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 \(3.01 \times 10^{12}\), which corresponds to option (a).

Step 1: Understand the Concept of NTP

NTP stands for Normal Temperature and Pressure, which is typically defined as a temperature of 0°C (273 K) and a pressure of 1 atm. At NTP, 1 mole of an ideal gas occupies a volume of 22.4 liters.

Step 2: Convert Volume to Liters

Given volume is \(1.12 \times 10^{-7} \space \text{cc}\). Convert this to liters by dividing by 1000 since 1 liter = 1000 cc.\[ \text{Volume in liters} = \frac{1.12 \times 10^{-7}}{1000} = 1.12 \times 10^{-10} \text{ liters} \]

Step 3: Calculate the Moles of Gas

Using the fact that 1 mole of gas occupies 22.4 liters at NTP, calculate the number of moles in the given volume.\[ \text{Moles of gas} = \frac{1.12 \times 10^{-10}}{22.4} \approx 5 \times 10^{-12} \text{ moles} \]

Step 4: Determine the Number of Molecules

Use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules per mole, to determine the number of molecules. \[ \text{Number of molecules} = 5 \times 10^{-12} \times 6.022 \times 10^{23} \]Calculate the above to find the number of molecules.\[ = 3.01 \times 10^{12} \text{ molecules} \]

Step 5: Choose the Correct Answer

The calculated number of molecules is \(3.01 \times 10^{12}\). Therefore, the correct answer is option (a) \(3.01 \times 10^{12}\).

Key Concepts

NTP (Normal Temperature and Pressure)

NTP, or Normal Temperature and Pressure, is an important concept in chemistry, especially when dealing with gases. It is a standard set of conditions used to compare the behavior of gases.
At NTP, the temperature is set to 0°C, which is equivalent to 273 Kelvin. Meanwhile, the pressure is set at 1 atmosphere (atm). These conditions are significant as they simplify calculations involving gases, providing a common reference point.

• Temperature: 0°C or 273K
• Pressure: 1 atm

At NTP, 1 mole of an ideal gas occupies a volume of 22.4 liters. This relationship is crucial when determining quantities of gases in chemical reactions or when converting between moles and volumes.

Moles of gas

Understanding moles in chemistry can make calculations easier, especially for reactions and when working with gases. The mole is a unit that measures the amount of substance.
A mole is defined as containing exactly 6.022 x 10²³ entities (atoms, molecules, etc.). This number is known as Avogadro's number, and it quantifies how many particles are in 1 mole of a substance.
When it comes to gases at NTP, 1 mole occupies 22.4 liters.
This simple linkage lets you convert a gas's volume into moles using the formula:

• Moles of gas = Volume in liters \(\div\) 22.4

For example, if you have a gas volume of 1.12 x 10^-10 liters, you can find the moles by dividing it by 22.4, which gives you approximately 5 x 10^-12 moles.

Avogadro's number

Avogadro's number is a fundamental concept crucial for understanding quantities of chemical substances in reactions. Named after Amedeo Avogadro, it is precisely 6.022 x 10²³ per mole, which indicates the number of atoms or molecules in one mole of a substance.
This number allows chemists to bridge the gap between the macroscopic world we observe and the microscopic world of atoms and molecules.

• Avogadro's number: 6.022 x 10²³ particles/mole

This concept is mostly used when converting between moles and number of molecules. For example, when calculating how many molecules exist in a certain amount of gas, you multiply the moles of gas by Avogadro's number.
Therefore, if you have 5 x 10^-12 moles of a gas, the number of molecules would be calculated as follows:
Number of molecules = (5 x 10^-12) x (6.022 x 10²³) = 3.01 x 10¹² molecules.