 393.5 kJmol-1
W(s) + 3/2 O2(g) = WO3(s) - 837.5 kJmol-1
WC(s) + 5/2 O2(g) = WO3(s) + CO2(g) - 1195.8 kJmol-1___________________________________________________________________
W(s) + C(s) = WC(s) - 35.2 kJmol-1
Caution!Small difference of large absolute values⇒ large relative error!
Water
T-measurement
Shielding
IsolationBomb
ΔCH: Enthalpy of combustione.g. 2Al + 3/2 O2 = Al2O3
⇒ Direct determination of reaction enthalpies!
4 Thermodynamic Data
Solvent
Solute
Simple Solution Calorimetry
Aqueous solutions at room temperature:
Solvent: WaterSolute: e.g. Salt
Measurement of ΔHSolv
Usually strong concentration dependence.
Extrapolation to c → 0Solute
Solvent
Solute
Experimental setup:Isoperibol, near adiabatic
5 Thermodynamic Data
High Temperature Solution Calorimetry“Drop Experiment”
* Solvent: Al(l), Sn(l), Cu(l),…* Solute: pure element or compound* Evacuated or inert gas condition * Crucible material: Al2O3, MgO, etc.
Experimental setup: Isoperibol
⇒ Determination of ΔmH (enthalpy of mixing) for liquid alloys⇒ Indirect determination of the enthalpy of formation ΔfH
The heat of solution in liquid metals is usually small!
Solvent
Solute
Furnace
Thermocouple
6 Thermodynamic Data
Typical experimental setup
Tmax= 1000 °C
Setaram High Temperature Calorimeter
7 Thermodynamic Data
Heat flow twin cell technique
Tian – Calvet Calorimeter
High reproducibility (two calorimetric elements)
Highest sensitivity (multiple thermocouple; thermo pile)
Effective heat flow (metal block)
sample reference
heating unit metal block
thermocouple
8 Thermodynamic Data
Example: Enthalpy of Mixing Bi-Cu (1)
24000 600 1200 1800 2400
-210
-510
-450
-390
-330
-270
39000
-1000
7000
15000
23000
31000
Single drop of a small peace of Cu(s) at drop temperature (Td) into a reservoir of Bi(l) at the measurement temperature (Tm).
The enthalpy of the signal is evaluated by peak integration. It is connected with the enthalpy of mixing by:
With Hm as molar enthalpy
Cu
reactionCumix
reactionTdCumTmCumCusignal
nHH
HHHnHΔ
=Δ
Δ+−=Δ )( ,,,,
Calibration:
Drop of reference substance with well known molar heat capacity (e.g. single crystalline Al2O3; sapphire)
9 Thermodynamic Data
Example: Enthalpy of mixing Bi-Cu (2)
xBi
0.0 0.2 0.4 0.6 0.8 1.0
ΔM
ixH
/ J.
mol
-1
-6000
-4000
-2000
0
2000
4000
6000
BiCu
1000 °C
800 °C
Two measurement series at different temperatures. The data points represent single drops. The values are combined to integral enthalpies of mixing in liquid Bi-Cu alloys.
→ L[L + Cu] ←
10 Thermodynamic Data
Vapor pressure methods
Thermodynamic Activity: 00i
i
i
ii p
pffa == pi…partial pressure of i
pi0..partial pressure of pure i
iii aRTG ln=Δ=μ
TGS i
i ∂Δ∂
−=Δ
)/1()/(
TTGH i
i ∂Δ∂
=Δ
Partial molar thermodynamic functions are obtained:
direct: chemical potentialindirect: entropy and enthalpy
Equilibrium constants: A(s) + B(g) = AB(s)Bp
k 1=
11 Thermodynamic Data
Gibbs-Duhem Integration
Calculation of the integral Gibbs energy from the activity data
x(B)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a(B
)
0.00.10.20.30.40.50.60.70.80.91.0
0lnln =+ BBAA adxadx
∫ −==
=
AA
A
xx
xB
AB
A adxxa
1lnln⇒
x(B)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
G/J
-8000
-6000
-4000
-2000
0
iiii dx
GdxGG Δ−+Δ=μ=Δ )1(
∫μ
=ΔBx
BA
BA dx
xxG
02⇒
12 Thermodynamic Data
Vapor pressure measurements - overview
1) Static: Closed system, constant temperature. Pressure determination by mechanical gauges or optical absorption.
2) Dynamic: Constant flow of inert gas as carrier of the gas species for measurement (transpiration method).
3) Equilibration: Condensed sample is equilibrated with the vapor of a volatile component. The pressure is kept constant by an external reservoir.
