Problem 90 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statements about polymers is correct? (1) The ratio of the weight average and number average molecular masses, \(\bar{M}_{\text {. }} / \bar{M}_{n}\), is called poly dispersion index (P.D.I.) (2) The number average molecular mass, \(\overline{\mathrm{M}} \mathrm{n}\) and the weight average molecular mass \(\bar{M}_{w}\) are expressed by the equations \(\overline{\mathrm{M}}_{\mathrm{w}}=\frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}\) and \(\overline{\mathrm{M}}_{\mathrm{w}}=\frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}^{2}}{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}\) Here, \(\mathrm{N}_{i}\) is the number of molecules of molecular mass \(\mathrm{M}_{i}\) of the species \(\mathrm{i}\). (3) Some physical properties such as freezing point, vapour pressure and osmotic pressure are related directly to \(\mathrm{M}\) whereas some other physical properties like light scattering, sedimentation and diffusion constant are related directly to \(\mathrm{M}_{w}\). (a) 1 and 2 (b) 2 and 3 (c) 1 and 3 (d) 1,2 and 3

Step-by-Step Solution

Answer

The correct statements are 1 and 3, so the answer is (c).

Step 1: Understanding the Task

We need to determine which statements about polymers are correct among the given options. There are three statements to evaluate about the concepts involving polymer molecular masses and their effects on physical properties.

Step 2: Analyzing Statement 1

Statement 1 claims that the ratio \( \bar{M}_{w} / \bar{M}_{n} \) is known as the poly dispersion index (P.D.I.). In polymer chemistry, this is indeed correct. P.D.I. is defined as \( \frac{\bar{M}_{w}}{\bar{M}_{n}} \), which indicates the distribution of molecular mass in a given polymer sample.

Step 3: Analyzing Statement 2

Statement 2 provides equations for number average molecular mass \( \bar{M}_{n} \) and weight average molecular mass \( \bar{M}_{w} \). However, the provided equation \( \bar{M}_{w}=\frac{\sum N_{i} M_{i}}{\sum N_{i} M_{i}} \) is incorrect because it simplifies to 1. The correct formula for \( \bar{M}_{n} \) is \( \bar{M}_{n} = \frac{\sum N_{i} M_{i}}{\sum N_{i}} \), and for \( \bar{M}_{w} \), it is \( \bar{M}_{w} = \frac{\sum N_{i} M_{i}^2}{\sum N_{i} M_{i}} \). Therefore, Statement 2 is incorrect.

Step 4: Analyzing Statement 3

Statement 3 discusses physical properties affected by the number average molecular mass (\( M_n \)) and weight average molecular mass (\( M_w \)). It correctly states that properties like freezing point, vapor pressure, and osmotic pressure depend on \( M_n \), while properties like light scattering are related to \( M_w \). This statement aligns with scientific understanding, making it correct.

Step 5: Determining the Correct Answer

Statement 1 and Statement 3 are validated as correct based on polymer chemistry concepts. Therefore, the choice with both correct statements, which is (c) 1 and 3, is the correct answer.

Key Concepts

Poly Dispersion Index

In the world of polymer chemistry, the Poly Dispersion Index (P.D.I.) is a crucial concept. It provides insight into the distribution of molecular masses within a polymer sample. In simple terms, P.D.I. is calculated as the ratio of the weight average molecular mass, \( \bar{M}_w \), to the number average molecular mass, \( \bar{M}_n \). This ratio is expressed mathematically as:\[ \text{P.D.I.} = \frac{\bar{M}_w}{\bar{M}_n}\]

• If the P.D.I. is equal to 1, it indicates that all polymer chains have the same molecular mass, implying a monodisperse polymer.
• For most naturally occurring and synthetic polymers, the P.D.I. is greater than 1, showing that there is a distribution of different molecular masses within the sample.

The P.D.I. helps polymer scientists understand the dispersity or uniformity of polymer chains. A lower P.D.I. signifies more uniformity, which can affect how the polymer behaves in applications such as plastic manufacturing and biomedical uses.

Molecular Mass Averages

Polymers consist of long chains of repeating units of molecules, which can vary greatly in length and mass. To describe these mass distributions, two primary averages are used: the number average molecular mass (\(\bar{M}_n\)) and the weight average molecular mass (\(\bar{M}_w\)).

• Number Average Molecular Mass (\(\bar{M}_n\)): It is defined as:\[\bar{M}_n = \frac{\sum N_{i} M_{i}}{\sum N_{i}}\]Here, \(N_i\) is the number of polymer molecules with mass \(M_i\). This average is useful for deducing colligative properties, such as osmotic pressure, which depend on the number of molecules.
• Weight Average Molecular Mass (\(\bar{M}_w\)): It is defined as:\[\bar{M}_w = \frac{\sum N_{i} M_{i}^2}{\sum N_{i} M_{i}}\]This average places more emphasis on the mass of each molecule and is important for properties like light scattering and tensile strength.

These two metrics provide contrasting insights and are used to understand and predict how polymers will perform in different settings.

Physical Properties of Polymers

Understanding the relationship between molecular mass averages and physical properties of polymers helps in optimizing their use in various applications. The different physical properties of polymers are often associated with either \(M_n\) or \(M_w\).

• Properties Related to \(M_n\): The number average molecular mass can predict properties that are affected by the number of particles. This includes:
  • Freezing point
  • Vapor pressure
  • Osmotic pressure
  These colligative properties depend on how many polymer molecules are present in a given quantity of solvent.
• Properties Related to \(M_w\): The weight average molecular mass is more influential on properties that depend on the mass or extent of polymerization, such as:
  • Light scattering
  • Sedimentation
  • Diffusion
  These properties are particularly important in manufacturing and material science, where durability and performance are crucial.

Improving our knowledge of these properties allows researchers to tailor polymers for specific industrial and commercial uses.