Thermodynamics in Materials Engineering

Mat E 212 - Course Notes

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

Nonideal mixing and the Regular Solution Model

Mat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-1Recall our general expressions for the Gibbs free energy of a binary solution phase.The Gibbs free energy vs compositionXBXA or in terms of the partial molar quantities:where:and is the partial molar enthalpy of mixing.

The Ideal SolutionMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-2For an ideal solution, the enthalpy of mixing is zero. or in terms of the partial molar quantities:where:For , we see that .

Ideal mixingMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-3Lets examine a simple example. Consider an unmixed binary system (below left). Now consider the randomly mixed condition (below right). The enthalpy of mixing is given by: State 1If each particle has no preference regarding which type of particle resides in its nearest neighbor shell (i.e. the energy of A-A, A-B, B-B pairs are equal), the mixing is said to be ideal.

Pair potentialsMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-4For any pair of particles, we can express the potential energy as a function of the distance between the two particles. Generally, there will be a net repulsion at short distances and a net attraction at long distances. xEThe minimum in energy represents the most likely (i.e. average) distance between the two particles and the average pair potential.

Pair potentialsMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-5Generally, the potential associated with each type of particle (or atom) pair will be different. xEThe minimum in energy represents the most likely (i.e. average) distance between the two particles and the average pair potential.

Pair potentialsMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-6Now we compare the A-A and B-B potentials with the A-B potential.xEHere we see that, in general, the equilibrium distances and the associated potentials are different for each pair type. (Note that EAB=EBA.)

Ideal MixingMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-7xEEAA = EBB = EABFor the case of ideal mixing, we assume that mixing is random and that all three pair potentials are identical. Thus, the enthalpy change associated with mixing is equal to zero.Thus, the change in the total interaction enthalpy associated with MIXING is zero.

Nonideal mixingMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-8If the potentials for the different pairs are not equal, then the mixing will generally NOT be random and (even if it is random) the enthalpy of mixing will depend on the differences in the pair potentials.xEThere are many ways to compute the enthalpy of mixing based on these pair potentials. Here, we will examine a few of the simplest.

A simple solution modelMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-9Assumptions: - we neglect the differences in equilibrium interparticle distances- we assume that mixing is random- we account only for nearest neighbor interactions- we assume that all pair potentials are equal to that for the equilibrium pair distance. xETherefore, the enthalpy of mixing is computed as a function of only {EAA, EBB, EAB }.

A simple solution modelMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-10State 1Number of A-A bonds broken = nAA = 4Number of B-B bonds broken = nBB = 4Number of A-B bonds created = nAB = 8In this case, the enthalpy change = In general:

A simple solution modelMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-11For a randomly mixed solution, the number of A-B pairs created per mole of solution is given by: For every A-B bond created, one A-A and one B-B bond must be broken:Substitution yields:

A simple solution modelMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-11For a randomly mixed solution, we must now determine the number of A-B pairs created per mole of solution: The number of atoms The number of A atomsThe number of nearest neighbors around A atoms The number of B-atom nearest neighbors around A atoms

A simple solution modelMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-12xEEBBEAAEABWhat governs the enthalpy of mixing in this simple model is just the difference between the A-B potential and the average of the the A-A and B-B potentials.Lets look at this more closely.

A simple solution modelMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-13EABE0EAAEBBIf e is positive, the enthalpy of mixing is positive.

If e is negative, the enthalpy of mixing is negative.

A simple solution modelMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-14Because DHm is a function only of composition, we introduce a constant, W, and write: This is known as the regular solution model.

The excess Gibbs free energyMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-15For the regular solution model, the excess Gibbs free energy depends only on XB, T, and W. For the RS Model, DHm may be positive or negative but does not vary with temperature.

The excess Gibbs free energyMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-16For a positive DHm, the regular solution model can predict immiscibility in a given phase. T1

Phase diagram predictions using the RS modelMat E 212 - Thermodynamics in Materials Engineering - R.E. Napolitano19-16HW#9 (Teams):

Use the regular solution model for two phases and indicated parameters to compute the highlighted phase diagrams (left).Develop a 3-phase model that shows a peritectic invariant in the binary phase diagram.

For both problems, submit:Your codeThe phase diagram(s)Representative G(X) curves

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