Problem 97 - A Complete Resource Book in Chemistry for JEE Main

Question

\(10^{21}\) molecules are removed from \(200 \mathrm{mg}\) of \(\mathrm{CO}_{2}\). The moles of \(\mathrm{CO}_{2}\) left are (a) \(2.88 \times 10^{-3}\) (b) \(28.8 \times 10^{-3}\) (c) \(288 \times 10^{-3}\) (d) \(28.8 \times 10^{3}\)

Step-by-Step Solution

Answer

The moles of CO2 left are approximately \(2.88 \times 10^{-3}\), corresponding to option (a).

Step 1: Calculate the Molecules in 200 mg of CO2

First, let's calculate the number of molecules in 200 mg of CO2. We know that the molar mass of CO2 is 44 g/mol. To find the number of moles in 200 mg, convert the mass to grams: \( 200 \text{ mg} = 0.2 \text{ g} \). Now, calculate the moles of CO2: \[ \text{Moles of } \mathrm{CO}_2 = \frac{0.2}{44} \approx 4.55 \times 10^{-3} \text{ mol} \]Using Avogadro's number \(6.022 \times 10^{23}\), calculate the number of molecules: \[ \text{Molecules of } \mathrm{CO}_2 = 4.55 \times 10^{-3} \times 6.022 \times 10^{23} \approx 2.74 \times 10^{21} \text{ molecules} \]

Step 2: Subtract the Removed Molecules

Remove \(10^{21}\) molecules from the total number of molecules calculated in Step 1. Perform the subtraction: \[ 2.74 \times 10^{21} - 10^{21} = 1.74 \times 10^{21} \text{ molecules remaining} \]

Step 3: Calculate Moles Left from Remaining Molecules

Now, let's calculate the moles from the remaining molecules by using Avogadro's number:\[ \text{Moles left} = \frac{1.74 \times 10^{21}}{6.022 \times 10^{23}} \approx 2.89 \times 10^{-3} \text{ moles} \]

Step 4: Select the Correct Option

Compare the moles calculated to the provided options:The moles calculated, \(2.89 \times 10^{-3}\) moles, is closest to option (a) \(2.88 \times 10^{-3}\). Thus, the correct answer is option (a).

Key Concepts

Avogadro's number

In chemistry, Avogadro's number is essential for converting between the number of atoms or molecules and the amount in moles. This number, approximately \(6.022 \times 10^{23}\), represents the number of constituent particles (usually atoms or molecules) in one mole of a substance. Think of it as a bridge between the atomic scale and the macroscopic world.

When solving problems involving molecules or atoms, Avogadro's number allows us to translate the microscopic scale of individual particles into the molar mass we use in everyday laboratory settings. It's an indispensable tool, especially when dealing with large counts of particles, which are too cumbersome to count one by one.

Molar mass calculation

Molar mass is the mass, in grams, of one mole of a substance and is usually expressed in units of grams per mole (g/mol). For any compound, the molar mass can be calculated by summing the atomic masses of all the atoms in the molecule.

Let's consider carbon dioxide (\(\mathrm{CO}_2\)). The molar mass of carbon (C) is about 12 g/mol and the molar mass of oxygen (O) is approximately 16 g/mol. So, the molar mass of \(\mathrm{CO}_2\) is calculated as:

• Carbon: \(1 \times 12 = 12 \text{ g/mol}\)
• Oxygen: \(2 \times 16 = 32 \text{ g/mol}\)
• Total: \(12 + 32 = 44 \text{ g/mol}\)

This calculation allows us to easily convert between mass and moles, a critical step in stoichiometric calculations.

Subtraction in chemistry

Subtraction in chemistry can be as important as addition, particularly when tracking changes in a chemical system. It involves determining how the number of particles (like atoms or molecules) changes after an action, such as a reaction or removal.

In the given problem, subtraction is used to find the number of molecules remaining after removing some from a given quantity. Starting with a total calculated from Avogadro's number, subtracting the specified number of molecules, \(10^{21}\), from the total gives the remaining entities.

This operation is crucial for obtaining accurate results in experiments and computations. It allows chemists to adjust quantities and assess how a system behaves under new conditions. Understanding subtraction helps in creating balanced equations and precise material calculations.