Measurement of Thermodynamic Properties
Literature: O.Kubaschewski, C.B.Alcock and P.J.Spencer: Materials
Thermochemistry, Pergamon 1993.
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
Standard-entropy
Molar heat capacity
Enthalpies of transformation
Methods:
• Calorimetry
• Vapor pressure measurements
• Electromotive force measurements
1
ΔfH(298)
S0(298)
cP(T)
ΔtrH(Ttr)
}
Thermodynamic Data
ΔfH, ΔtrH, cp, S0,….
pi(T), ai(T), K, ΔG, ..
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:
ΔQ
C (T ) = lim
ΔT a 0 ΔT
Classification of methods:
1)
2)
3)
Tc = Ts = const.; variation of Q
Tc = Ts ≠ const.; variation of Tc, Ts with Q
Ts = const.; Tc varies with Q
Tc…temperature of the calorimeter
Ts…temperature of the surrounding
Q…heat produced per unit of time
2
Thermodynamic Data
⇒ Isothermal Cal.
⇒ Adiabatic Cal.
⇒ Isoperibol Cal.
Observed thermal effects
T
ΔT
∫ c × ΔT dt
ΔT
t, time
Adiabatic
Isoperibol, near adiabatic
Isoperibol
Q& = ΔT × c
Q = ΔT × c
Constant “c” obtained from calibration!
3
Thermodynamic Data
Bomb calorimetry
Can also be used for the indirect
determination of ΔfH(298)
T-measurement
Water
Shielding
Bomb
Isolation
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)
ΔCH: Enthalpy of combustion
e.g. 2Al + 3/2 O2 = Al2O3
⇒ Direct determination of reaction
enthalpies!
4
Caution!
Small difference of large
absolute values
⇒ large relative error!
Thermodynamic Data
- 35.2 kJmol-1
Simple Solution Calorimetry
Aqueous solutions at room
temperature:
Solvent: Water
Solute: e.g. Salt
Measurement of ΔHSolv
Solvent
Solvent
Solute
Solute
Solute
Usually strong concentration
dependence.
Extrapolation to c → 0
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.
Solute
Experimental setup: Isoperibol
Solvent
⇒ Determination of ΔmH (enthalpy of mixing) for
liquid alloys
⇒ Indirect determination of the enthalpy of
formation ΔfH
Furnace
Thermocouple
The heat of solution in liquid metals is usually
small!
6
Thermodynamic Data
Typical experimental setup
Setaram High Temperature Calorimeter
Tmax= 1000 °C
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)
thermocouple
sample
heating unit
8
reference
metal block
Thermodynamic Data
Example: Enthalpy of Mixing Bi-Cu (1)
-210
39000
-270
31000
-330
23000
-390
15000
-450
7000
-510
0
600
1200
1800
Calibration:
Drop of reference substance
with well known molar heat
capacity (e.g. single
crystalline Al2O3; sapphire)
9
-1000
2400
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:
ΔH signal = nCu (H m ,Cu ,Tm − H m ,Cu ,Td ) + ΔH reaction
Δ mix HCu =
ΔH reaction
nCu
With Hm as molar enthalpy
Thermodynamic Data
Example: Enthalpy of mixing Bi-Cu (2)
6000
1000 °C
ΔMixH / J.mol
-1
4000
2000
800 °C
0
-2000
-4000
[L + Cu] ←
→L
-6000
0.0
Cu
0.2
0.4
0.6
0.8
1.0
Bi
xBi
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.
