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, \(\bar{M}_{\alpha} / \overline{M n}\), 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 \(\bar{M}_{w}=\frac{\sum N_{i} M_{i}}{\sum N_{i} M_{i}}\) and \(\bar{M}_{w}=\frac{\sum N_{i} M_{i}^{2}}{\sum N_{i} M_{i}}\) 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 \(\mathrm{M}_{\mathrm{n}}\) whereas some other physical properties like light scattering, sedimentation and diffusion constant are related directly to \(\mathrm{M}_{\mathrm{w}}\). (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.

Step 1: Understanding Polydispersity Index (P.D.I.)

The polydispersity index (P.D.I.) is defined as the ratio of the weight average molecular mass \( \bar{M}_{w} \) to the number average molecular mass \( \overline{M_n} \). Thus, statement (1) is correct in mentioning that \( \bar{M}_{w} / \overline{M_n} \) indicates the polydispersity of a polymer.

Step 2: Analyzing the Given Equations

Statement (2) provides two equations. The first part of statement (2) mentioning \( \bar{M}_{w} = \frac{\sum N_{i} M_{i}}{\sum N_{i} M_{i}} \) is incorrect because simplification results in a constant value of 1, not the weight average molecular mass. The correct formula is \( \bar{M}_{w} = \frac{\sum N_{i} M_{i}^2}{\sum N_{i} M_{i}} \), which is correctly provided in the second part. Therefore, statement (2) is incorrect.

Step 3: Understanding Related Physical Properties

Statement (3) points out that certain physical properties such as freezing point, vapor pressure, and osmotic pressure depend on the number average molecular mass \( \overline{M_n} \), while properties like light scattering and others depend on the weight average molecular mass \( \bar{M}_{w} \). This statement is correct because showing dependence on specific molecular averages reflects how different phenomena react based on molecular distribution.

Step 4: Determine the Correct Statements

To finalize the correct set of statements, observe that statement (1) and statement (3) are correct. Statement (2) contains inaccuracies within the provided equations. Therefore, based on the analysis, the correct combination of true statements is 1 and 3.

Key Concepts

Polydispersity Index

Polydispersity Index (PDI) is a crucial parameter in polymer chemistry. It helps describe the distribution of molecular mass in a polymer sample. Specifically, PDI is defined as the ratio of the weight average molecular mass (\( \bar{M}_{w} \)) to the number average molecular mass (\( \overline{M}_n \)). This ratio provides insight into the uniformity of the polymer chains.
A PDI value of 1 indicates a perfectly uniform polymer, where all chains are of equal length. Values greater than 1 imply that the polymer sample contains chains of varying lengths. The higher the PDI, the broader the distribution of molecule sizes within the sample. This parameter is essential for understanding the properties and performance of polymer materials in various applications.

Weight Average Molecular Mass

Weight Average Molecular Mass (\( \bar{M}_{w} \)) is one of the key molecular mass descriptors of polymers. It accounts for the mass of the polymer chains, weighting longer chains more heavily because they impact the physical properties of the polymer.
The correct formula for calculating \( \bar{M}_{w} \) is:
\[ \bar{M}_{w} = \frac{\sum N_{i} M_{i}^2}{\sum N_{i} M_{i}} \]
where \( N_{i} \) is the number of molecules of a particular molecular mass \( M_{i} \).

• This formula emphasizes larger molecular masses more significantly, reflecting their larger contribution to the polymer's properties like tensile strength, viscosity, and melting point.

Having an accurate measurement of \( \bar{M}_{w} \) is essential for applications where mechanical strength and heat resistance are crucial.

Number Average Molecular Mass

Number Average Molecular Mass (\( \overline{M}_n \)) focuses more on the count of polymer molecules in a sample rather than the mass. It is calculated using the formula:
\[ \overline{M}_n = \frac{\sum N_{i} M_{i}}{\sum N_{i}} \]
This calculation provides an average molecular mass that gives equal weight to all chains, regardless of their size.

• This makes \( \overline{M}_n \) particularly important in applications dealing with colligative properties, such as freezing point depression and boiling point elevation, where the number of molecules matters more than their size.

By focusing on the molecular count, \( \overline{M}_n \) provides valuable insight into applications that require precise control over quantitative molecular properties.

Physical Properties of Polymers

The physical properties of polymers are fundamentally linked to different molecular mass averages. Some properties are more influenced by the number average molecular mass (\( \overline{M}_n \)), while others depend on the weight average molecular mass (\( \bar{M}_{w} \)).

• Properties such as freezing point, vapor pressure, and osmotic pressure primarily depend on \( \overline{M}_n \).
• These properties are known as colligative properties and are influenced by the number of polymer molecules, not their size.

Conversely, phenomena such as light scattering, sedimentation, and diffusion constants are more responsive to \( \bar{M}_{w} \), as they relate to the size and shape of polymer coils in solution.
Understanding these dependencies helps chemists and material scientists tailor polymers for specific functions, ensuring that they meet the desired performance criteria for a wide range of practical applications.