4) Effusion: Effusion of the vapor through a small hole into a high vacuum chamber (Knudsen cell technique)
Pressure range: p ≥ 10-5 – 10-7 Pa
13 Thermodynamic Data
Static Methods - Example
Atomic Absorption technique
Determination of the pressure by specific atomic absorption
k…..constantd…..optical path length
Pressure range down to 10–7 Pa(gas species dependent)
Sample
VaporLight Path
Photo-meter
Heating
Vacuum Chamber
dkTIIpi ×
×=
)/ln( 0
14 Thermodynamic Data
Transpiration Method
Inert gas flow (e.g. Ar) carries the vapor of the volatile component away
Argon
Furnace
Sample
Condensate
Exhaust
Under saturation conditions:Nn
nPpi
ii +
×=
e.g.: CaTeO3(s) = CaO(s) + TeO2(g)Measurement of p(TeO2) ⇒ ΔGf(CaTeO3)
15 Thermodynamic Data
Equilibration Method
Isopiestic Experiment:Equilibration of several samples (non-volatile) with the vapor of the volatile component in a temperature gradient
)()(
)()()( 0
0
0Si
Ri
Si
Sisi Tp
TpTpTpTa ==
Activity Calculation:
TS….Temperature at the sampleTR….Temperature in the reservoirpi
0….pressure of the pure volatile component
e.g.: Fe(s) + Sb(g) = Fe1±xSb(s)
⇒ Antimony activity as a function of composition and temperature
Tem
pera
ture
Gra
dien
t
16 Thermodynamic Data
Example: Isopiestic Experiment Fe-Sb (1)
17 Thermodynamic Data
Experimental Fe-Sb Phase Diagram. Phase boundaries from IP already included. Equilibration Experiment: Fe(s) in quartz glass crucibles + Sb from liquid Sbreservoir.
Example: Isopiestic Experiment Fe-Sb (2)
18 Thermodynamic Data
Several experiments at different reservoir temperatures
The principal result of the experiments are the “equilibrium curves”
One curve for each experiment: T/x data
The composition of the samples after equilibration is obtained from the weight gain.
Kinks in the equilibrium curves can be used fro the determination of phase boundaries
Isopiestic Experiment Fe-Sb (3)
19 Thermodynamic Data
Antimony in the gas phase:Temperature dependent pressure known from literature (tabulated values):
Experimental temperature: 900-1350K
Relevant species: Sb2 and Sb4
(1) Ptot = pSb2 + pSb4 (fixed in experiment)
Gas equilibrium: Sb4 = 2Sb2
(2) k(T) = pSb22/pSb4
Activity formulated based on Sb4:
(3)
4/1
40
4
)()()( ⎟⎟⎠
⎞⎜⎜⎝
⎛=
sSb
sSbsSb Tp
TpTa
Isopiestic Experiment Fe-Sb (4)
20 Thermodynamic Data
The pressure of Sb4 at different temperatures in the reaction vessel pSb4(T) can be obtained by combining (1) and (2):
(4)
Analytical expressions for ptot(T), p0Sb4(T), p0
Sb2(T) and k(T) can be derived from the tabulated values by linear regression in the form ln(a) versus 1/T
2)(4)(2)(
)(2
4tottot
SbpTkTkpTk
TP+−+
=
TKatmptot 13940883.6)/ln( −=
TKatmp Sb 12180005.5)/ln( 4
0 −=
TKK 3011099.17)ln( −=
TKatmp Sb 2114049.11)/ln( 2
0 −=
Example: Isopiestic Experiment Fe-Sb (5)
21 Thermodynamic Data
Run 5 reservoir temperature: 969 K 32 days
Nr. at% Sb Tsample/K lna(Tsample) Δ⎯H/kJmol-1 lna(1173K)
1 48.04 1015 -0.222 -18.0 0.065 2 47.79 1032 -0.281 -20.6 0.006 3 47.58 1050 -0.345 -22.5 -0.075 4 47.35 1068 -0.409 -24.4 -0.163 5 47.09 1087 -0.479 -26.5 -0.265 6 46.81 1107 -0.556 -28.5 -0.382 7 46.39 1127 -0.636 -31.3 -0.506 8 45.97 1152 -0.743 -33.6 -0.681 9 45.45 1180 -0.873 -36.1 -0.895
10 44.50 1207 -1.008 -39.6 -1.122 11 43.68 1232 -1.140 -41.7 -1.345 12 42.64 1253 -1.256 -43.7 -1.543 13 41.05 1271 -1.357 -46.4 -1.724 14 40.13 1285 -1.437 -47.9 -1.865 15a) 34.63 1295 -1.494 - -16a) 33.65 1304 -1.545 - -17a) 32.75 1311 -1.585 - -18a) 30.55 1316 -1.614 - -
Each single sample contributes one data point. Steps of evaluation: 1) a(Ts), 2) partial enthalpy from T-dependence, 3) conversion to common temperature
Example: Isopiestic Experiment Fe-Sb
22 Thermodynamic Data
lnaSb
Plotting lna versus 1/T for selected compositions, the partial enthalpy can be obtained
Gibbs-Helmholtz:
Partial enthalpy evaluated from the slope of the curves for the different compositions.
Different symbols mark different experiments.
RH
Td
ad SbSb Δ=1
ln
Example: Isopiestic Experiment Fe-Sb
23 Thermodynamic Data
Δ⎯HSb/Jmol-1
If the agreement of results in different experiments is reasonable, a smooth curve of Δ⎯HSb versus composition is observed.
