The room-temperature tensile strengths of pure copper and pure silver are 209 and \(125 \mathrm{MPa}\), respectively. (a) Make a schematic graph of the room-temperature tensile strength versus composition for all compositions between pure copper and pure silver. (Hint: You may want to consult Sections \(9.10\) and 9.11, as well as Equation \(9.24\) in Problem 9.79.) (b) On this same graph, schematically plot tensile strength versus composition at \(600^{\circ} \mathrm{C}\). (c) Explain the shapes of these two curves as well as any differences between them.

Understanding the relationship between a material's composition and its tensile strength is essential in Materials Science. Tensile strength is a measure of the force required to pull something such as rope, wire, or a structural beam to the point where it breaks. The composition of a material refers to the elements and compounds it consists of and their respective proportions.

In the case of alloys, which are mixtures of two or more metals, tensile strength can vary significantly based on the composition. For pure metals like copper and silver, we note specific tensile strengths, 209 MPa and 125 MPa respectively. As we begin to mix these metals, the tensile strength at room temperature generally follows a predictable pattern. Most simplistically, one might assume a linear relationship between tensile strength and composition when creating a schematic graph. This linear model suggests that by gradually changing the amount of copper and silver, the tensile strength would adjust proportionally.

However, the actual relationship can be more complex. Alloys can exhibit changes in tensile strength due to factors such as the formation of solid solutions, intermetallic compounds, or due to the presence of impurities. Such intricacies may cause deviations from a strictly linear relationship, prompting materials scientists to conduct experiments and use more sophisticated predictive models in order to accurately determine the tensile strength of different compositions.

Temperature plays a critical role in influencing the tensile strength of materials. Tensile strength, indicating how much tensile stress a material can withstand before failure, decreases with increasing temperature for many materials, including metals.

At higher temperatures, the atoms in a material vibrate more vigorously, reducing the forces that hold them together. This phenomenon leads to a decrease in tensile strength because the material can more easily deform and ultimately break under stress. The graph of tensile strength versus temperature typically shows a downward slope, illustrating that as temperature rises, the ability of a material to resist tensile stress diminishes.

For instance, pure copper and pure silver, which have notable tensile strengths at room temperature, would display lower tensile strength at 600°C. When plotting tensile strength versus composition at this elevated temperature, the resulting line or curve would sit beneath the room-temperature line. This visual comparison signals how materials weaken as temperatures climb, reinforcing the importance of accounting for temperature when selecting materials for high-temperature applications.

Creating a schematic graph can be invaluable for visualizing the relationship between different variables such as composition and tensile strength under varying temperatures. In the context of our problem, two graphs will be constructed: one for room temperature and another for 600°C.

The room-temperature graph begins by plotting the known tensile strengths of pure copper and pure silver on a coordinate system. With the tensile strength along the y-axis and the composition along the x-axis, a line connecting the two points provides an initial approximation of the relationship. This is a simplified assumption of linearity between two points of known composition—pure copper and pure silver.

For the 600°C graph, the tensile strength of each composition is expected to be lower than that at room temperature, indicated by a line or curve positioned below the former. While the first graph can be straightforwardly linear, the second graph may showcase a curve due to the more complex influence of temperature on the tensile strength of the alloy. By comparing these graphs, one can assess the materials' performance across a range of compositions and temperatures, aiding in the selection process for specific engineering applications.