The number average molecular mass and mass average molecular mass of a polymer are respectively 30,000 and 40,000 . The poly dispersity index of the polymer is: (a) \(<1\) (b) \(>1\) (c) 1 (d) 0

In the study of polymer science, the number average molecular mass, denoted by \(M_n\), is a way to quantify the size of polymer molecules. It's basically an average that considers the number of molecules present.

Imagine you have a crowd of people and want to know the average height. If you measure everyone's height and then divide by the total number of people, you've calculated the number average height. Applying the same logic to polymers, \(M_n\) is calculated by summing the molecular masses of all the polymer molecules and dividing by the number of molecules. The formula is: \[ M_n = \frac{\sum (n_i \cdot M_i)}{\sum n_i} \] where \(n_i\) represents the number of molecules of a particular mass \(M_i\).

Students may sometimes confuse it with other averages, but remember, \(M_n\) is all about the number of molecules, much like the idea of an 'average count'. A polymer with a narrow molecular weight distribution would have a number average molecular mass close to the mass average molecular mass, indicating uniformity in the size of the molecules.

Contrasting the number average molecular mass is the mass average molecular mass, denoted as \(M_w\). This average takes into account not just the number of molecules, but also how much mass each molecule contributes.

Returning to our height analogy, this would be like if you calculated the average height by considering how much space each person takes up in a room—in other words, you're weighting the average by mass or volume. In polymer terms, larger molecules contribute more to the \(M_w\) than smaller ones, since they have more 'heft'. The formula is: \[ M_w = \frac{\sum (n_i \cdot M_i^2)}{\sum (n_i \cdot M_i)} \] where, again, \(n_i\) is the number of molecules of mass \(M_i\).

Understanding \(M_w\) is crucial when considering the strength and mechanical properties of polymers because those properties can be significantly affected by the presence of high-mass molecules.

Now, let's delve into the polymer molecular weight distribution, a term that represents the spread of different molecular masses within a polymer sample.

A polymer sample rarely consists of molecules of all the same size, so we often see a distribution—a mix of various sizes, some small, some large. This variety in molecular size can significantly influence the material properties of the polymer, such as its viscosity, melting point, and tensile strength.

• If the polymer has mostly molecules of about the same size, we say it has a narrow molecular weight distribution.
• Conversely, a wide variety means a broad distribution, reflecting a variety of molecule sizes.

Remember the Polydispersity Index (PDI) from the exercise? It's a numerical value that helps to describe this distribution. If PDI is greater than 1, it indicates a broad distribution; a PDI of 1 would suggest all molecules are of the same mass thereby indicating a narrow distribution. In our original problem, a PDI greater than 1 confirms our polymer has a range of molecular weights. This insight is crucial for understanding the polymer's behavior in different applications.