The partial Enthalpy is considered to be independent from temperature.
Δ⎯HSb is used to convert the activity data to a common intermediate temperature:
(Integrated Gibbs-Helmholtz Equation)
⎟⎟⎠
⎞⎜⎜⎝
⎛−
Δ=−
2121
11)(ln)(lnTTR
HTaTa SbSbsb
Example: Isopiestic Experiment Fe-Sb
24 Thermodynamic Data
lnaSb
Final activity data for all experiments converted to the common temperature of 1173 K
Due to the strong temperature dependence of the phase boundary of the NiAs-type phase, not all data lie within the homogeneity range of FeSb1+/-x at 1173 K
Equilibration with gas mixtures
⇒ The partial pressure of a component is fixed indirectly by use of an external equilibrium
e.g.: H2S(g) = H2(g) + ½ S2(g)
⇒ The partial pressure of S in the system can be fixed by the H2S / H2
ratio in the system
)()()()(
2
2/122
SHpSpHpTK ×
=
)()()()(
22
22
22 Hp
SHpTKSp =
Can be used for a number of different gas equilibria:
• H2 / H2O ⇒ p(O)• CO / CO2 ⇒ p(C)• H2 / NH3 ⇒ p(N)• H2 / HCl ⇒ p(Cl)etc.
Good for low pressures!
25 Thermodynamic Data
Effusion Method: Knudsen Cell
Kinetic Gas Theory:
Detection System:• Mass Loss (Thermobalance)• Condensation of Vapor• Torsion• Mass Spectroscopy
The vapor pressure is determined from the evaporation rate
Detection System
Small hole
Knudsen Cell
High Vacuum Chamber
pi
Effusion
ii M
TRfAt
mp ××π××
=2
26 Thermodynamic Data
Electromotive Force (EMF)Well known basic principle:
ZnSO4 CuSO4
Zn Cu
ΔE
porous barrier
Cell reaction: Zn + Cu2+ = Cu + Zn2+
EMF = reversible potential difference(for I → 0)
Convention for cell notation:
Zn(s) | Zn2+(aq) | Cu2+(aq) | Cu(s)
EFzGR Δ××−=Δ
27 Thermodynamic Data
EMF as thermodynamic methodThe most important challenge is, to find a suitable cell arrangement and electrolyte for the reaction in question.
Most commonly used: B,BX|AX|C,CX (AX….ionic electrolyte)
Example for evaluation:
Cell arrangement:
A(s) | Az+(electrolyte) | A in AxBy(s)
left: A(s) = Az+ + z e-
right: Az+ + z e- = A in AxBy(s)total: A(s) = A in AxBy(s)
zFEaRTGG AA −==Δ=Δ ln
TEzFS
TG
AA
∂∂
=Δ−=∂Δ∂
TEzFTzFE
HT
TGA
A
∂∂
+−=
Δ=∂Δ∂
)/1()/(
28 Thermodynamic Data
Molten Salt Electrolytes
Electrolyte e.g. LiCl / KCl –eutecticFor temperatures larger than 350°CDoped by MClz⇒ Mz+ is the charge carrier
Example: Reference: liquid ZnSample: liquid Ag-Sn-Zn
Zn(l) | Zn2+(LiCl + KCl) | Ag-Sn-Zn(l)
Cell reaction: Zn(l) = Zn in Ag-Sn-Zn(l)
⇒ ΔGZn, ΔSZn, ΔHZn in liquid Ag-Sn-Zn
29 Thermodynamic Data
Solid Electrolytes
At the operating temperature the solid electrolytes show high ionic conductivity and negligible electronic conductivity (tion ≅ 1).⇒ Large electronic bandgap in combination with an ion migration mechanism
• Oxide ion conductors: ZrO2 (CaO or Y2O3) “Zirconia”ThO2 (Y2O3) “Thoria”
• Sodium ion conductor: Na2O • 11 Al2O3 “Sodium - β Alumina”• Fluoride ion conductor: CaF2
Example: “Exchange cell” [Ni, NiO] | ZrO2(CaO) | [(Cu-Ni), NiO]left: Ni + O2- = NiO + 2e-
right: NiO + 2e- = Ni (Cu-Ni) + O2-
total: Ni = Ni(Cu-Ni)⇒ ΔGNi in (Cu-Ni) alloy
30 Thermodynamic Data
Oxide Electrolytes - Mechanism
Thoria and Zirconia: Fluorite type structureDefect Mechanism: Oo = O2-
i + V2+o ⇒ formation of charge carriers!
low pO2: Oo = ½ O2(g) + V2+o + 2e-
high pO2: ½ O2(g) = O2-i + 2h+
medium pO2: pure ionic mechanism
log σ
log pO2(schematic)
undoped ZrO2
ZrO2 – Y2O3
Y2O3 – Doping:Y2O3 = 2Y-
Zr + 3Oo + V2+o
⇒ increasing ionic conductivity⇒ shift to lower po2
31 Thermodynamic Data