Problem 38 - A Complete Resource Book in Chemistry for JEE Main

Question

One mole of \(\mathrm{CH}_{4}\) contains (a) \(4.0 \mathrm{~g}\) atoms of hydrogen (b) \(3.0 \mathrm{~g}\) atom of carbon (c) \(6.02 \times 10^{23}\) atoms of hydrogen (d) \(1.81 \times 10^{23}\) molecules of \(\mathrm{CH}_{4}\)

Step-by-Step Solution

Answer

Option (a) is correct: "One mole of \(\mathrm{CH}_4\) contains 4.0 g atoms of hydrogen."

Step 1: Understanding the Mole Concept

The concept of a mole is crucial to solving this problem. One mole of any substance contains Avogadro's number of entities, which is approximately \(6.02 \times 10^{23}\). This might be atoms, molecules, ions, etc. Therefore, one mole of \(\mathrm{CH}_4\) contains \(6.02 \times 10^{23}\) molecules of \(\mathrm{CH}_4\).

Step 2: Molecular Composition of Methane

\(\mathrm{CH}_4\) is methane, and each molecule comprises 1 carbon atom and 4 hydrogen atoms. Therefore, in one mole of \(\mathrm{CH}_4\), you have 1 mole of carbon atoms and 4 moles of hydrogen atoms.

Step 3: Calculating Number of Hydrogen Atoms per Mole

Since 1 mole of \(\mathrm{CH}_4\) contains 4 moles of hydrogen atoms and each mole contains \(6.02 \times 10^{23}\) atoms, simply multiply these numbers to find the total hydrogen atoms. \(4 \times 6.02 \times 10^{23} = 2.408 \times 10^{24}\) hydrogen atoms, confirming that choice (c) is incorrect.

Step 4: Molar Masses of Elements

Calculate the molar masses required to determine the number of grams per atomic weight. The molar mass of hydrogen is approximately \(1 \text{ g/mol}\) and for carbon is \(12 \text{ g/mol}\). Thus, one mole of hydrogen atoms weighs 1g, and one mole of carbon atoms weighs 12g.

Step 5: Evaluating Each Option

Analyze each statement:(a) One mole of \(\mathrm{CH}_4\) contains 4 moles of hydrogen atoms, weighing \(4 \times 1\, \text{g} = 4\, \text{g}\), making this statement true. (b) There are 12g per mole of carbon, but since we're considering 1 mole of carbon atom within \(\mathrm{CH}_4\), \(3\, \text{g}\) is incorrect. (c) As calculated, \(2.408 \times 10^{24}\) atoms of hydrogen, so \(6.02 \times 10^{23}\) is incorrect. (d) One mole of any compound includes \(6.02 \times 10^{23}\) molecules, not \(1.81 \times 10^{23}\).

Step 6: Final Conclusion

Verification shows option (a) is correct. It states that one mole of \(\mathrm{CH}_4\) truly contains \(4.0\, \text{g}\) atoms of hydrogen.

Key Concepts

Avogadro's Number

Avogadro's Number is a fundamental concept in chemistry that connects the macroscopic scale of substances to the microscopic scale of atoms and molecules. When we talk about a "mole" of a substance, it refers to a specific quantity - precisely, Avogadro's Number of entities. This number is approximately \(6.02 \times 10^{23}\). Just as a "dozen" means 12, a "mole" means \(6.02 \times 10^{23}\).This concept is incredibly useful in chemistry because it allows us to count actual atoms or molecules by weighing them out in the laboratory. For example, one mole of methane \(\mathrm{CH}_4\) contains \(6.02 \times 10^{23}\) molecules of methane.To grasp its importance:

• It provides a bridge between the atomic world and the world we can see and measure.


• Makes it possible to use the mass of a substance to determine how many molecules or atoms are present.

Molecular Composition

The molecular composition of a substance tells us what types of atoms and how many of each are in a single molecule. For methane, \(\mathrm{CH}_4\), this involves a straightforward breakdown:- It consists of one carbon (C) atom.- Surrounding the carbon are four hydrogen (H) atoms.This simple molecular composition indicates that methane has a tetrahedral shape, which is key to understanding its properties and reactions.In terms of moles:

• One mole of \(\mathrm{CH}_4\) will contain 1 mole of carbon atoms.


• Also contains 4 moles of hydrogen atoms.

Understanding molecular composition is essential in determining how much of each element is present in substances. It is also foundational in figuring out molecular interactions and stoichiometry in reactions.

Molar Mass Calculation

Calculating the molar mass of a substance is a crucial step in stoichiometry, helping us convert moles into grams and vice versa. The molar mass is the mass of one mole of a given substance and can be obtained from the periodic table.The calculation involves:

• Adding up the atomic masses of each atom in the molecule.


• For methane \(\mathrm{CH}_4\), carbon has an atomic mass of approximately 12 g/mol, and hydrogen's is 1 g/mol.

Thus, the molar mass of methane \(\mathrm{CH}_4\) is calculated as:\[molar \ mass \ = (1 \times 12) + (4 \times 1) = 16 \ \text{g/mol}\]This means that one mole of \(\mathrm{CH}_4\) weighs 16 grams. Knowing this helps chemists measure out the right amount of substance needed for experiments and to predict product yields accurately.