10
Thermodynamic Data
Vapor pressure methods
Thermodynamic Activity:
μ i = ΔGi = RT ln ai
ΔSi
ΔHi
∂ΔGi
=−
∂T
∂ (ΔGi / T )
=
∂ (1/ T )
Equilibrium constants:
11
f
p
ai = i0 = 0i
fi
pi
pi…partial pressure of i
pi0..partial pressure of pure i
Partial molar thermodynamic
functions are obtained:
direct: chemical potential
indirect: entropy and enthalpy
A(s) + B(g) = AB(s)
Thermodynamic Data
1
k=
pB
Gibbs-Duhem Integration
Calculation of the integral Gibbs energy from the activity data
x Ad ln a A + x Bd ln aB = 0
⇒ ln aA =
xA = xA
xB
d ln aB
−
∫
xA
x A =1
dΔG
ΔGi = μ i = ΔG + (1 − x i )
dx i
xB μ
⇒ ΔG = x A ∫ B2 dx B
0 xA
1.0
0
0.9
0.8
-2000
0.6
G/J
a(B)
0.7
0.5
-4000
0.4
0.3
-6000
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-8000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x(B)
x(B)
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
Vacuum
Chamber
Determination of the pressure by
specific atomic absorption
Light
Path
Vapor
Sample
Heating
14
Photo
-meter
pi =
ln(I0 / I ) × T
k ×d
k…..constant
d…..optical path length
Pressure range down to 10–7 Pa
(gas species dependent)
Thermodynamic Data
Transpiration Method
Inert gas flow (e.g. Ar) carries the vapor of the volatile component away
Argon
Sample
Exhaust
Condensate
Furnace
Under saturation conditions:
pi = P ×
ni
ni + N
e.g.: CaTeO3(s) = CaO(s) + TeO2(g)
Measurement of p(TeO2) ⇒ ΔGf(CaTeO3)
15
Thermodynamic Data
Equilibration Method
Temperature Gradient
Activity Calculation:
pi (TS )
ai (Ts ) = 0
= 0
pi (TS ) pi (TS )
TS….Temperature at the sample
TR….Temperature in the reservoir
pi0….pressure of the pure volatile
component
Isopiestic Experiment:
Equilibration of several samples
(non-volatile) with the vapor of the
volatile component in a temperature
gradient
16
pi0 (TR )
e.g.: Fe(s) + Sb(g) = Fe1±xSb(s)
⇒ Antimony activity as a function
of composition and temperature
Thermodynamic Data
Example: Isopiestic Experiment Fe-Sb (1)
Experimental Fe-Sb Phase Diagram. Phase boundaries from IP already included.
Equilibration Experiment: Fe(s) in quartz glass crucibles + Sb from liquid Sb
reservoir.
17
Thermodynamic Data
Example: Isopiestic Experiment Fe-Sb (2)
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
18
Thermodynamic Data
Isopiestic Experiment Fe-Sb (3)
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)
19
Thermodynamic Data
⎛ p (T ) ⎞
aSb (Ts ) = ⎜⎜ 0Sb 4 s ⎟⎟
⎝ p Sb 4 (Ts ) ⎠
1/ 4
Isopiestic Experiment Fe-Sb (4)
The pressure of Sb4 at different temperatures in the reaction vessel pSb4(T) can be
obtained by combining (1) and (2):
k (T ) + 2 ptot − k 2 (T ) + 4k (T ) ptot
(4) PSb 4 (T ) =
2
Analytical expressions for ptot(T), p0Sb4(T), p0Sb2(T) and k(T) can be derived from the
tabulated values by linear regression in the form ln(a) versus 1/T
20
ln( p 0 Sb 4 / atm ) = 5.005 − 12180
K
T
ln( p 0 Sb 2 / atm ) = 11.49 − 21140
K
T
ln( ptot / atm ) = 6.883 − 13940
Thermodynamic Data
ln(K ) = 17.99 − 30110
K
T
K
T
Example: Isopiestic Experiment Fe-Sb (5)
Run 5
reservoir temperature:
Nr.
at% Sb
Tsample/K
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15a)
16a)
17a)
18a)
48.04
47.79
47.58
47.35
47.09
46.81
46.39
45.97
45.45
44.50
43.68
42.64
41.05
40.13
34.63
33.65
32.75
30.55
1015
1032
1050
1068
1087
1107
1127
1152
1180
1207
1232
1253
1271
1285
1295
1304
1311
1316
969 K
lna(Tsample)
-0.222
-0.281
-0.345
-0.409
-0.479
-0.556
-0.636
-0.743
-0.873
-1.008
-1.140
-1.256
-1.357
-1.437
-1.494
-1.545
-1.585
-1.614
32 days
Δ⎯H/kJmol-1
-18.0
-20.6
-22.5
-24.4
-26.5
-28.5
-31.3
-33.6
-36.1
-39.6
-41.7
-43.7
-46.4
-47.9
-
lna(1173K)
0.065
0.006
-0.075
-0.163
-0.265
-0.382
-0.506
-0.681
-0.895
-1.122
-1.345
-1.543
-1.724
-1.865
-
Each single sample contributes one data point. Steps of evaluation: 1) a(Ts),
2) partial enthalpy from T-dependence, 3) conversion to common temperature
21
Thermodynamic Data
Example: Isopiestic Experiment Fe-Sb
Plotting lna versus 1/T for selected
compositions, the partial enthalpy can
be obtained
Gibbs-Helmholtz:
d ln aSb ΔHSb
=
1
R
d
T
lnaSb
Partial enthalpy evaluated from the
slope of the curves for the different
compositions.
