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 \(M_{1}, M_{2}, M_{3} \ldots M_{i}\) 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) \(\frac{\sum \mathrm{N}_{\mathrm{i}} \mathrm{M}_{\mathrm{i}}}{\sum \mathrm{N}_{\mathrm{i}}}\) (c) \(\frac{\sum \mathrm{M}_{\mathrm{i}}^{2}}{\sum \mathrm{N}_{\mathrm{i}}}\) (d) \(\frac{\sum \mathrm{N}_{i} \mathrm{M}_{\mathrm{i}}}{\sum \mathrm{M}_{\mathrm{i}}}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The question asks for the formula of the number average molecular mass, denoted as \( \bar{M}_w \). This is calculated from the given masses and number of molecules.

Step 2: Define Number Average Molecular Mass

The number average molecular mass \( \bar{M}_n \) is defined as the total mass of all groups of molecules divided by the total number of molecules. It is given by the formula \( \bar{M}_n = \frac{\sum N_i M_i}{\sum N_i} \).

Step 3: Identify the Correct Formula

Match the formula for the number average molecular mass \( \bar{M}_n = \frac{\sum N_i M_i}{\sum N_i} \) with the given options to identify the correct one.

Step 4: Select the Correct Option

Option (b) \( \frac{\sum N_i M_i}{\sum N_i} \) aligns with the definition of the number average molecular mass. Therefore, option (b) is correct.

Key Concepts

Molecular Masses

The molecular mass of a substance is essentially the sum of the masses of all the atoms in a molecule. Knowing the molecular mass is vital for understanding many chemical reactions, as it allows us to calculate the amounts of substances required or produced in a reaction. Every element has a characteristic atomic mass, and the molecular mass is the total of these atomic masses for all the atoms present. For example, water (H₂O) has a molecular mass calculated by adding the atomic masses of two hydrogen atoms and one oxygen atom.

• Hydrogen's atomic mass is approximately 1 u (atomic mass unit).
• Oxygen's atomic mass is about 16 u.
• The molecular mass of water is thus 2(1) + 16 = 18 u.

Understanding molecular masses helps in converting between moles of substances and grams, further allowing precise formulation and analysis in experiments.

Number of Molecules

When we talk about the "number of molecules," we are referring to how many individual molecules are present in a sample or reaction. This is typically expressed in terms of moles. A mole is a unit that represents a whopping 6.022 x 10²³ molecules (Avogadro's number). This large number is used because molecules and atoms are extremely small, making individual counting impractical. For instance, if we say we have 1 mole of water, it means there are 6.022 x 10²³ molecules of H₂O present.

• The number of molecules is directly proportional to the number of moles.
• It helps in calculating the stoichiometry of reactions, an essential concept in chemistry.
Calculating the number of molecules involved in a chemical process allows chemists to predict how much product will result or how much of a reactant they need.

Average Molecular Mass Formula

The average molecular mass helps in understanding the overall properties of a sample composed of different molecules with varying masses. It is especially useful in polymer chemistry where samples consist of molecules of various sizes, each present in different quantities.The formula used to calculate the number average molecular mass, denoted \( \bar{M}_n \), takes into account both the quantity and masses of the molecules:\[\bar{M}_n = \frac{\sum N_i M_i}{\sum N_i}\]Where:

• \(N_i\) is the number of molecules with mass \(M_i\).
• The numerator \(\sum N_i M_i\) represents the total mass of all molecules.
• The denominator \(\sum N_i\) represents the total number of molecules.

This formula thus averages the masses by giving weight to the number of molecules, ensuring that the contribution of each type of molecule is proportionate to its abundance.