Problem 45 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(\mathrm{N}_{1}, \mathrm{~N}_{2}, \mathrm{~N}_{3} \ldots \mathrm{N}_{\mathrm{i}}\) are the number of molecules with molecular masses \(\mathrm{M}_{1}, \mathrm{M}_{2}, \mathrm{M}_{3} \ldots \mathrm{M}_{\mathrm{i}}\) respectively, then the number average molecular mass \(\left(\bar{M}_{w}\right.\) ) is given by (a) \(\frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}^{2}}{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}\) (b) \(\frac{\sum N_{i} M_{i}}{\sum N_{i}}\) (c) \(\sum_{\sum_{i}^{2}}\) (d) \(\frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}{\sum \mathrm{M}_{\mathrm{i}}}\)

Step-by-Step Solution

Answer

Option (b): \( \frac{\sum N_{i} M_{i}}{\sum N_{i}} \).

Step 1: Understanding the Number Average Molecular Mass

Number average molecular mass, often denoted as \( \bar{M}_n \), is used to calculate the average molecular mass of a polymer system, considering the number of molecules of each type in the mixture.

Step 2: Formula for Number Average Molecular Mass

The formula for number average molecular mass is \( \bar{M}_n = \frac{\sum N_i M_i}{\sum N_i} \), where \( N_i \) is the number of molecules with molecular mass \( M_i \).

Step 3: Identifying the Correct Option

Comparing the given options to the formula for number average molecular mass:- Option (a): \( \frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}^{2}}{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}} \) is not correct as it uses \( M_i^2 \).- Option (b): \( \frac{\sum N_{i} M_{i}}{\sum N_{i}} \) matches the formula for the number average molecular mass.- Option (c): \( \sum_{\sum_{i}^{2}} \) does not make sense as an equation.- Option (d): \( \frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}{\sum \mathrm{M}_{\mathrm{i}}} \) is incorrect as it divides by \( \sum M_i \) instead of \( \sum N_i \).

Step 4: Conclusion

Therefore, the correct answer based on the formula for the number average molecular mass is option (b): \( \frac{\sum N_{i} M_{i}}{\sum N_{i}} \).

Key Concepts

Molecular Masses

Molecular mass is a key concept in chemistry that refers to the mass of a single molecule of a substance. It is calculated by summing the atomic masses of all atoms included in the molecular formula.
Understanding molecular masses is crucial for applications that range from designing drugs and chemicals to synthesizing polymers and other materials.

• Examples: Water (H₂O) has a molecular mass calculated from the mass of two hydrogen atoms and one oxygen atom.
• Importance: Knowing molecular masses allows chemists to predict the behavior of molecules in reactions and mixtures.

It's integral in calculations for experiments and industrial applications, ensuring the correct proportions of substances are used. Molecular mass is generally expressed in unified atomic mass units (u). This fundamental concept provides a base for more complex topics like polymer chemistry.

Polymer Chemistry

Polymer chemistry focuses on the chemical synthesis and properties of polymers, which are large molecules composed of repeating structural units called monomers. This field is critical in developing materials that have a wide range of uses, from plastics to textiles.

• Applications: Polymers are essential in everyday products like packaging materials, rubber, and clothing.
• Structure: The repeating units in polymers affect their mechanical properties, such as strength, elasticity, and durability.

In polymer chemistry, understanding how the number average molecular mass influences the properties of the final polymer is vital. This average gives insight into the distribution of molecular weights in a polymer sample, which determines the polymer's characteristics, such as melting temperature and viscosity.
By selecting and manipulating monomer units, chemists can design polymers tailored for specific applications, revolutionizing industries like automotive and biotechnology.

Average Molecular Mass Calculation

Calculating the average molecular mass, especially in polymer systems, involves determining the mean mass of molecules present. The number average molecular mass, often symbolized as \( \bar{M}_n \), is calculated to obtain meaningful interpretations of a polymer's characteristics.
This formula is represented as:

• \( \bar{M}_n = \frac{\sum N_i M_i}{\sum N_i} \)

Where:

• \( N_i \) is the number of molecules of a particular mass
  
• \( M_i \) is the mass of those molecules.
  

This formula ensures we correctly average the molecular masses by accounting for the number of molecules present at each mass. By utilizing this average, we get an insight into the distribution and diversity of polymer chains in the sample.
This type of calculation is essential because it helps chemists and material scientists understand how polymers will perform, ensuring the proper selection of materials for specific applications. The correct understanding of molecular mass distributions can lead to optimized product performance and efficiency.