Problem 119 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(3.02 \times 10^{19}\) molecules are removed from \(98 \mathrm{mg}\) of \(\mathrm{H}_{2} \mathrm{SO}_{4}\), then the number of moles of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) left are (a) \(0.1 \times 10^{-3}\) (b) \(5 \times 10^{-4}\) (c) \(1.2 \times 10^{-4}\) (d) \(1.5 \times 10^{-3}\)

Step-by-Step Solution

Answer

The closest answer by calculation is (d) \(1.5 \times 10^{-3}\), though there may be an answer key error.

Step 1: Calculate Moles in Original Sample

First, we find out the number of moles in the original 98 mg of \( \mathrm{H}_{2} \mathrm{SO}_{4} \). The molar mass of \( \mathrm{H}_{2} \mathrm{SO}_{4} \) is approximately \( 98 \) g/mol. Since we have 98 mg, which is \( 0.098 \) g, we can calculate the initial moles as follows:\[\text{Initial moles} = \frac{0.098}{98} = 0.001 \text{ moles}\]

Step 2: Calculate Moles Removed

Now, calculate the moles of \( \mathrm{H}_{2} \mathrm{SO}_{4} \) removed. Avogadro's number \( (6.022 \times 10^{23}) \) is the number of molecules per mole. Thus, the moles removed is:\[\text{Moles removed} = \frac{3.02 \times 10^{19}}{6.022 \times 10^{23}} \approx 5.02 \times 10^{-5} \text{ moles}\]

Step 3: Calculate Remaining Moles

To find the remaining moles of \( \mathrm{H}_{2} \mathrm{SO}_{4} \), subtract the moles removed from the initial moles:\[\text{Remaining moles} = 0.001 - 5.02 \times 10^{-5} \approx 0.0009498 \text{ moles}\]Rounding this result to two significant figures based on the given options gives approximately \( 0.00095 \) moles or \( 9.5 \times 10^{-4} \) moles.

Step 4: Compare to Answer Choices

Finally, convert your calculated result \( 9.5 \times 10^{-4} \) moles to match one of the provided answer choices. None of the options directly match this result, but \( 9.5 \times 10^{-4} \) is closest to option (d), \( 1.5 \times 10^{-3} \), indicating a possible variation or error in answer key matching.

Key Concepts

Molar Mass Calculation

Molar mass is a fundamental concept in stoichiometry that allows us to relate the mass of a substance to the number of moles. Calculating the molar mass involves summing up the masses of all the atoms in a molecule. For example, sulfuric acid, \(\mathrm{H}_2\mathrm{SO}_4\), consists of 2 hydrogen atoms, 1 sulfur atom, and 4 oxygen atoms. The atomic masses are approximately 1 g/mol for hydrogen, 32 g/mol for sulfur, and 16 g/mol for oxygen. To find the molar mass, add these up:

• Hydrogen: 2 atoms \( \times 1 \text{ g/mol} = 2 \text{ g/mol} \)
• Sulfur: 32 g/mol
• Oxygen: 4 atoms \( \times 16 \text{ g/mol} = 64 \text{ g/mol} \)

Thus, the molar mass of \(\mathrm{H}_2\mathrm{SO}_4\) is \(2 + 32 + 64 = 98 \text{ g/mol}\). Understanding this calculation is crucial when converting between mass and moles, as we can use this value as a conversion factor. For example, if we have 98 mg (which is 0.098 g), to find the number of moles, we divide mass by molar mass:\[\text{moles} = \frac{0.098 \text{ g}}{98 \text{ g/mol}} = 0.001 \text{ moles}\]

Avogadro's Number

Avogadro's number is critical in understanding the mole concept because it provides a bridge between the macroscopic and microscopic worlds. Specifically, Avogadro's number \( (6.022 \times 10^{23}) \) tells us the number of particles, such as atoms or molecules, in one mole of a substance. This allows us to convert between the number of molecules and the amount of substance in moles. For instance, in this particular problem, 3.02 \(\times\) 10^{19} molecules of \(\mathrm{H}_2\mathrm{SO}_4\) are considered. To find the corresponding number of moles, we use Avogadro's number: \[\text{Moles} = \frac{3.02 \times 10^{19}}{6.022 \times 10^{23}} \approx 5.02 \times 10^{-5} \text{ moles}\]Avogadro's number aids not only in such calculations but also helps us comprehend vast numbers by providing a scale with practical size. This number essentially allows chemists and students to count very tiny entities by grouping them into manageable quantities.

Mole Concept

The mole is a basic unit in chemistry, used to express amounts of a chemical substance. The mole concept is essential for stoichiometry, as it makes quantitative chemistry equations conducive to practical calculations. One mole of any substance contains Avogadro's number of particles, be they atoms, molecules, or ions.Imagine you have one dozen eggs. Just as dozen helps count mundane catalogues efficiently, the mole allows chemists to precisely address the minuscule tally of molecules or atoms. For instance, when given 98 mg of \(\mathrm{H}_2\mathrm{SO}_4}\), we convert this into moles to determine the substance's measurable quantities:\[\text{Moles of } \mathrm{H}_2\mathrm{SO}_4} = 0.001 \text{ moles}\]This mole concept helps in breaking down all chemical reactions and calculations into molar terms, enabling chemists to predict the results of reactions accurately. Whether determining reactants needed or predicting products formed, moles form the core unit in these calculations, ensuring accurate and feasible chemical experimentation and study.