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