Mixing A and B

Consider entropy and energy involved with mixing <math>A</math> and <math>B</math>.

Entropy

Below is the ideal solution entropy.

<center>

<br>

<math>\Omega = \frac

</math>

<br>

<math>S = k_B \ln \Omega</math>

<br>

<math>\frac

= k \left [x_A \ln x_A + x_B \ln x_B \right]</math>

<br>

</center>

Energy model

<center>

<br>

<math>U = N_

+ N_

+ N_

</math>

<br>

</center>

Count <math>A-A</math> bonds.

<center>

<br>

<math>\mbox

</math>

<br>

<math>N_

= \frac

\right ) \right) \right] x_A x_B</math>

<br>

</center>

Regular Solution parameter

<center>

<br>

<math>\Delta \underline U_

= wx_A x_B + RT \left [x_A \ln x_A + x_B \ln x_B \right]</math>

<br>

</center>

When the term <math>w</math> is less than zero, the change of enthalpy when mixing is less than zero. High energy bonds are replaced with low energy bonds, and heat is released when <math>A</math> and <math>B</math> mix. Below is an expression when this condition is true.

<center>

<br>

<math>w_

> \frac{w_

+ w_{BB}}

</math>

<br>

</center>

Assumption and how to use model

An assumption when calculating the change in internal energy is that there is a same phase in the pure state and mixture. For instance, two fcc materials mix to form an fcc solid solution, or two bcc materials form a bcc solid solution.

<p>
</p>

There are two pieces of the model, enthalpy and entropy. Consider the case when one component changes phase. Since the state function is independent of path, add a positive energy term in a calculation.

<center>

<br>

<math>\Delta \underline G_

^

+ X_B \left (\underline H_

^B - \underline H_

^B}</math>

<br>

</center>

Vibrational and electronic entropy are smaller entropy terms and follow a different spectra than configurational entropy. Entropy may decrease when moving from bcc to fcc. It is a good approximation to set other terms to zero.

Fundamental flaw in model

There is inconsistency in solution model. A fundamental flaw is that all states in the model are equally likely. The regular solution model is more accurate at higher temperature.

When is a material an ideal solution

When AB interaction is the same as the average bond energy.

<p>
</p>

Atoms must interact in a solid. Look at solution behavior that follows from the regular solution model. It is a pedagogical tool. Most systems are more complicated. Small changes lead to very different phase behavior.

Phase stability in regular solution model

Consider when <math>w</math> is less than zero. The enthalpy and entropy work in the same direction, and there is always mixing.

<p>
</p>

When <math>w</math> is greater than zero, there is competition. Other entropy terms can overcome configurational entropy. Consider difference in slopes and logarithmic singularity. Slope is infinite at sides; the curve of energy always starts going down.

<p>
</p>

Consider when the free energy curve is convex and concave. This leads to mixing. Everything could be symmetric. Mix at a certain composition, and unmixing lowers free energy. The lowest possible chord gives the lowest possible free energy. A common tangent does not need to be horizontal.

How do solution limits change with temperature?

At low temperature, common tangent points move out, and there is complete miscibility at high temperature. Below a critical temperature, there are two phases. There is a competition between enthalpy and entropy. There is miscibility in end members, and there are systems that demonstrate this behavior. In the miscibility gap, the system separates into two different solutions.

Crucial divide

Regular solution model shows crucial divide. Materials show either <math>w>0</math> or <math>w<0</math> The phsyics of energy and entropy are considered.