The theoretical background of

GTT-Technologies

The theoretical background of FactSage

The following slides give an abridged overviewof the major underlying principles of the calculational modules of FactSage.

GTT-Technologies

The Gibbs Energy Tree

Mathematical methods are used to derive more information from the Gibbs energy ( of phase(s)or whole systems )

GibbsEnergy

Minimisation

Gibbs-Duhem

Legendre Transform.Partial Derivativeswith Respect tox, T or P

Equilibria

Phase DiagramMaxwellH, U, F mi,cp(i),H(i),S(i),ai,vi

Mathematical Method

Calculational result derived

from G

GTT-Technologies

Thermodynamic potentials and their natural variablesVariables

Gibbs energy: G = G(T, p, ni ,...) Enthalpy: H = H

(S, P, ni ,...) Free energy: A= A (T,V, ni ,...)

Internal energy: U = U(S, V, ni ,...)

Interrelationships:A = U TSH = U PVG = H TS =

U PV TS

GTT-Technologies

PTii n

Gµ,

VTin

A

,

PSin

H

,

VSin

U

,

Maxwell-relations:

Thermodynamic potentials and their natural variables

VPH

STG

PP TT

S U V

H A

G

S V

and

GTT-Technologies

...nV,S,const.for0 i,

dUU min

...np,T,const.for0 i,

dGG min

Thermodynamic potentials and their naturalEquilibrium condition:

...nU,T,const.for0 i,

dTA min

...np,S,const.for0 i,

dHH min

...nV,U,const.for0 i,

dSS max

GTT-Technologies

Temperature

Composition

ii

i

i

npnpp

np

np

TGT

THc

TGTGSTGH

TGS

,2

2

,

,

,

Use of model equations permits to start at either end!

Gibbs-Duhem integrationPartial Operator

Integral quantity: G, H, S, cp

Partial quantity: µi, hi, si, cp(i)

Thermodynamic propertiesfrom the Gibbs-energy

GTT-Technologies

With (G is an extensive property!)

one obtains

T,pinG

i

mJ.W. Gibbs defined the chemical potential of a component as:

mi GnG

Thermodynamic propertiesfrom the Gibbs-energy

mi

im

mii

i

Gn

nG

Gnn

m

GTT-Technologies

Transformation to mole fractions :

mi

imi

mi Gx

xGx

G

mi

ii x

xx

1 = partial operator

ii xn

Thermodynamic propertiesfrom the Gibbs-energy

mpCipc mpC

mpC

mi

imi

mi Hx

xHx

Hh

mSis mS mS

GTT-Technologies

Gibbs energy functionfor a pure substance• G(T) (i.e. neglecting pressure terms) is calculated from the

enthalpy H(T) and the entropy S(T) using the well-knownGibbs-Helmholtz relation:

• In this H(T) is

• and S(T) is

• Thus for a given T-dependence of the cp-polynomial (for example after Meyer and Kelley) one obtains for G(T):

TSHG

T

p dTcHH298298

T

p dTTcSS298298

232ln TFTETDTTCTBAG(T)

GTT-Technologies

Gibbs energy functionfor a solution• As shown above Gm(T,x) for a solution consists

of three contributions: the reference term, the ideal term and the excess term.

• For a simple substitutional solution (only one lattice site with random occupation) one obtains using the well-known Redlich-Kister-Muggianu polynomial for the excess terms:

)/())()()((

))((ln),( )(,

kjii j k

ijkkk

ijkjj

ijkiikji

i j

n

jiijjii

iii

oiiim

xxxTLxTLxTLxxxx

xxTLxxxxRTGxxTGij

0

GTT-Technologies

Equilibrium condition: or

Reaction : nAA + nBB + ... = nSS + nTT + ...Generally :

For constant T and p, i.e. dT = 0 and dp = 0,and no other work terms:

min G 0 dG

i

iiB 0

i

iidndG m

Equilibrium considerationsa) Stoichiometric reactions

GTT-Technologies

For a stoichiometric reaction the changes dni are given by the stoichiometric coefficients ni and the change in extend of reaction dx.

