(a) Determine the ratio of butadiene to acrylonitrile repeat units in a copolymer having a number-average molecular weight of \(250,000 \mathrm{~g} / \mathrm{mol}\) and a degree of polymerization of 4640 . (b) Which type(s) of copolymer(s) will this copolymer be, considering the following possibilities: random, alternating, graft, and block? Why?

Understanding the number-average molecular weight, denoted as \( \bar{M}_n \), is crucial when studying the properties of polymers. It is defined as the total mass of a polymer sample divided by the total number of moles of polymer molecules in that sample. This is mathematically expressed as:

\[ \bar{M}_n = \frac{\text{Sum}(m_i)}{\text{Sum}(n_i)} \]
where \( m_i \) is the mass and \( n_i \) is the number of moles of each polymer molecule. When considering copolymers, which are composed of different types of repeat units, \( \bar{M}_n \) gives us an average that accounts for the variation in mass due to the differing molecular weights of the repeat units.

This average is particularly useful because it reflects the distribution of molecular sizes in a polymer sample, impacting properties like melting point, tensile strength, and viscosity. When looking at a copolymer's \( \bar{M}_n \), understanding how it relates to the repeat units provides insight into the polymer's overall structure and potential applications.

The degree of polymerization, represented as \( N \), is a fundamental characteristic of polymer chains, denoting the number of repeat units in a single polymer molecule. In essence, it is the count of how many monomers have linked together to form the polymer. The degree of polymerization is related to molecular weight as follows:

\[ N = \frac{\bar{M}_n}{\bar{M}_u} \]
where \( \bar{M}_n \) is the number-average molecular weight and \( \bar{M}_u \) is the average molecular weight of a repeating unit. In copolymers, this concept is significant because it helps determine the polymer chain length and, by extension, its properties such as durability, flexibility, and strength. By calculating the degree of polymerization, as seen in the exercise example, you gain insight into these physical attributes and can make predictions about how the copolymer might behave in different applications.

The repeat unit ratio in copolymers is a critical piece of the polymer puzzle. It refers to the proportion of each type of repeat unit within the copolymer structure. Determining this ratio gives crucial information about the copolymer composition and properties.

For instance, in a copolymer made from two monomers, such as butadiene and acrylonitrile, the ratio of these repeat units can be represented by \( x : (1-x) \). This proportion determines a unique set of characteristics, including glass transition temperature, elasticity, and chemical resistance. Hence, by knowing this ratio, we can predict how a copolymer will perform in specific environments or applications.

As the exercise example illustrates, calculating the repeat unit ratio involves balancing the average molecular weights of the copolymer and its constituents. The result, expressed as a straightforward ratio, allows us to understand the copolymer's overall makeup and assists with the engineering of tailored polymeric materials for diverse uses.