The tensile strength and number-average molecular weight for two poly(methyl methacrylate) materials are as follows: $$\begin{array}{cc}\hline \begin{array}{c}\text { Tensile Strength } \\\\\text { (MPa) }\end{array} & \begin{array}{c}\text { Number-Average } \\ \text { Molecular Weight (g/mol) }\end{array} \\\\\hline 50 & 30,000 \\\150 & 50,000 \\\\\hline\end{array}$$ Estimate the tensile strength at a numberaverage molecular weight of \(40,000 \mathrm{g} / \mathrm{mol}\).

The concept of linear interpolation, a fundamental technique in mathematics and engineering, comes into play when we need to estimate unknown values within a range of known data points. At its core, linear interpolation assumes that the change between two points is consistent and can be described by a straight line. By using the formula provided in the exercise,

\[ T = T_1 + (T_2 - T_1) \frac{(MW - MW_1)}{(MW_2 - MW_1)} \],
we can predict unknown quantities by creating a proportion based on this linearity. To use the method effectively, it's essential to recognize the two known points,

• Tensile Strength: 50 MPa, Molecular Weight: 30,000 g/mol
• Tensile Strength: 150 MPa, Molecular Weight: 50,000 g/mol

and then apply the formula, making an educated guess between these points. In this exercise, the interpolation helped determine the tensile strength at \(40,000 \text{ g/mol} \).

Molecular weight is a critical measure in chemistry and materials science that indicates the mass of a molecule. In polymers like poly(methyl methacrylate), understanding molecular weight is key to predicting their physical properties, such as tensile strength. We use unit grams per mole (g/mol) to express the average molecular weight across a sample of molecules.
In the exercise, molecular weights given for the data points are 30,000 g/mol and 50,000 g/mol, representing the number-average molecular weight of distinct poly(methyl methacrylate) samples. This average is calculated by summing the molecular weights of all the molecules in a sample and dividing by the number of molecules, giving us a glimpse into the typical size of the polymer chains within the material.

Commonly known as PMMA or by its trade names like Plexiglas and Lucite, poly(methyl methacrylate) is a synthetic polymer prized for its transparency and light-transmitting properties, being a glass alternative. Its applications are vast, from windows and lenses to acrylic paint and medical devices.

The tensile strength of PMMA, like many materials, generally increases with the molecular weight due to longer polymer chains which can more effectively share the load. However, at some point, the gains level off, and other factors such as defects and chain entanglement start to dominate. In educational contexts, PMMA serves as an excellent model for studying molecular weight versus property relationships in a material.

Materials science is an interdisciplinary field that studies the properties of matter and its applications to various areas of science and engineering. This field involves understanding the structure of materials at atomic or molecular scales and how this structure gives rise to their macroscopic properties.

In the context of the given exercise, materials science principles explain how changes in the molecular structure, like the molecular weight of a polymer, affect its mechanical properties, including tensile strength. Tensile strength, a crucial mechanical property, is the resistance of a material to breaking under tension. It's what allows bridges to span great distances and aircraft to withstand the stress of flight. Understanding these material property relationships is foundational for engineers and scientists when designing new materials or enhancing existing ones.