The room-temperature tensile strengths of pure lead and pure tin are \(16.8 \mathrm{MPa}\) and \(14.5\) \(\mathrm{MPa}\), respectively. (a) Make a schematic graph of the roomtemperature tensile strength versus composition for all compositions between pure lead and pure tin. (Hint: you may want to consult Sections \(9.10\) and 9.11, as well as Equation \(9.24\) in Problem 9.64.) (b) On this same graph schematically plot tensile strength versus composition at \(150^{\circ} \mathrm{C}\). (c) Explain the shapes of these two curves, as well as any differences between them.

A phase diagram is a graphical representation that shows the phases of a material as a function of temperature and composition. It serves as a map for predicting the structure of an alloy and its physical properties at various temperatures and compositions. For instance, in finding the tensile strength of alloys made from lead (Pb) and tin (Sn), we rely on the phase diagram to understand how these materials will interact.

When combining two pure metals, such as lead and tin, the diagram indicates the resulting alloy's phase – whether it is a solid solution or consists of multiple phases. The phase regions can give insight into the tensile strength of the alloy. The 'U-shaped' curve mentioned in the exercise solution typically corresponds to the presence of solid solution phases, where a mixture of both elements results in either a stronger or weaker material compared to the pure components. Curves like this can tell us that at certain compositions, the alloy has a higher tensile strength due to the intermixing of atoms, creating an obstacle to dislocation movement (a mechanism for materials’ deformation) and hence leading to a strengthening effect.

In the case of the schematic graph required by the exercise, the phase diagram would assist in predicting the tensile strength at different compositions, indicating where the alloy is strongest and at what composition it is weakest.

The composition of an alloy—which is essentially a mixture of two or more elements, mainly metals—significantly affects its properties, including tensile strength. The exercise draws attention to compositions varying between pure lead and pure tin. Alloy composition is typically expressed in terms of weight percent or atom percent, and this concentration affects the phase structure and, consequently, the mechanical properties.

As the proportion of one metal increases relative to another, the physical properties, such as tensile strength, hardness, and ductility, will vary. In a binary alloy system like lead-tin, the room-temperature tensile strengths of the pure metals are given as starting points. The physical properties at intermediate compositions often do not follow a linear relationship and can show complex behavior due to the interactions between the different types of atoms within the alloy.

Alloys are developed with specific compositions to achieve desired characteristics, such as increased tensile strength, corrosion resistance, or electrical conductivity. The schematic graph mentioned in the exercise is used to predict how altering the composition of lead and tin would affect the tensile strength of the alloy. Such information is vital for designing materials with optimal properties for specific applications.

Temperature plays a significant role in determining the properties of materials, including their tensile strength. As temperature increases, materials generally become softer and less strong—a trend clearly expected in the exercise solution for the alloy at 150°C.

The reduction in tensile strength with temperature can be attributed to the increased atomic vibrations, which can make it easier for dislocations—defects in the crystal structure that allow deformation—to move. This allows the material to deform more readily under stress, leading to a lower tensile strength. Moreover, elevated temperatures can lead to changes in phase stability within an alloy, possibly resulting in a different microstructure that could be less capable of resisting applied forces.

This general trend of decreasing tensile strength with increasing temperature must be considered when designing components that will experience varied thermal environments. Understanding the temperature effects on tensile strength is crucial in materials engineering, as it determines the materials' performance and reliability over a range of operating conditions.