Problem 59 - A Complete Resource Book in Chemistry for JEE Main

Question

Ratio of \(\mathrm{C}_{\mathrm{p}}\) and \(\mathrm{C}_{\mathrm{v}}\) of a gas ' \(\mathrm{X}\) ' is \(1.4\). The number of atoms of the gas 'X' present in \(11.2\) litres of it at NTP will be (a) \(6.02 \times 10^{21}\) (b) \(60.2 \times 10^{23}\) (c) \(6.02 \times 10^{23}\) (d) \(1.02 \times 10^{23}\)

Step-by-Step Solution

Answer

The number of atoms is \( 6.02 \times 10^{23} \), option (c).

Step 1: Understand the problem

The problem is asking for the number of atoms in 11.2 liters of gas 'X' at NTP given that the ratio of \( \mathrm{C}_p \) to \( \mathrm{C}_v \) is 1.4. This value is characteristic of diatomic gases. Understand that at NTP, 1 mole of any ideal gas occupies 22.4 liters.

Step 2: Determine the type of gas

The ratio of specific heat capacities \( \frac{C_p}{C_v} = \gamma = 1.4 \) indicates that the gas is diatomic. Common diatomic gases include \( \text{O}_2, \text{N}_2, \text{H}_2 \), etc.

Step 3: Calculate the moles of gas

At NTP, 1 mole of any ideal gas occupies 22.4 liters. Therefore, the number of moles of gas 'X' in 11.2 liters is calculated as follows:\[ \text{Number of moles} = \frac{11.2 \text{ L}}{22.4 \text{ L/mol}} = 0.5 \text{ mol} \]

Step 4: Calculate the number of molecules

Using Avogadro's number, \( 1 \text{ mol} = 6.02 \times 10^{23} \text{ molecules} \). Hence, the number of molecules in 0.5 moles is: \[ 0.5 \text{ mol} \times 6.02 \times 10^{23} \text{ molecules/mol} = 3.01 \times 10^{23} \text{ molecules} \]

Step 5: Calculate the number of atoms

Since the gas is diatomic, each molecule consists of 2 atoms. Therefore, the number of atoms is:\[ 3.01 \times 10^{23} \times 2 = 6.02 \times 10^{23} \text{ atoms} \]

Key Concepts

Diatomic Gases

Diatomic gases are a type of gas that consists of molecules made up of two atoms. These atoms may be of the same element, like nitrogen \((\text{N}_2)\) or oxygen \((\text{O}_2)\), which are among the most commonly found diatomic gases. Having only two atoms in their molecular structure gives these gases certain physical and chemical properties. One of the significant properties of diatomic gases is their specific heat ratio, represented by the symbol \(\gamma\), or \(\frac{C_p}{C_v}\). This is the ratio of the heat capacity at constant pressure \((C_p)\) to the heat capacity at constant volume \((C_v)\). For diatomic gases, this value typically figures around 1.4.

Understanding the properties of diatomic gases is crucial because it offers insights into their behavior under different conditions. For instance, when you know the specific heat ratio of a gas, you can determine whether it's monoatomic or diatomic, which impacts how the gas will behave when it changes temperature or pressure.

• Examples of diatomic gases include Oxygen \((\text{O}_2)\) and Nitrogen \((\text{N}_2)\).
• The specific heat ratio \(\gamma = 1.4\) indicates a diatomic gas.

Moles Calculation

Calculating the moles of a gas helps us understand how much of a substance is present under given conditions of temperature and pressure. At normal temperature and pressure (NTP), which is generally taken as 0°C and 1 atm, one mole of any ideal gas occupies 22.4 liters. Knowing this allows us to convert between the volume of a gas and the amount precisely.

In the specific problem, you need to determine how many moles of gas 'X' are present in 11.2 liters. By using the fact that 1 mole of gas at NTP occupies 22.4 liters, the moles of gas can be calculated with the formula:
\[\text{Number of moles} = \frac{\text{Volume of gas (L)}}{\text{Volume per mole at NTP (22.4 L/mol)}}\]
Thus, for 11.2 liters:
\[0.5 \text{ moles} = \frac{11.2 \text{ L}}{22.4 \text{ L/mol}}\]

• One mole of any gas at NTP occupies 22.4 liters.
• To find moles from volume at NTP, use the formula: \(\text{Number of moles} = \frac{\text{Volume}}{22.4}\).

Avogadro's Number

Avogadro's Number is a fundamental constant used to relate the number of entities (atoms, molecules, etc.) in a mole of any substance to the quantity of matter. It is approximately \(6.02 \times 10^{23}\). This means that one mole of any substance contains exactly that many molecules or atoms. This number provides a bridge between the macroscopic scale (things you can see, like the amount of substance) and the microscopic scale (like the number of atoms or molecules).

In our problem, we use Avogadro's Number to determine the actual number of molecules in a half mole of a gas. By multiplying the moles by Avogadro's Number, we find the number of molecules:
\[0.5 \text{ mol} \times 6.02 \times 10^{23} \text{ molecules/mol} = 3.01 \times 10^{23} \text{ molecules}\]
Since the gas is diatomic, each of those molecules comprises two atoms. Thus, the number of atoms is twice the number of molecules:
\[3.01 \times 10^{23} \times 2 = 6.02 \times 10^{23} \text{ atoms}\]

• Avogadro's number links moles to number of molecules or atoms.
• One mole equals \(6.02 \times 10^{23}\) entities.
• Multiplying moles by Avogadro's Number gives the number of molecules or atoms.