There are a few methods of determining the enthalpy of mixing between two elements theoretically.

One method is to use the ideal solution model. In this model, it is assumed that the two elements mix completely and randomly, and that there are no interactions between the molecules. The enthalpy of mixing in an ideal solution is zero.

Another method is to use the regular solution model. This model takes into account the interactions between the molecules of the two elements. The enthalpy of mixing in a regular solution can be positive or negative, depending on the strength of the interactions.

A third method is to use the lattice model. This model treats the two elements as occupying lattice sites, and it takes into account the energetics of the different ways that the molecules can be arranged on the lattice. The enthalpy of mixing in a lattice model can be positive, negative, or zero.

The choice of which method to use depends on the specific system being studied. The ideal solution model is often a good approximation for systems where the two elements mix easily, such as mixtures of non-polar gases. The regular solution model is more accurate for systems where there are strong interactions between the molecules, such as mixtures of polar molecules. The lattice model is the most accurate, but it is also the most complex.

In addition to these theoretical methods, the enthalpy of mixing can also be determined experimentally. This is done by measuring the heat released or absorbed when the two elements are mixed.

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One possible method of determining the enthalpy of mixing between two elements theoretically is to use the regular solution model, which assumes that the atoms are randomly distributed and that the enthalpy of mixing depends on the difference in the atomic sizes and the bond energies of the elements. The regular solution model can be expressed as follows:

ΔH mix = zNε(1 - x)x

where ΔH mix is the enthalpy of mixing per mole of solution, z is the coordination number, N is the Avogadro’s number, ε is the interaction energy between unlike atoms, and x is the mole fraction of one of the elements.

The interaction energy ε can be estimated from the bond energies of the pure elements using the following formula:

ε = (E A + E B )/2 - E AB

where E A and E B are the bond energies of the pure elements A and B, and E AB is the bond energy of the compound AB.

The bond energies can be obtained from experimental data or calculated from theoretical methods, such as density functional theory or molecular dynamics.

For more details and examples of this method, you can refer to these sources:

Enthalpy of Mixing in Binary Alloys: A Simple Model

Thermodynamic Modeling of Alloys

there are other possible methods to determine the enthalpy of mixing between two elements theoretically. Some of them are:

• The subregular solution model, which is a modification of the regular solution model that allows for different interaction energies between unlike atoms depending on their relative positions in the solution. This model can account for the deviation from ideality and asymmetry in the enthalpy of mixing1.
• The quasi-chemical model, which is based on the statistical mechanics of lattice gases and considers the short-range order in the solution. This model can describe the enthalpy of mixing as a function of the coordination number, the nearest-neighbor interactions, and the degree of order in the solution2.
• The cluster variation method, which is an extension of the quasi-chemical model that incorporates higher-order clusters of atoms and their interactions. This method can capture the long-range order and phase transitions in the solution3.

These are some examples of alternative methods to calculate the enthalpy of mixing theoretically. However, each method has its own assumptions, limitations, and parameters that need to be determined or fitted from experimental data or first-principles calculations. Therefore, no single method can be universally applicable or accurate for all kinds of solutions.

Here is a summary of the methods:

• The regular solution model assumes that the atoms are randomly distributed and that the enthalpy of mixing depends on the difference in the atomic sizes and the bond energies of the elements. The enthalpy of mixing is given by:

ΔH mix = zNε(1 - x)x

where ΔH mix is the enthalpy of mixing per mole of solution, z is the coordination number, N is the Avogadro’s number, ε is the interaction energy between unlike atoms, and x is the mole fraction of one of the elements.

The source for this method is Lecture 3: Models of Solutions - University of Cambridge.

• The subregular solution model is a modification of the regular solution model that allows for different interaction energies between unlike atoms depending on their relative positions in the solution. This model can account for the deviation from ideality and asymmetry in the enthalpy of mixing. The enthalpy of mixing is given by:

ΔH mix = zNε(1 - x)x + zNβ(1 - x)x(1 - 2x)

where ΔH mix is the enthalpy of mixing per mole of solution, z is the coordination number, N is the Avogadro’s number, ε is the average interaction energy between unlike atoms, β is a parameter that measures the deviation from regularity, and x is the mole fraction of one of the elements.

The source for this method is Lecture 7: Quasichemical Solution Models - University of Cambridge.

• The quasi-chemical model is based on the statistical mechanics of lattice gases and considers the short-range order in the solution. This model can describe the enthalpy of mixing as a function of the coordination number, the nearest-neighbor interactions, and the degree of order in the solution. The enthalpy of mixing is given by:

ΔH mix = −zN(NA2AA + NB2BB − NABω)

where ΔH mix is the enthalpy of mixing per mole of solution, z is the coordination number, N is the Avogadro’s number, NA , NB , and NAB are the numbers of AA , BB , and AB bonds per atom, respectively, and ω = 2AA + 2BB − 22AB is a parameter that measures the bond energy difference between like and unlike atoms.

The source for this method is Cluster Variation Method Analysis of Correlations and … - Springer.

• The cluster variation method is an extension of the quasi-chemical model that incorporates higher-order clusters of atoms and their interactions. This method can capture the long-range order and phase transitions in the solution. The enthalpy of mixing is given by:

ΔH mix = −zN(NA2AA + NB2BB − NABω) − zN(NABCωABC + NABDωABD + …)

where ΔH mix is the enthalpy of mixing per mole of solution, z is the coordination number, N is the Avogadro’s number, NA , NB , NAB , NABC , NABD , etc. are the numbers of AA , BB , AB , ABC , ABD , etc. clusters per atom, respectively, ω = 2AA + 2BB − 22AB is a parameter that measures the bond energy difference between like and unlike atoms, and ωABC , ωABD , etc. are parameters that measure the cluster interaction energies.

The source for this method is Cluster Variation Method | SpringerLink.

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