For a 76 wt\(\%\) \(\mathrm{Pb}-24\) wt\% \(\mathrm{Mg}\) alloy, make schematic sketches of the microstructure that would be observed for conditions of very slow cooling at the following temperatures: \(575^{\circ} \mathrm{C}\) \(\left(1070^{\circ} \mathrm{F}\right), 500^{\circ} \mathrm{C}\left(930^{\circ} \mathrm{F}\right), 450^{\circ} \mathrm{C}\left(840^{\circ} \mathrm{F}\right),\) and \(300^{\circ} \mathrm{C}\left(570^{\circ} \mathrm{F}\right) .\) Label all phases and indicate their approximate compositions.

Understanding how a material changes with temperature is crucial for material science. A phase diagram is a valuable tool in this context; it visually summarizes the conditions under which certain phases of a material exist at equilibrium. Specifically for alloys, it indicates what phase or phases will be present at a given composition and temperature.

For example, a phase diagram for a lead-magnesium (Pb-Mg) system will show lines and areas demarcating solid, liquid, and the combinations of solid and liquid phases that exist at various temperatures and compositions. When interpreting a phase diagram, a vertical line at a specific composition can be imagined, with temperature changing as you move up and down this line. As you do so, you can predict what phases your alloy will consist of at each temperature, which is crucial for understanding properties such as strength, malleability, and conductivity.

A eutectic reaction is a specific type of phase transition in alloy systems. When a liquid alloy cools down to a particular temperature (the eutectic temperature), it solidifies into two or more different solid phases simultaneously. A common misconception is that eutectic structures involve the separation of pure components, but this is not always the case—multiple phases of varying compositions can form from the same liquid.

In the 76 wt% Pb-24 wt% Mg alloy, at the eutectic temperature, pure lead (Pb) and magnesium (Mg) phases form alternately, resulting in a layered or 'lamellar' microstructure. This eutectic structure is typically very fine, which could lead to properties such as higher corrosion resistance and lower electrical conductivity when compared to the individual metals.

The solidification process is how a material changes from a liquid to a solid as it loses heat. For alloys, solidification is highly dependent on the local composition and the cooling rate. During very slow cooling, which allows for equilibrium conditions, an alloy at a higher temperature will follow the phase boundaries on the diagram as it cools and transforms.

At each temperature, different microstructures will form: above the liquidus line, the alloy is fully liquid; between the liquidus and solidus lines, it is a mixture of solid and liquid with solid grains starting to precipitate out; and below the solidus, it is a completely solid mix. If the solidification process is too fast, nonequilibrium structures such as dendrites or a glassy phase might form. These can alter the mechanical properties of the alloy, which is why cooling rates play such a pivotal role in materials processing.

Alloy compositions are expressed as a percentage by weight (wt%) of each element in the alloy. The composition determines the alloy's properties, including density, melting temperature, ductility, hardness, and corrosion resistance, among others.

Using the alloy in question, 76 wt% Pb-24 wt% Mg, one can predict its behavior by consulting its phase diagram and understanding how those compositions interact as the alloy cools. For instance, at high temperatures, the Pb-rich liquid can dissolve a certain amount of Mg, leading to different observable phenomena during the cooling process. At lower temperatures, however, the solubility limit may be surpassed, and a secondary phase or phases will precipitate. This highlights the importance of precisely controlling composition to achieve the desired material characteristics for industrial applications.