2 The theoretical background of FactSage The following slides give anabridged overviewof the majorunderlying principlesof thecalculational modulesofFactSage.
3 Calculational result derived from G The Gibbs Energy TreeGibbsEnergyMinimisationGibbs-DuhemLegendre Transform.Partial Derivativeswith Respect tox, T or PEquilibriaPhase DiagramMaxwellH, U, Fmi,cp(i),H(i),S(i),ai,viMathematical MethodCalculational result derived from GMathematical methods are used to derive more information from the Gibbs energy ( of phase(s) or whole systems )
4 Thermodynamic potentials and their natural variables Variables 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 TS H = U PV G = H TS = U PV TS
5 Thermodynamic potentials and their natural variables GMaxwell-relations:and
6 Thermodynamic potentials and their natural Equilibrium condition:
7 Thermodynamic properties from the Gibbs-energy Temperature CompositionIntegral quantity: G, H, S, cpPartial OperatorGibbs-Duhem integrationPartial quantity: µi, hi, si, cp(i)Use of model equations permits to start at either end!
8 Thermodynamic properties from the Gibbs-energy J.W. Gibbs defined the chemical potentialof a component as:With (G is an extensive property!)one obtains
9 Thermodynamic properties from the Gibbs-energy Transformation to mole fractions := partial operator
10 Gibbs energy function for 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-known Gibbs-Helmholtz relation:In this H(T) isand S(T) isThus for a given T-dependence of the cp-polynomial (for example after Meyer and Kelley) one obtains for G(T):
11 Gibbs energy function for 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:
12 Equilibrium considerations a) Stoichiometric reactions Equilibrium condition:orReaction : nAA + nBB = nSS + nTT + ...Generally :For constant T and p, i.e. dT = 0 and dp = 0,and no other work terms:
13 Equilibrium considerations a) Stoichiometric reactions 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) ]
14 Equilibrium considerations a) Stoichiometric reactions Extent of Reaction xGibbs energy G0.00.10.20.184.108.40.206.220.127.116.11T = 400KT = 500KT = 550KGibbs Energy as a function of extent of the reaction 2NH3<=>N2 + 3H2 for various temperatures. It is assumed, that the changes of enthalpy and entropy are constant.
15 Equilibrium considerations a) Stoichiometric reactions Separation of variables results in :Thus the equilibrium conditionfor a stoichiometric reaction is:Introduction of standard potentials mi° and activities ai yields: One obtains:
16 Equilibrium considerations a) Stoichiometric reactions It follows the Law of Mass Action:where the product or is the well-known Equilibrium Constant.The REACTION module permits a multitude ofcalculations which are based on the Law of Mass Action.
17 Equilibrium considerations b) Multi-component multi-phase approach Complex EquilibriaMany components, many phases (solution phases), constant T and p :withor
18 Equilibrium considerations b) Multi-component multi-phase approach Massbalance constraintj = 1, ... , n of components bLagrangeian Multipliers Mj turn out to be the chemical potentials of the system components at equilibrium:
19 Equilibrium considerations b) Multi-component multi-phase approach aijjiExample of a stoichiometric matrix for the gas-metal-slag system Fe-N-O-C-Ca-Si-Mg
20 Equilibrium considerations b) Multi-component multi-phase approach Modelling of Gibbs energy of (solution) phasesPure Substance (stoichiometric) Solution phaseChoose appropriate reference state and ideal term, then check for deviations from ideality.See Page 11 for the simple substitutional case.
21 Equilibrium considerations Multi-component multi-phase approach 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 (suchas fixed heat balances, or the occurrence of apredefined phase) makes this module evenmore versatile.
22 Phase diagrams as projections of Gibbs energy plots Hillert 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.)
23 Phase diagrams as projections of Gibbs energy plots a+bb+ga+gUnary system: projection from m-T-p diagram
24 Phase diagrams as projections of Gibbs energy plots 3004005006007001.00.50.0-0.5-1.00.80.60.40.2TCuxNiNiGBinary system: projection from G-T-x diagram, p = const.
25 Phase diagrams as projections of Gibbs energy plots Ternary system: projection from G-x1-x2 diagram, T = const and p = const
26 Phase diagrams generated with FactSage 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 ofT, P, m (as RT ln a), a (as ln a), mol (x) or weight (w)fraction as axis variables. Multi-component phase diagramsrequire 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.
27 N-Component System (A-B-C-…-N) Extensive variablesS V nA nB nNT -P µA µB µNCorresponding potentialsGibbs-Duhem:
28 Choice of Variables which always give a True Phase Diagram 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:[ 1, 2, … , n; Qn+1, Qn+2, …, QN+1] are then the (N+1) variablesof which 2 are chosen as axesand the remainder are held constant.
29 MgO-CaO Binary System Extensive variables and corresponding potentials Chosen axes variables and constantsS TV -PnMgO µMgOnCaO µCaO1 = T for y-axis2 = -P constant for x-axis
30 Fe - Cr - S - O System S T V -P nFe mFe nCr mCr f1 = T (constant) f2 = -P (constant)x-axis(constant)
31 Fe - Cr - C System - improper choice of axes variables V -PnC mCnFe mFenCr mCrf1 = T (constant)f2 = -P (constant)f3 = mC aC for x-axis and Q4 for y-axis (NOT OK)(OK)Requirement:
32 Fe - Cr - C System - improper choice of axes variables M23C6M7C3bccfcccementitelog(ac)Mole fraction of Cr0.10.20.30.18.104.22.168.80.91.0-3-2-112This 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.