Thermodynamics in Materials Engineering

Mat E 212 - Course Notes

R. E. NapolitanoDepartment of Materials Science & EngineeringIowa State University

An Introduction to Binary Phase Diagrams

A binary phase diagram is a thermodynamic map, indicating the most stable (i.e. equilibrium) phases and phase compositions for any average binary composition at a selected temperature. Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoBinary Phase Diagrams

Multiple invariants can give binary phase diagrams a more complex appearance. However, they can always be broken down into the individual invariant reactions.Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoMultiple invariants

In summary, the auxiliary functions express the intercepts as functions of P,V,S, and T.Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoAuxiliary Functions

These functions provide useful conditions for equilibrium under certain constraints.Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoAuxiliary Functions

Realizing that there may be other sources of energyMat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoOther sources of energy where these may be electrical, magnetic, chemical, etc.

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoChemical PotentialConsidering the chemical energy, we realize that there will be an energy contribution from each component in the material. where C is the number of components in the system.We now define the chemical potential for the ith component asand the Gibbs free energy becomes:

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoFor any binary solution01Gibbs free energy

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoOur StrategyRecall that the Gibbs free energy is related to the experimentally measurable quantity of heat capacity:The problem is that the above relationship gives G for a particular composition. We could choose to measure the heat capacity for many compositions, and compute G(T,ni) using the above relation. A more reasonable (and more useful) approach is to use reference compositions for which we measure the heat capacity and compute a reference value of G(T). We then compute the free energy for a mixture or solution of the reference components by combining the appropriate reference free energies and then adding the energy associated with mixing. This free energy of mixing must include all enthalpic and entropic contributions

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoOur StrategyFor convenience, we introduce a different composition variable, the mole fraction, X:For example, in the simple case of a binary solution (C=2):Furthermore, because XA=1-XB, we can reduce the composition to a single variable, X=XB.And rewrite our expression for the Gibbs free energy.

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoOur StrategyGiven that the system is comprised of a set of components for which the heat capacity can be (or has been) measured, the free energy of each reference state can be computed. Our primary challenge then, is to formulate the appropriate description of both the enthalpy of mixing and the entropy of mixing, i.e. we seek the free energy associated with combining the reference states into the mixture of interest.

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoTo summarize our approach:

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGibbs Free Energy vs TemperatureGT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGibbs Free Energy vs TemperatureGTTm

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoT1Gibbs Free Energy in a binary solution

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoTXBT2GGibbs Free Energy in a binary solution

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGibbs Free Energy in a binary solutionGTXBT3

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGibbs Free Energy vs Composition: the simplest caseGXBThe Gibbs free energy of the unmixed mixture.If there are no atom-atom interactions, then Hmix=0.What about Smix?

Generally, we are interested in Smix as a function of composition. 1 1Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoEntropy of Mixing

Mat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano4-23Entropy of Mixing(Entropy of mixing - per mole of the mixture)NOTE:No = Avogadros #n/ No = # moles = NNok=R(Now Apply Sterlings approx.)Now we define the mole fraction, Xi=ni /n.

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoEntropy of Mixing

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGibbs Free Energy vs CompositionGXB

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoThe Ideal SolutionThis is a simple model for G(XB) for a single phase.

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoComparing G(X) for two phasesGXB

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGComparing G(X) for two phases

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGComparing G(X) for two phases

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGComparing G(X) for two phases

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGComparing G(X) for two phases

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGComparing G(X) for two phases

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGComparing G(X) for two phases

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGWAIT!There is another variable to considerComparing G(X) for two phases

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoCompositional StabilitySuppose we have a particular overall composition, defined by XB. Will the system remain homogeneously mixed at equilibrium?XB

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoSuppose we have a particular overall composition, defined by XB. Will the system remain homogeneously mixed at equilibrium?Compositional StabilityXB

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoSuppose we have a particular overall composition, defined by XB. Will the system remain homogeneously mixed at equilibrium?Compositional StabilityXB

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoHere we consider fluctuations in local composition. Realizing that mass must be conserved, a local increase in XB must be accompanied by a decrease in XB elsewhere.Compositional StabilityXBXB+XB-There is a net increase in G!

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGPhase Stability+

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoPhase StabilityGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoA Note on Reference StatesGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano4-12A Note on Reference StatesGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano4-12A Note on Reference StatesGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano4-12A Note on Reference StatesGA vs TGB vs TGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoA simple binary phase diagramGGLGXBT

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTA simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTA simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTA simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTA simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTA simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTA simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTLA simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTAnother simple binary phase diagramG

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTGAnother simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTGAnother simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTGAnother simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTGAnother simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTG+Another simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTG-Another simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTGAnother simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTGAnother simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTLGAnother simple binary phase diagram

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoGGLGXBTLHomework #8 (by team) Due Friday 3/9/07G1. Create a ppt schematic like this for your project system. Be as quantitative as possible with temperature and composition values.

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoHomework #8 (by team) Due Friday 3/9/072. Assume ideal solutions for the solid and liquid phases and write a short program to compute and plot the binary phase diagram for the Cu-Ni system over the temperature range from 1080 to 1460 C.

Mat E 212 - Thermodynamics in Materials Engineering - R.E. NapolitanoHomework #8 (by team) Due Friday 3/9/072. Assume ideal solutions for the solid and liquid phases and write a short program to compute and plot the binary phase diagram for the Cu-Ni system over the temperature range from 1080 to 1460 C.SUBMIT BOTH OF THESE ITEMS ELECTRONICALLY (EMAIL) BY 11am ON FRIDAY, 3/9.