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