The density and associated percent crystallinity for two poly(ethylene terephthalate) materials are as follows: $$\begin{array}{cc} \hline \rho\left(g / c m^{3}\right) & \text { Crystallinity (\%) } \\ \hline 1.408 & 74.3 \\ 1.343 & 31.2 \\ \hline \end{array}$$ (a) Compute the densities of totally crystalline and totally amorphous poly(ethylene terephthalate). (b) Determine the percent crystallinity of a specimen having a density of \(1.382 \mathrm{g} / \mathrm{cm}^{3}\)

The density of a polymer is a key property that significantly affects its overall characteristics, including strength, stiffness, and thermal behavior. In simple terms, density can be defined as the mass of a polymer per its volume. This property is particularly important in applications where weight savings are crucial, such as in the aerospace and automotive industries.

Polymers are unique materials that can exist in different states of matter, specifically regarding their crystalline and amorphous regions. The density of a polymer changes based on its level of crystallinity. Generally, the greater the crystallinity in a polymer, the higher its density will be. This is because the orderly arrangement of polymer chains in crystalline regions takes up less space than the disordered chains in amorphous regions, allowing more mass to occupy the same volume.

In the context of exercises like the one given, understanding how the polymer density changes with varying levels of crystallinity is fundamental to solving problems related to the material's properties.

The rule of mixtures is a principle applied in materials science to estimate the properties of composite materials based on the properties of their constituents and their volume fractions. For polymers, this rule is often used to determine the overall density of a material with both crystalline and amorphous phases.

The formula for the rule of mixtures for density is given by: \[\rho = \rho_c V_c + \rho_a V_a\]. where \(\rho\) represents the overall density, \(\rho_c\) and \(\rho_a\) are the densities of the crystalline and amorphous phases, respectively, and \(V_c\) and \(V_a\) are their corresponding volume fractions.

With this rule, it's possible to model the polymer as a mixture of two phases and to calculate unknown variables as long as the other values are provided. This process is integral for engineers and material scientists when designing materials for specific applications and predicting their behavior in real-world scenarios.

Percent crystallinity is a measure of the amount of crystalline structure within a polymer. This measurement is critical because it influences the physical properties of the material such as tensile strength, resistance to solvents, and optical properties. Calculating the percent crystallinity is a matter of comparing the density of the semi-crystalline polymer to the densities of its fully crystalline and fully amorphous states.

From the rule of mixtures, we can express the density of a polymer as a weighted average of its crystalline and amorphous densities. The percent crystallinity (\(V_c\)) can then be determined by the ratio of the difference between the total density (\(\rho\)) and the amorphous density (\(\rho_a\)) over the difference between the crystalline and amorphous densities: \[\% \text{Crystallinity} = \left(\frac{\rho - \rho_a}{\rho_c - \rho_a}\right) \times 100\].

By using this equation, students can solve problems where they are given the density of the polymer and either the crystalline or amorphous density, to find the remaining density or to calculate the percent crystallinity of the polymer. Understanding how to calculate percent crystallinity is essential in polymer science, as this value can greatly impact the functionality and application of a material.