Is it possible to have a poly(vinyl chloride) homopolymer with the following molecular weight data, and a degree of polymerization of \(1120 ?\) Why or why not? $$\begin{array}{ccc} \hline \text {Molecular Weight} & & \\ \text {Range }(\mathrm{g} / \mathrm{mol}) & \boldsymbol{w}_{\boldsymbol{i}} & \boldsymbol{x}_{\boldsymbol{i}} \\ \hline 8,000-20,000 & 0.02 & 0.05 \\ 20,000-32,000 & 0.08 & 0.15 \\ 32,000-44,000 & 0.17 & 0.21 \\ 44,000-56,000 & 0.29 & 0.28 \\ 56,000-68,000 & 0.23 & 0.18 \\ 68,000-80,000 & 0.16 & 0.10 \\ 80,000-92,000 & 0.05 & 0.03 \\ \hline \end{array}$$

The degree of polymerization (DP) is a fundamental concept used to describe the size of a polymer molecule. It represents the number of monomer units that are linked together within a polymer chain. This number is especially important because it directly correlates with the overall properties of the polymer, such as strength, melting temperature, and solubility.

For instance, in the context of poly(vinyl chloride), or PVC, the degree of polymerization can be thought of as the number of vinyl chloride units that are chemically bonded to form the PVC polymer. It is calculated by the equation:

\[\begin{equation}DP = \frac{\overline{M}_n}{M_{\text{monomer}}} \end{equation}\]

where \(\overline{M}_n\) stands for the number-average molecular weight of the polymer and \(M_{\text{monomer}}\) is the molecular weight of a single monomer unit. In general, higher DP values indicate that the chains forming the polymer are longer, giving rise to different physical properties. Understanding this relationship helps us comprehend why the degree of polymerization is such an essential parameter in polymer science.

Weight-average molecular weight (\(\overline{M}_w\)) is a way to express the molecular weight of a polymer that takes into account the weight of each fraction of the polymer's molecular weight distribution. What distinguishes this measure is its emphasis on the heavier molecules within a sample – those with a higher molecular weight exert a stronger influence on \(\overline{M}_w\) than they do on other averages.

The weight-average molecular weight is calculated using the formula:

\[\begin{equation}\overline{M}_w = \sum_{i=1}^n w_i \times M_i \end{equation}\]

In this formula, \(w_i\) denotes the weight fraction of the segment of the polymer population falling within a specific molecular weight range, and \(M_i\) is the midpoint of that range. This method of calculation provides us with a weighted mean that better reflects the contribution of heavier, longer chains present in the polymer sample. As a result, \(\overline{M}_w\) is often greater than the number-average molecular weight (\(\overline{M}_n\)), especially in polymers with a broad molecular weight distribution.

Number-average molecular weight (\(\overline{M}_n\)) is a statistical measure that signifies the average molecule size of a polymer. Unlike weight-average molecular weight, this calculation assigns equal significance to each molecule, regardless of its size. Essentially, this average is more representative of the majority of molecules present, particularly when there is an abundance of lower molecular weight species.

To calculate \(\overline{M}_n\), the following equation is used:

\[\begin{equation}\overline{M}_n = \sum_{i=1}^n x_i \times M_i \end{equation}\]

Here, \(x_i\) denotes the number fraction of molecules within a certain molecular weight range and \(M_i\) is the midpoint molecular weight of that range. The number-average molecular weight is crucial for understanding polymer characteristics like solution viscosity and osmotic pressure. When comparing \(\overline{M}_n\) to \(\overline{M}_w\), one can gain insight into the molecular weight distribution of a polymer sample.