A \(0.10-\mathrm{g}\) sample of a polymer, dissolved in \(100 \mathrm{~mL}\). of toluene, had an osmotic pressure of \(5.4\) Torr at \(20^{\circ} \mathrm{C}\). What is the molar mass of the polymer?

Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a substance. It is crucial to determining the molecular weight of a compound, which can then be used to calculate the number of moles in a given sample. In the context of the exercise, we are interested in finding the molar mass of a polymer.

To find the molar mass, we first need to understand the relationship between mass, molar mass, and the number of moles. The general formula is:
\[ M = \frac{mass}{n} \]
Where \( M \) is the molar mass, mass is the given mass of the substance, and \( n \) is the number of moles.

In practice, to determine the molar mass of the polymer from the exercise, we calculate the number of moles using the known osmotic pressure, volume, and temperature with the formula for osmotic pressure, and then we use the above formula to find the molar mass. This approach is typical in polymer chemistry where direct measurement might be challenging due to the large size and complexity of the molecules.

Colligative properties are properties of solutions that depend on the ratio of the number of solute particles to the solvent molecules and not on the nature of the chemical species present. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.

Osmotic pressure is a colligative property that is particularly important for molar mass determination of large molecules such as polymers. It is the pressure required to stop the flow of solvent molecules through a semipermeable membrane from a region of low solute concentration to one of high concentration.

The formula to calculate osmotic pressure is given by: \[\Pi = \frac{nRT}{V}\]
where \( \Pi \) symbolizes the osmotic pressure, \( n \) represents the number of moles of solute, \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( V \) is the volume of the solution.

Understanding the concept of osmotic pressure is critical, as it applies to various disciplines, including biology (e.g., plant water uptake and kidney function) and materials science (e.g., membrane technologies).

Polymer chemistry is a branch of chemistry that focuses on the chemical synthesis and chemical properties of polymers or macromolecules. Polymers consist of repeating structural units called monomers, and their large size makes their physical properties significantly different from those of small molecules.

Aspects of Polymer Behavior

Polymers exhibit unique behaviors, such as viscoelasticity and the ability to form more complex structures like block copolymers. Their properties can be tailored for specific uses, ranging from everyday products like plastic containers to high-performance materials in aerospace.

In the context of the textbook problem, the polymer solution's osmotic pressure can provide insights into the polymer's molar mass, which is crucial for understanding its physical and chemical properties. Due to the large size of polymer molecules, typical methods of molar mass determination, such as freezing point depression, are often not viable. That's why osmotic pressure measurements are particularly useful in polymer chemistry for molar mass determination. Moreover, the study of polymers as a subject intertwines physics, engineering, and materials science, making it an interdisciplinary and dynamic field.