Problem 102 - A Complete Resource Book in Chemistry for JEE Main

Question

Which has maximum number of molecules? (a) \(7 \mathrm{~g} \mathrm{~N}_{2}\) (b) \(2 \mathrm{~g} \mathrm{H}_{2}\) (c) \(18 \mathrm{~g} \mathrm{NO}_{2}\) (d) \(16 \mathrm{~g} \mathrm{O}_{2}\)

Step-by-Step Solution

Answer

2 g of \(\mathrm{H}_{2}\) contains the maximum number of molecules.

Step 1: Calculate Molar Mass

First, calculate the molar mass of each substance. - Nitrogen gas (\(\mathrm{N}_{2}\)) has a molar mass of \(28\, \mathrm{g/mol}\) (since 1 nitrogen atom has a molar mass of \(14\, \mathrm{g/mol}\)).- Hydrogen gas (\(\mathrm{H}_{2}\)) has a molar mass of \(2\, \mathrm{g/mol}\) (since 1 hydrogen atom has a molar mass of \(1\, \mathrm{g/mol}\)).- Nitrogen dioxide (\(\mathrm{NO}_{2}\)) has a molar mass of \(46\, \mathrm{g/mol}\) ((14 for N + 16×2 for O)).- Oxygen gas (\(\mathrm{O}_{2}\)) has a molar mass of \(32\, \mathrm{g/mol}\) (since 1 oxygen atom has a molar mass of \(16\, \mathrm{g/mol}\)).

Step 2: Calculate Moles of Each Substance

Use the formula \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\) to calculate moles:- For \(7\, \mathrm{g}\) of \(\mathrm{N}_{2}\), moles \(= \frac{7}{28} = 0.25\, \text{moles}\).- For \(2\, \mathrm{g}\) of \(\mathrm{H}_{2}\), moles \(= \frac{2}{2} = 1\, \text{moles}\).- For \(18\, \mathrm{g}\) of \(\mathrm{NO}_{2}\), moles \(= \frac{18}{46} \approx 0.391\, \text{moles}\).- For \(16\, \mathrm{g}\) of \(\mathrm{O}_{2}\), moles \(= \frac{16}{32} = 0.5\, \text{moles}\).

Step 3: Use Avogadro's Number to Find Molecules

Multiply the number of moles by Avogadro's number (\(6.022 \times 10^{23}\)) to find the number of molecules:- \(\mathrm{N}_{2}\): \(0.25 \times 6.022 \times 10^{23} = 1.5055 \times 10^{23}\, \text{molecules}\).- \(\mathrm{H}_{2}\): \(1 \times 6.022 \times 10^{23} = 6.022 \times 10^{23}\, \text{molecules}\).- \(\mathrm{NO}_{2}\): \(0.391 \times 6.022 \times 10^{23} \approx 2.3546 \times 10^{23}\, \text{molecules}\).- \(\mathrm{O}_{2}\): \(0.5 \times 6.022 \times 10^{23} = 3.011 \times 10^{23}\, \text{molecules}\).

Step 4: Compare the Number of Molecules

Compare the number of molecules in each substance to find the maximum:- For \(\mathrm{N}_{2}\): \(1.5055 \times 10^{23}\, \text{molecules}\).- For \(\mathrm{H}_{2}\): \(6.022 \times 10^{23}\, \text{molecules}\), which is the highest.- For \(\mathrm{NO}_{2}\): \(2.3546 \times 10^{23}\, \text{molecules}\).- For \(\mathrm{O}_{2}\): \(3.011 \times 10^{23}\, \text{molecules}\).

Key Concepts

Molar Mass Calculation

Calculating the molar mass is a vital first step in understanding chemical quantities. The molar mass is the mass of one mole of a given substance, expressed in grams per mole (g/mol). This is calculated by summing the atomic masses of all the atoms that make up a molecule. For instance, in the molecule of nitrogen gas \( \mathrm{N}_2 \), each nitrogen atom has an atomic mass of 14 g/mol. Since there are two nitrogen atoms, the molar mass of \( \mathrm{N}_2 \) is \( 28 \, \mathrm{g/mol} \).

Here’s how to calculate the molar mass for other molecules:

• \( \mathrm{H}_2 \): Two hydrogen atoms, each with a mass of 1 g/mol, results in a molar mass of 2 g/mol.
• \( \mathrm{NO}_2 \): This consists of one nitrogen (14 g/mol) and two oxygen atoms (each 16 g/mol), totaling to 46 g/mol.
• \( \mathrm{O}_2 \): Two oxygen atoms with a mass of 16 g/mol each, leading to a combined mass of 32 g/mol.

Molar masses help us convert between the mass of a substance and the number of moles, paving the way for deeper calculations such as determining the number of molecules.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry symbolized as \( 6.022 \times 10^{23} \), representing the number of atoms, ions, or molecules in one mole of a substance. Named after the scientist Amedeo Avogadro, this number allows chemists to easily calculate the quantity of fundamental units present in a given amount of material.

For example, when you calculate the number of molecules in certain masses of different gases, you multiply the number of moles by Avogadro's number. Let's say we have 1 mole of hydrogen \( \mathrm{H}_2 \); using Avogadro's number, you would have \( 6.022 \times 10^{23} \) molecules of hydrogen. This constant provides a bridge between the mass of substances and the microscopic particles they comprise, thus establishing a link between the macroscopic and the atomic scale in chemistry.

Calculating Moles

The mole is a unit of measurement that plays a critical role in chemistry, used for expressing amounts of a chemical substance. To find the number of moles in a sample given its mass, you use the formula:\[\text{moles} = \frac{\text{mass}}{\text{molar mass}}\]This conversion is essential because it enables chemists to relate the mass of a substance to the number of atoms or molecules in it. For example, if you have 7 grams of nitrogen gas \( \mathrm{N}_2 \), you first calculate the moles by dividing the mass (7 g) by the molar mass (28 g/mol), which equals 0.25 moles.

Let’s look at another instance for hydrogen gas. With a molar mass of 2 g/mol, a 2-gram sample of \( \mathrm{H}_2 \) is exactly 1 mole. Knowing how to find the moles allows you to apply Avogadro's number in determining the number of molecules, further bridging the gap between mass and the number of particles within a substance.