Different symbols mark different
experiments.
22
Thermodynamic Data
Example: Isopiestic Experiment Fe-Sb
If the agreement of results in
different experiments is
reasonable, a smooth curve of
Δ⎯HSb versus composition is
observed.
Δ⎯HSb/Jmol-1
The partial Enthalpy is considered
to be independent from
temperature.
Δ⎯HSb is used to convert the
activity data to a common
intermediate temperature:
ln asb (T1 ) − ln aSb (T2 ) =
ΔHSb
R
⎛1 1⎞
⎜⎜ − ⎟⎟
⎝ T1 T2 ⎠
(Integrated Gibbs-Helmholtz
Equation)
23
Thermodynamic Data
Example: Isopiestic Experiment Fe-Sb
Final activity data for all
experiments converted to the
common temperature of 1173 K
lnaSb
24
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
Thermodynamic Data
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)
• H2 / H2O ⇒ p(O)
• CO / CO2 ⇒ p(C)
• H2 / NH3 ⇒ p(N)
• H2 / HCl ⇒ p(Cl)
etc.
p(H2 ) × p(S2 )1/ 2
K (T ) =
p(H2S )
2
p(S2 ) = K (T )
Can be used for a number of
different gas equilibria:
p2 (H2S )
p 2 (H2 )
Good for low pressures!
⇒ The partial pressure of S in the
system can be fixed by the H2S / H2
ratio in the system
25
Thermodynamic Data
Effusion Method: Knudsen Cell
The vapor pressure is determined from the evaporation rate
Kinetic Gas Theory:
High Vacuum
Chamber
m
pi =
t × A×f
Detection
System
Effusion
Small
hole
pi
Detection System:
• Mass Loss (Thermobalance)
• Condensation of Vapor
• Torsion
• Mass Spectroscopy
Knudsen Cell
26
2π × R × T
Mi
Thermodynamic Data
Electromotive Force (EMF)
Well known basic principle:
EMF = reversible potential difference
(for I → 0)
ΔE
Zn
Cu
Δ RG = − z × F × Δ E
Convention for cell notation:
ZnSO4
Zn(s) | Zn2+(aq) | Cu2+(aq) | Cu(s)
CuSO4
porous barrier
Cell reaction: Zn + Cu2+ = Cu + Zn2+
27
Thermodynamic Data
EMF as thermodynamic method
The 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:
ΔG = ΔG A = RT ln a A = − zFE
Cell arrangement:
∂E
∂ΔG A
= − ΔS A = zF
∂T
∂T
A(s) | Az+(electrolyte) | A in AxBy(s)
left:
right:
total:
28
A(s) = Az+ + z eAz+ + z e- = A in AxBy(s)
A(s) = A in AxBy(s)
∂ (ΔG A / T )
= ΔH A
∂ (1/ T )
∂E
= − zFE + zFT
∂T
Thermodynamic Data
Molten Salt Electrolytes
Electrolyte e.g. LiCl / KCl –eutectic
For temperatures larger than 350°C
Doped by MClz
⇒ Mz+ is the charge carrier
Example:
Reference: liquid Zn
Sample: 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 + 2eright: NiO + 2e- = Ni (Cu-Ni) + O2total:
Ni = Ni(Cu-Ni)
⇒ ΔGNi in (Cu-Ni) alloy
30
Thermodynamic Data
Oxide Electrolytes - Mechanism
Thoria and Zirconia: Fluorite type structure
Defect Mechanism: Oo = O2-i + V2+o ⇒ formation of charge carriers!
log σ
low pO2: Oo = ½ O2(g) + V2+o + 2ehigh pO2: ½ O2(g) = O2-i + 2h+
medium pO2: pure ionic mechanism
Y2O3 – Doping:
ZrO2 – Y2O3
Y2O3 = 2Y-Zr + 3Oo + V2+o
⇒ increasing ionic conductivity
undoped ZrO2
⇒ shift to lower po2
(schematic)
31
Thermodynamic Data
log pO2