Problem 6 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The ratio of masses of oxygen and nitrogen in a particular gaseous mixture is \(1: 4\). The ratio of number of their molecules is: (a) \(1: 4\) (b) \(7: 32\) (c) \(1: 8\) (d) \(3: 16\)

Step-by-Step Solution

Answer

The ratio of the number of molecules of oxygen to nitrogen is 7:32 (Option b).

Step 1: Understand the problem

We need to find the ratio of the number of molecules of oxygen to nitrogen, given the mass ratio of 1:4. This involves using molecular weights to calculate the number of moles and then molecules.

Step 2: Determine Molar Masses

The molar mass of oxygen (O_2) is 32 g/mol and nitrogen (N_2) is 28 g/mol. These values will help us convert mass into moles.

Step 3: Calculate Mole ratio from Mass ratio

Given the mass ratio of oxygen to nitrogen is 1:4, convert these into moles: moles of O_2 = 1/32 and moles of N_2 = 4/28.

Step 4: Simplify Moles to Molecule Ratio

The number of molecules is proportional to the number of moles. Calculate the ratio: \(\text{moles of O}_2 : \text{moles of N}_2 = \frac{1/32}{4/28} = \frac{28}{128} = \frac{7}{32}\).

Step 5: Conclusion

Convert the simplified mole ratio into the answer choice format. The ratio of the number of molecules of O_2 to N_2 is 7:32, correlating to answer choice (b).

Key Concepts

Mole Concept

The mole is a fundamental concept in chemistry used to quantify the amount of substance. One mole corresponds to Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles, such as atoms, molecules, or ions. This concept aids in relating microscopic properties of molecules to macroscopic measurements we use in the laboratory. When dealing with gaseous mixtures, knowing the mole concept helps us understand how different gases will behave when they are mixed. The number of molecules in a substance is directly proportional to the number of moles of that substance. So, to find the number of molecules given a certain mass of gas, we first convert that mass into moles using the substance's molar mass. This allows chemists to use a standard unit of measurement when doing calculations, like predicting reaction yields or balancing chemical equations.

Molar Mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole. It is a critical conversion factor when transforming mass amounts to mole quantities. For example, the molar mass of oxygen (O\(_2\)) is 32 g/mol, and nitrogen (N\(_2\)) is 28 g/mol. These values mean that one mole of O\(_2\) has a mass of 32 grams, while one mole of N\(_2\) has a mass of 28 grams.In the context of a gaseous mixture, molar mass helps in determining how much of each gas is present, both in terms of moles and in terms of the molecular count. By converting the mass ratio (like 1:4 for oxygen and nitrogen in the exercise) to a mole ratio using their molar masses, this helps to elucidate the actual number of molecules present in a sample.

Molecule Ratio

Understanding the ratio of molecules in a mixture involves two main steps: converting known mass ratios into mole ratios, and, subsequently, into molecular ratios. Since the number of molecules is directly proportional to the number of moles, the initial mass ratio gives us a start point.Given a mass ratio of oxygen to nitrogen as 1:4, we use the molar masses to find how many moles each component represents:

• For oxygen (O\(_2\)), with a mass of 1 part and molar mass of 32 g/mol, the moles = \( 1/32 \).
• For nitrogen (N\(_2\)), with a mass of 4 parts and molar mass of 28 g/mol, the moles = \( 4/28 \).

Simplifying the ratio of these moles gives the ratio of molecules. From this calculation, we find the ratio \( 7:32 \), which represents the count of molecules, matching the solution choice b in the exercise. This ratio is key for anyone needing to understand the relative abundance of different molecules in a given sample.