Molecular weight data for some polymer are tabulated here. Compute (a) the number- average molecular weight and (b) the weightaverage molecular weight. (c) If it is known that this material's degree of polymerization is 710 , which one of the polymers listed in Table \(14.3\) is this polymer? Why? \begin{tabular}{rcc} \hline \multicolumn{3}{|c}{ Molecular Weight Range \((g /\) mol \()\)} & \(\boldsymbol{x}_{\boldsymbol{i}}\) & \(\boldsymbol{w}_{\boldsymbol{i}}\) \\ \hline \(15,000-30,000\) & \(0.04\) & \(0.01\) \\ \(30,000-45,000\) & \(0.07\) & \(0.04\) \\ \(45,000-60,000\) & \(0.16\) & \(0.11\) \\ \(60,000-75,000\) & \(0.26\) & \(0.24\) \\ \(75,000-90,000\) & \(0.24\) & \(0.27\) \\ \(90,000-105,000\) & \(0.12\) & \(0.16\) \\ \(105,000-120,000\) & \(0.08\) & \(0.12\) \\ \(120,000-135,000\) & \(0.03\) & \(0.05\) \\ \hline \end{tabular}

Understanding the number-average molecular weight (\(M_n\)) is crucial in polymer science. It represents the arithmetic mean of the molecular weights of the individual polymers within a sample. To calculate it, we use the formula \(M_n = \sum_{i} x_i M_i\), where \(x_i\) is the mole fraction of the polymer species and \(M_i\) is the molecular weight of that species.

Consider a playing field where different teams have different numbers of players; \(M_n\) would tell us the average team player count. In the context of polymers, if each team is a batch of molecules, \(M_n\) gives us an average molecule's size based on quantity, not mass. This understanding helps when comparing polymers, as a high \(M_n\) indicates a greater average size of the polymer chains, which can affect properties like solubility and strength.

When calculating \(M_n\), we multiply the fraction of molecules in a given size range by the midpoint molecular weight of that range, and then sum up these products across all ranges. Using the midpoint ensures that we take into account the distribution of different sized molecules within each range rather than assuming they all have the same weight.

The weight-average molecular weight (\(M_w\)), another fundamental metric, gives us a pondered average by considering the weight contribution of each polymer species to the total weight. We calculate it using \(M_w = \sum_{i} w_i M_i\), wherein \(w_i\) is the weight fraction and \(M_i\) is the molecular weight of the species.

Going back to the analogy of teams, if instead of the number of players, we focus on their total weight, \(M_w\) gives us an average weight per team. In polymers, this means \(M_w\) reflects the size of the molecules in terms of their mass contribution to the mixture. A high \(M_w\) is often indicative of a significant presence of heavier, possibly longer, polymer chains, which can confer superior mechanical properties like toughness.

It's important to note that \(M_w\) is always greater than or equal to \(M_n\) because it's more influenced by the presence of higher molecular weight species. Thus, \(M_w\) provides a more mass-centric lens to understanding the distribution of molecular weights in a polymer sample.

The degree of polymerization (DP) is a simple yet insightful number that tells us about the average number of repeating units in a polymer chain. It's calculated by dividing the molecular weight of the polymer (\(M_n\) or \(M_w\)) by the molecular weight of the repeating unit (\(M_{repeat}\)).

\frac{M_n or M_w}{M{repeat}} = DP

The DP can be viewed as the average length of a chain in terms of monomer units. Like a train composed of a certain number of cars, the DP would tell us how many cars are on average in a train. In a polymer context, a higher DP means longer chains, which can enhance characteristics like the melting point and tensile strength, while a lower DP might make the polymer more suitable for applications requiring less rigidity.

The concept of DP is fundamental when it comes to understanding the properties of polymers and is directly connected to the molar mass of the repeating unit. Identifying a polymer based on DP involves calculating the molar mass of the repeating unit and comparing it to known values to determine the polymer's identity.

The molar mass of the repeating unit (\(M_{repeat}\)) in a polymer holds the key to identifying the type of the polymer. This is because this molar mass is unique for each type of polymer, given that it is the weight of one mole of the repeating structural unit that makes up the polymer chain.

To establish this molar mass, we use the formula \(M_{repeat} = \frac{M_n}{DP}\), where DP stands for the degree of polymerization. You could view \(M_{repeat}\) as the building block size of a structure; knowing the size of one block can help you deduce the entire structure's makeup. For a polymer, by figuring out the molar mass of a single repeating unit, we can match it with known polymers to identify which one we have in hand.

In the step-by-step solution, \(M_{repeat}\) was obtained using \(M_n\) and the given DP, allowing us to estimate which polymer corresponds to a known repeating unit's molar mass. When determining the molar mass of a repeating unit, it's essential to ensure our calculations reflect accurate measurements from molecular weight data, which in turn informs the polymer's chemical identity.