Thus the problem becomes one-dimensional.One obtains:

[see the following graph for an example of G = G(x) ]

x d dn ii

0i

id dG xm i

Equilibrium considerationsa) Stoichiometric reactions

GTT-Technologies

Gibbs Energy as a function of extent of the reaction2NH3<=>N2 + 3H2 for various temperatures. It is assumed,that the changes of enthalpy and entropy are constant.

Extent of Reaction x

Gib

bs e

nerg

y G

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T = 400K

T = 500K

T = 550K

Equilibrium considerationsa) Stoichiometric reactions

GTT-Technologies

Separation of variables results in :

Thus the equilibrium condition for a stoichiometric reaction is:

Introduction of standard potentials mi° and activities ai yields:

One obtains:

0i

ii µdξdG

0 i

ii µG

iii aRTµµ ln

0 i

iii

ii aRTµ ln

Equilibrium considerationsa) Stoichiometric reactions

GTT-Technologies

It follows the Law of Mass Action:

where the product

or

is the well-known Equilibrium Constant.

i i

iiiiaRTµG ln

i

iiaK

Equilibrium considerationsa) Stoichiometric reactions

RTGK

exp

The REACTION module permits a multitude of calculations which are based on the Law of Mass Action.

GTT-Technologies

Complex EquilibriaMany components, many phases (solution phases), constant T and p :

with

or

i

ioi

iiii aRTnnG lnmm

m

m

im GnG

p

minG

Equilibrium considerationsb) Multi-component multi-phase approach

GTT-Technologies

Massbalance constraint

j = 1, ... , n of components b

Lagrangeian Multipliers Mj turn out to be the chemical potentials of the system components at equilibrium:

i

jiij bna

j

jjMbG

Equilibrium considerationsb) Multi-component multi-phase approach

GTT-Technologies

System ComponentsPhase ComponentsFe N O C Ca Si Mg

Fe 1 0 0 0 0 0 0N2 0 2 0 0 0 0 0O2 0 0 2 0 0 0 0C 0 0 0 1 0 0 0CO 0 0 1 1 0 0 0CO2 0 0 2 1 0 0 0Ca 0 0 0 1 0 0 0CaO 0 0 1 0 1 0 0Si 0 0 0 0 0 1 0SiO 0 0 1 0 0 1 0

Gas

Mg 0 0 0 0 0 0 1SiO2 0 0 2 0 0 1 0Fe2O3 2 0 3 0 0 0 0CaO 0 0 1 0 1 0 0FeO 1 0 1 0 0 0 0

Slag

MgO 0 0 1 0 0 0 1Fe 1 0 0 0 0 0 0N 0 1 0 0 0 0 0O 0 0 1 0 0 0 0C 0 0 0 1 0 0 0Ca 0 0 0 0 1 0 0Si 0 0 0 0 0 1 0

Liq. Fe

Mg 0 0 0 0 0 0 1

Example of a stoichiometric matrix for the gas-metal-slag system Fe-N-O-C-Ca-Si-Mg

aij j

i

Equilibrium considerationsb) Multi-component multi-phase approach

GTT-Technologies

Modelling of Gibbs energy of (solution) phases

Pure Substance (stoichiometric)

Solution phase

,pT,nGG imm

),(,, pTGG oom

m

ex

m

idm

idm

refmm

GSTG

GG

,

,

,

Equilibrium considerationsb) Multi-component multi-phase approach

Choose appropriate reference state and ideal term, then check for deviations from ideality.See Page 11 for the simple substitutional case.

GTT-Technologies

Use the EQUILIB module to execute a multitude of calculations based on the complex equilibrium approach outlined above, e.g. for combustion of carbon or gases, aqueous solutions, metal inclusions, gas-metal-slag cases, and many others .

NOTE: The use of constraints in such calculations (such as fixed heat balances, or the occurrence of a predefined phase) makes this module even more versatile.

