For a 64 wt\(\%\) \(\mathrm{Zn}-36\) wt\(\%\) \(\mathrm{Cu}\) alloy, make schematic sketches of the microstructure that would be observed for conditions of very slow cooling at the following temperatures: \(900^{\circ} \mathrm{C}\) \(\left(1650^{\circ} \mathrm{F}\right), 820^{\circ} \mathrm{C}\left(1510^{\circ} \mathrm{F}\right), 750^{\circ} \mathrm{C}\left(1380^{\circ} \mathrm{F}\right),\) and \(600^{\circ} \mathrm{C}\left(1100^{\circ} \mathrm{F}\right) .\) Label all phases and indicate their approximate compositions.

Zinc (Zn) and Copper (Cu) alloys are commonly used in various industries due to their unique combination of properties such as strength, ductility, and electrical conductivity. The Zn-Cu alloys are often referred to as brass, and their properties can be tuned by adjusting the ratio of zinc to copper. This is why understanding the microstructure at different temperatures is important for material scientists and engineers.

The microstructure of an alloy, which comprises the arrangement and distribution of phases within the material, is highly dependent on its composition and the thermal treatment it undergoes. As temperature changes, so does the solid-liquid state of the metal components within the alloy, leading to the formation of different phases. Each phase has its own physical and mechanical properties which affect the overall characteristics of the alloy.

For the case of a 64 wt% Zn - 36 wt% Cu alloy, very slow cooling allows for equilibrium conditions, meaning the microstructure has ample time to develop according to the phase diagram. This results in distinct phase transformations at specific cooling temperatures, which can be predicted with the aid of a Zn-Cu phase diagram, as seen in the provided step-by-step solution.

In material science, temperature plays a crucial role in determining the phases present within an alloy. The rate at which an alloy is cooled from its molten state affects the size, shape, and distribution of the resulting phases in the solid material. Slow cooling rates, such as those in the exercise, typically lead to the formation of a more orderly and homogeneous microstructure.

When a Zn-Cu alloy undergoes very slow cooling, several significant temperatures delineate different events in the alloy's phase transition: At temperatures above the liquidus line, the alloy is entirely in the liquid phase. Between the liquidus and the solidus lines, the material exists as a mix of solid and liquid phases. Below the solidus line, the alloy becomes entirely solid, containing single or multiple solid phases based on the alloy's composition and the temperature.

Cooling temperatures like the ones described (900°C, 820°C, 750°C, and 600°C) fall within different regions of the phase diagram and result in different microstructures. Understanding these cooling temperatures and how they affect the Zn-Cu alloy's microstructure is fundamental in designing materials with desired mechanical and physical properties.

The phase diagram is a powerful tool used to determine the phases present in an alloy at various temperatures and compositions. Phase regions on a diagram indicate the stability of different phases under equilibrium conditions. Each region is bounded by phase boundaries (lines or curves), which represent the limits of a phase's stability. When an alloy's composition intersects these lines, phase changes occur.

In cooling a Zn-Cu alloy, the phase regions provide a visual guide for identifying the phases at specific temperatures. At high temperatures, the alloy resides in the single-phase liquid region. As temperature decreases, the alloy may enter a two-phase region where liquid and solid phases coexist. Eventually, at lower temperatures, the alloy transitions into a single-phase solid region.

The compositions of the phases, as seen in the step-by-step solution for temperatures such as 820°C, can also be derived from the phase diagram. For instance, the liquid would have higher zinc content whereas the solid \(\alpha\) phase's composition would be closer to the original alloy's composition. These insights into phase compositions are critical in predicting the material's properties and tailoring them to specific applications by manipulating cooling rates and alloy compositions.