Calculate the number-average molecular weight of a random nitrile rubber [poly (acrylonitrile-butadiene) copolymer] in which the fraction of butadiene repeat units is \(0.30\) assume that this concentration corresponds to a degree of polymerization of 2000 .

The degree of polymerization (DP) is a fundamental concept in polymer chemistry that indicates the number of monomeric units in a polymer chain. Knowing the DP is essential to understanding the structure and properties of a polymer. In simple terms, it represents how big a single molecule of a polymer is in terms of the number of repeat units it contains.

For instance, a higher DP signifies a longer polymer chain which can affect the material's strength, toughness, and viscosity. Calculating the DP is particularly useful when assessing the characteristics of a copolymer, like in our exercise where a random nitrile rubber comprises units of acrylonitrile and butadiene.

To determine the DP, you may be provided with a value, or you may need to calculate it by examining the polymer's molecular weight and the weight of the repeating unit. In the exercise solution, a DP of 2000 was given, which helps in determining the copolymer's number-average molecular weight.

Molecular weight calculations in polymer chemistry are pivotal for predicting how a polymer behaves in different conditions. The number-average molecular weight (Mn) is particularly important and it's obtained by multiplying the degree of polymerization by the molecular weight of the repeat unit. With copolymers, you consider the weights and proportions of each type of repeating unit.

In the provided solution, we see this process applied to calculate the Mn of a poly(isobutylene-isoprene) copolymer. By knowing the individual molecular weights of isobutylene and isoprene, along with their respective degrees of polymerization, we can find the copolymer's overall average molecular weight. The calculation gives us an insight into the material's average performance traits such as its mechanical properties, which can range from elasticity to tensile strength based on the Mn.

For a copolymer, the formula for Mn incorporates the weight fraction and DP of each monomer repeating unit, offering a nuanced look into the polymer's mass distribution. This approach emphasizes why molecular weight calculation is not just a mere mathematical exercise but a key to understanding polymers at a molecular level.

Polymer chemistry is the branch of chemistry that deals with the synthesis, processing, and properties of polymers, which are large molecules made up of repeating structural units. Knowledge of polymer chemistry is essential for creating new materials with tailored properties for various applications, from medical devices to automotive components.

Whether it's thermoplastics, thermosets, elastomers, or fibers, each class of polymers has unique characteristics that are a result of their structure and the types of monomers used. For instance, nitrile rubber - featured in the given exercise - is a type of synthetic rubber known for its resistance to chemicals and oils, which is attributable to the acrylonitrile content within the material.

It's through studying polymer chemistry that we develop the underlying principles governing polymer reactions and behaviors, allowing for the methodical design and fabrication of new products. This field is underscored by calculations like those shown in the exercise, which provide a glimpse into the practical aspects of polymer chemistry that drive innovation and application in a wide array of industrial sectors.