Problem 87 - A Complete Resource Book in Chemistry for JEE Main

Question

Volume of a gas at NTP is \(1.12 \times 10^{-7} \mathrm{cc}\). The number of molecules in it is (a) \(3.01 \times 10^{12}\) (b) \(3.01 \times 10^{18}\) (c) \(3.01 \times 10^{24}\) (d) \(3.01 \times 10^{30}\)

Step-by-Step Solution

Answer

The number of molecules is approximately \(3.01 \times 10^{12}\).

Step 1: Understand the Problem

We need to find the number of molecules in a given volume of gas at Normal Temperature and Pressure (NTP). The provided volume is \(1.12 \times 10^{-7} \mathrm{cc}\).

Step 2: Convert Volume to Moles

At NTP, 1 mole of an ideal gas occupies 22.4 liters, which is \(22.4 \times 10^3 \) cc. The number of moles \( n \) can be calculated as \( n = \frac{1.12 \times 10^{-7} \text{ cc}}{22.4 \times 10^3 \text{ cc/mol}} \).

Step 3: Calculate Number of Moles

Perform the division: \( n = \frac{1.12 \times 10^{-7}}{22.4 \times 10^3} \approx 5 \times 10^{-12} \) moles.

Step 4: Determine Number of Molecules

Use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules/mol, to find the number of molecules: \( \text{Number of molecules} = 5 \times 10^{-12} \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} \).

Step 5: Calculate Final Result

Multiply the values to find the total number of molecules: \( 3.011 \times 10^{12} \) molecules.

Key Concepts

NTP (Normal Temperature and Pressure)

Normal Temperature and Pressure, commonly abbreviated as NTP, is an important standard condition often used in chemistry and physics. It is defined as a temperature of 0°C (273.15 K) and a pressure of 1 atm. These conditions serve as a reference point for calculations involving gases, allowing scientists to compare measurements across different situations.
At NTP, gases behave predictably, conforming closely to the Ideal Gas Law. In this scenario, 1 mole of an ideal gas occupies exactly 22.4 liters (or 22.4 x 10³ cc). This standard volume, connected to the uniform behavior of gases, simplifies many calculations.
Understanding the concept of NTP is crucial for accurately calculating moles, volumes, and the number of molecules in a given sample of gas. By comprehending these conditions, you can more easily convert between gas volumes and moles, critical to solving many chemistry problems.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry that quantifies the number of entities, typically atoms or molecules, in one mole of a substance. Its value is approximately 6.022 x 10²³ molecules/mol. This large number is essential when converting between moles, mass, and the number of molecules.
To put it in perspective, Avogadro's Number bridges the gap between the macroscopic scale (what we can see and feel) and the molecular scale (individual atoms and molecules).
When you use Avogadro's Number in calculations, it allows for the precise determination of how many molecules are present in an ideal gas sample when given the number of moles. This makes it a crucial tool in stoichiometry, thermodynamics, and various chemistry and physics calculations.

Volume to Moles Conversion

Converting a gas volume to moles is a critical step in many chemical calculations, especially for gases at standard conditions like NTP. The key concept is that, under NTP, 1 mole of a gas occupies 22.4 liters.
The formula to convert volume to moles is relatively straightforward:

• Moles (n) = Volume of the gas (in cc) / (22.4 x 10³ cc/mol)

Applying this formula allows you to determine the number of moles from the volume information provided. This is an essential conversion in chemistry because it connects the volume of a gas, which can be easily measured, to the amount of substance, which is necessary for further calculations like determining the number of molecules present.

Number of Molecules Calculation

Once you have determined the number of moles in a gas sample, calculating the number of molecules is straightforward using Avogadro's Number.
This calculation involves multiplying the number of moles by Avogadro's Number (6.022 x 10²³ molecules/mol). It helps you find out the exact number of molecules present in your gas volume at NTP. For instance:

• Number of Molecules = Number of moles x 6.022 x 10²³ molecules/mol

This method is precise and allows scientists and students to scale measurements from the laboratory to the molecular level, providing a deeper understanding of chemical interactions and properties.