Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

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

Step-by-Step Solution

Answer

The polydispersity index is greater than 1, so the answer is (b) >1.

Step 1: Understand the Terms

The number average molecular mass (\( M_n \)) is the total mass of all polymer molecules divided by the total number of polymer molecules. The mass average molecular mass (\( M_w \)) is the average of the molecular masses weighted by the mass of each polymer. The polydispersity index (PDI) is the ratio of \( M_w \) to \( M_n \).

Step 2: Identify Given Data

Here, we are given \( M_n = 30,000 \) and \( M_w = 40,000 \). You will need this information to calculate the polydispersity index.

Step 3: Calculate the Polydispersity Index

The formula for the polydispersity index is \( \text{PDI} = \frac{M_w}{M_n} \). Substituting the given values, we have \[ \text{PDI} = \frac{40,000}{30,000} = \frac{4}{3}. \]

Step 4: Determine the Answer by Analyzing PDI Value

Since the calculated PDI is \( \frac{4}{3} \), which is equal to approximately 1.33, since \( \text{PDI} > 1 \), the correct option is (b) \(>1\).

Key Concepts

Number Average Molecular Mass

The number average molecular mass ( M_n ) is essential in the world of polymer chemistry. Think of it as the average molecular size of all the different molecules in a polymer sample. It's calculated by adding up the mass of each individual molecule, then dividing this total by the number of molecules present. This gives us a fair representation of the polymer's overall molecular size by considering each molecule equally. To conceptualize this, imagine you have a bowl of beads of different sizes, representing molecules. By taking the number average, you're determining the average size as if every bead contributed equally, regardless of its actual size. With a value of 30,000 in the given problem, it indicates that, on average, the polymer molecules have this molecular mass, offering insight into the polymer's composition without being biased by larger molecules.

Mass Average Molecular Mass

Mass average molecular mass ( M_w ) offers a slightly different perspective in understanding polymers. Unlike the number average molecular mass, the mass average considers the mass contribution of each molecule, giving more weight to larger molecules. This calculation is especially important because, in polymer chemistry, larger molecules often influence the material properties even more. The mass average is found by weighting each molecular mass by its actual mass and then taking the average. This means bigger molecules, or those that are more massive, have a greater influence on the final mass average. In our exercise, a value of 40,000 indicates that when mass is taken into account, the average molecular mass of the polymer increases, pointing to the presence of some particularly large molecules within the sample. This gives us more insight into the material's behavior and performance.

Polymer Chemistry

Polymer chemistry is a fascinating field that explores the makeup and behavior of polymers, which are large molecules composed of repeating subunits. Understanding concepts like number and mass average molecular mass is crucial in this field, as they help chemists determine polymer characteristics like strength, elasticity, and thermal stability. In polymers, the distribution of molecular masses can vary significantly, affecting the property and quality of the material produced. This is where the polydispersity index (PDI) becomes invaluable. It provides a measure of the molecular weight distribution of a polymer. A higher PDI indicates a wider distribution of molecular sizes, suggesting variations in the polymer's structural makeup. The calculation of the PDI in our example (by dividing the mass average by the number average) shows a value greater than 1, which is typical as it confirms that the polymer sample contains molecules of varying sizes. This understanding allows scientists and engineers to tailor polymers for specific uses, ensuring materials are well-suited for their intended application.