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

Question

If \(\mathrm{N}_{1}, \mathrm{~N}_{2}, \mathrm{~N}_{3} \ldots \mathrm{N}_{i}\) are the number of molecules with molecular masses \(\mathrm{M}_{1}, \mathrm{M}_{2}, \mathrm{M}_{3} \ldots \mathrm{M}_{1}\) 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) \(\frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}{\sum \mathrm{N}_{\mathrm{i}}}\) (c) \(\frac{\sum \mathrm{M}_{i}^{2}}{\sum \mathrm{N}_{\mathrm{i}}}\) (d) \(\frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}{\sum \mathrm{M}_{\mathrm{i}}}\)

Step-by-Step Solution

Answer

The correct answer is (b) \( \frac{\sum N_i M_i}{\sum N_i} \).

Step 1: Understanding Number Average Molecular Mass

The number average molecular mass \( \bar{M}_n \) is calculated by considering the total mass of all molecules divided by the total number of molecules. This is a way to average the molecular mass of molecules in a mixture.

Step 2: Formula for Number Average Molecular Mass

The formula to calculate the number average molecular mass \( \bar{M}_n \) is given by:\[\bar{M}_n = \frac{\sum N_i M_i}{\sum N_i}\]where \( N_i \) is the number of molecules with molecular mass \( M_i \).

Step 3: Identify the Correct Option

Compare the given options with the derived formula. The correct formula for \( \bar{M}_n \) is option (b):\[ \bar{M}_n = \frac{\sum N_i M_i}{\sum N_i}\]The other options do not match this formula.

Key Concepts

Molecular Mass

Molecular mass is a fundamental concept in chemistry and physics. It refers to the mass of a single molecule of a substance. This mass is derived from the sum of the masses of all the atoms present in the molecule. Each element's atoms have a specific atomic mass, which can be found on the periodic table. By adding the atomic masses of each atom type in a molecule, one obtains the molecular mass.
For example, water (H_2O) consists of two hydrogen atoms and one oxygen atom. Hydrogen has an atomic mass of approximately 1 atomic mass unit (amu), and oxygen has an atomic mass of about 16 amu. Thus, the molecular mass of water is calculated as:

• 2 Hydrogen atoms x 1 amu = 2 amu
• 1 Oxygen atom x 16 amu = 16 amu
• Total molecular mass of H2O = 18 amu

This concept is crucial for various scientific calculations, such as determining the amount of a substance required for a chemical reaction or understanding the behavior of gases under different conditions.

Average Molecular Mass

The average molecular mass is a way to understand the composition of molecular mixtures in a substance. While the molecular mass refers to a single molecule, average molecular mass considers the distribution of molecules with different masses in the sample.
This is particularly important for substances like polymers, where molecules can vary greatly in size and mass. For instance, if you have a mixture of polymers, each with different chain lengths, calculating the average can help determine the general characteristics of that mixture.
In our exercise, the number average molecular mass is specifically considered. It is calculated using the formula:

• \( \bar{M}_n = \frac{\sum N_i M_i}{\sum N_i} \)
• Where \( N_i \) is the number of molecules with molecular mass \( M_i \).

This formula sums up the product of molecule numbers and their respective masses, divided by the total number of molecules. This approach is essential in fields such as material science, where it helps to quantify and predict the properties of materials.

Molecular Weight Distribution

Molecular weight distribution refers to the variety in size or mass of molecules within a sample, rather than a single uniform mass. It is an important characteristic in determining the performance and processing behavior of substances like polymers.
Different molecules in a polymer sample can have a wide range of molecular masses. Understanding the distribution gives insights into the sample's physical properties, such as strength, elasticity, and viscosity.
One can visualize molecular weight distribution using different statistical measures:

• Number average molecular mass (\( \bar{M}_n \)) - calculated as described in previous sections.
• Weight average molecular mass - considers the weight fraction of each molecule in the sample.

Each measure can provide different information. While the number average considers each molecule equally regardless of size, the weight average considers larger molecules more significant. Understanding the nuances of molecular weight distribution is crucial for industries such as plastics and biopolymers, where specific properties are desired in end products.