The density and associated percent crystallinity ( for two poly(ethylene terephthalate) materials are as follows: \begin{tabular}{lc} \hline\(\rho\left(\mathrm{g} / \mathrm{cm}^{3}\right)\) & crystallinity \((\%)\) \\\ \hline \(1.408\) & \(74.3\) \\ \hline \(1.343\) & \(31.2\) \\ \hline \end{tabular} (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 \(1.382 \mathrm{~g} / \mathrm{cm}^{3}\).

Understanding the computation of density is vital in materials science and engineering. Density, expressed as mass per unit volume (\rho), is a fundamental physical property of a material. Materials with complex structures, like polymers, can have varying densities based on their degree of crystallinity.

In the context of poly(ethylene terephthalate) (PET), which is often used in textiles and packaging, calculating density involves considering both crystalline and amorphous forms. The crystalline regions are tightly packed and regularly ordered, which makes them denser. Contrarily, the amorphous regions are more randomly ordered and typically less dense.

When it comes to computation, the rule of mixtures is applied to estimate the overall density of a partially crystalline polymer. This method assumes linear mixing between the amorphous and crystalline regions, calculated by the formula: \[ \rho = \rho_a \times (1 - X_c) + \rho_c \times X_c \] where \rho_a represents the density of the amorphous phase, \rho_c is the density of the crystalline phase, and X_c is the fraction of the crystalline phase.

Percent crystallinity is a measure of how much of a polymer's volume is taken up by crystalline regions. High levels of crystallinity can affect a material's mechanical properties and transparency. For instance, higher crystallinity in PET can lead to increased strength and rigidity, making it suitable for beverage bottles and containers.

To calculate the percent crystallinity, we use the density values of the fully amorphous and fully crystalline states as reference points. Then we express the crystallinity of a specimen as a percentage of the difference between these reference states and the specimen's density.

Using the aforementioned rule of mixtures, the fraction crystallinity (X_c) can be found from a known density, \( \rho \), of the specimen. It's given by: \[ X_c = \frac{\rho - \rho_a}{\rho_c - \rho_a} \] Multiplying this fraction by 100 gives us the percent crystallinity. Solving exercises involving percent crystallinity is not only about performing calculations but also about understanding the material's structure-properties relationship.

In the study of polymers, particularly PET, it's crucial to differentiate between the amorphous and crystalline densities, as they play a significant role in determining the material's overall properties. The density of the amorphous phase (\( \rho_a \)) characterizes the less ordered, more flexible regions of the polymer, while the crystalline density (\( \rho_c \)) reflects the tightly packed, highly ordered structures.

These densities are not just theoretical numbers; they have practical implications in the application of polymers. For example, they influence thermal resistance, strength, and the diffusion rate of gases.

To find these values, we typically rely on laboratory data or direct measurement. However, in the absence of direct data, we can use mathematical relationships between known densities and percent crystallinities of given samples: \[ \rho = \rho_a \times (1 - X_c) + \rho_c \times X_c \] By working with systems of equations derived from these formulas, one can deduce both \( \rho_a \) and \( \rho_c \). This enables material scientists and engineers to tailor polymers for specific uses by varying their processing conditions to adjust their crystallinity.