Equilibrium considerationsMulti-component multi-phase approach

GTT-Technologies

Phase diagrams as projections of Gibbs energy plotsHillert has pointed out, that what is called a phase diagram is derivable from a projection of a so-called property diagram. The Gibbs energy as the property is plotted along the z-axis as a function of two other variables x and y.

From the minimum condition for the equilibrium the phase diagram can be derived as a projection onto the x-y-plane.

(See the following graphs for illustrations of this principle.)

GTT-Technologies

a

b g

P

Tab

bg

ag

a

b

g

ab

g

m

PT

Unary system: projection from m-T-p diagram

Phase diagrams as projections of Gibbs energy plots

GTT-Technologies

Binary system: projection from G-T-x diagram, p = const.

300

400

500

600

700

1.0

0.5

0.0

-0.5

-1.0

1.0 0.8 0.6 0.4 0.2 0.0

T

CuxNiNi

G

Phase diagrams as projections of Gibbs energy plots

GTT-Technologies

Ternary system: projection from G-x1-x2 diagram, T = const and p = const

Phase diagrams as projections of Gibbs energy plots

GTT-Technologies

Use the PHASE DIAGRAM module to generate a multitude of phase diagrams for unary, binary, ternary or even higher order systems.

NOTE: The PHASE DIAGRAM module permits the choice of T, P, m (as RT ln a), a (as ln a), mol (x) or weight (w)

fraction as axis variables. Multi-component phase diagrams

require the use of an appropriate number of constants, e.g. in a ternary isopleth diagram T vs x one molar ratio has to be kept constant.

Phase diagrams generated with FactSage

GTT-Technologies

0i i i iSdT VdP n d q dm Gibbs-Duhem:

i i i idU TdS PdV dn dqm

N-Component System (A-B-C-…-N)

SVnAnB nN

T-P µAµB µN

Extensive variables

Corresponding potentials

jqii q

U

iq

GTT-Technologies

N-component system(1) Choose n potentials: 1, 2, … , n (2) From the non-corresponding extensive variables

(qn+1, qn+2, … ), form (N+1-n) independent ratios(Qn+1, Qn+2, …, QN+1).

Example:

Choice of Variables which always give a True Phase Diagram

1Nn

11 Nin

2

1

N

nJj

ij

q

qQ

[ 1, 2, … , n; Qn+1, Qn+2, …, QN+1] are then the (N+1) variables of which 2 are chosen as axes

and the remainder are held constant.

GTT-Technologies

MgO-CaO Binary System

1 = T for y-axis

2 = -P constant

for x-axis

S T

V -P

nMgO µMgO

nCaO µCaO

Extensive variables and corresponding potentials

Chosen axes variables and constants

CaOMgO

CaO

CaO

MgO

nnnQ

nq

nq

3

4

3

GTT-Technologies

S T

V -P

nFe mFe

nCr mCr

f1 = T (constant)

f2 = -P (constant)

x-axis

x-axis

(constant)

Fe - Cr - S - O System

Fe

Cr

Fe

Cr

S

O

nnQ

nq

nq

5

6

5

4

3

2

2

m

m

2

2

S

O

m

m

2

2

S

O

n

n

GTT-Technologies

Fe - Cr - C System - improper choice of axes variablesS T

V -P

nC mC

nFe mFe

nCr mCr

f1 = T (constant)

f2 = -P (constant)

f3 = mC aC for x-

axis andQ4 for y-axis

(NOT OK)

(OK)

4

4

Cr

Fe C

Cr

e

r

F Cr C

nQn n n

nQn n

Requirement: 0 3j

i

dQfor i

dq

GTT-Technologies

This is NOT a true phase diagram.

Reason: nC must NOT be used in formula for mole fraction when aC is an axis variable.

NOTE: FactSage users are safe since they are not given this particular choice of axes variables.

M23C6

M7C3

bcc

fcc

cementitelog(ac)

Mol

e fr

actio

n of

Cr

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-3 -2 -1 0 1 2

Fe - Cr - C System - improper choice of axes variables