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 Δ f H(298) Standard-entropy S 0 (298) Molar heat capacity c P (T) Enthalpies of transformation Δ tr H(T tr ) Methods: • Calorimetry Δ f H, Δ tr H, c p , S 0 ,…. • Vapor pressure measurements • Electromotive force measurements } p i (T), a i (T), K, ΔG, .. 1 Thermodynamic Data Literature: O.Kubaschewski, C.B.Alcock and P.J.Spencer : Materials Thermochemistry, Pergamon 1993.
31
Embed
Measurement of Thermodynamic Properties - univie.ac.at · Measurement of Thermodynamic Properties For equilibrium calculations we need: • Equilibrium constant K or ΔG for a reaction
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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)
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)
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:
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