Problem 129 - A Complete Resource Book in Chemistry for JEE Main

Question

\(500 \mathrm{~mL}\) of \(\mathrm{NH}_{3}\) contains \(6.0 \times 10^{23}\) molecules at STP. How many molecules are present in \(100 \mathrm{~mL}\) of \(\mathrm{CO}_{2}\) at STP? (a) \(6 \times 10^{23}\) (b) \(1.5 \times 10^{23}\) (c) \(1.2 \times 10^{23}\) (d) none of these

Step-by-Step Solution

Answer

The correct number of molecules is \(1.2 \times 10^{23}\), corresponding to option (c).

Step 1: Understand the Given Conditions

We are given that 500 mL of NH₃ contains \(6.0 \times 10^{23}\) molecules at STP. We need to find out how many molecules are in 100 mL of CO₂ at the same conditions.

Step 2: Use Avogadro's Law

According to Avogadro's Law, equal volumes of gases at the same temperature and pressure contain the same number of molecules. This means that if you have the same conditions, the number of molecules is proportional to the volume of the gas.

Step 3: Set Up the Proportion

Given that 500 mL of NH₃ contains \(6.0 \times 10^{23}\) molecules, we can set up a proportion to find the number of molecules in 100 mL of CO₂. The proportion is: \( \frac{\text{Molecules in 100 mL of CO₂}}{6.0 \times 10^{23}} = \frac{100 \text{ mL}}{500 \text{ mL}} \).

Step 4: Solve the Proportion

Solving this proportion, we find the number of molecules in 100 mL of CO₂ at STP:\[ \text{Molecules in 100 mL of CO₂} = \left(\frac{100}{500}\right) \times 6.0 \times 10^{23} \]\[ = 0.2 \times 6.0 \times 10^{23} \]\[ = 1.2 \times 10^{23} \]

Step 5: Determine the Correct Answer Choice

The number of molecules in 100 mL of CO₂ at STP is \(1.2 \times 10^{23}\), which corresponds to option (c).

Key Concepts

Molecules

Molecules are the smallest units of a chemical compound that retain its chemical properties. In the context of gases, the behavior of molecules is particularly interesting at the molecular level. When dealing with gases, each molecule moves in random, straightforward paths until it collides with another molecule or with the walls of its container. These collisions are perfectly elastic, meaning no energy is lost. Thus, the energy of a gas is the result of the collective kinetic energy of all its molecules. Understanding the number of molecules within a gas helps determine its properties at a macroscopic level. For example:

• Increasing the number of molecules while keeping the container the same will increase the pressure of the gas.
• Avogadro's Law states that equal volumes of gases under the same conditions contain an equal number of molecules.

This concept is central to solving many gas-related problems, including the one presented in the exercise, where the number of molecules of different gases is compared.

Standard Temperature and Pressure (STP)

In chemistry, the concept of Standard Temperature and Pressure (STP) is essential for comparability and consistency in experiments. STP is defined as a temperature of 0°C (273.15 K) and a pressure of 1 atmosphere (101.3 kPa). At STP, one mole of any ideal gas occupies a volume of 22.4 liters. This relationship is crucial for performing calculations related to gas volumes and comparing different gases. Usefulness of STP includes:

• Standardizing conditions so that measurements are consistent and comparable.
• Simplifying calculations since properties of gases like volume and number are directly proportional at STP.

Understanding gases under STP allows chemists to predict and explain behaviors in chemical reactions efficiently, using familiar rules like Avogadro's Law, which we applied in our exercise.

Gas Volume

Gas volume refers to the amount of space that a gas occupies. It is usually measured in units like liters or milliliters. Gas volume is sensitive to changes in pressure and temperature. Several principles govern the relationship between gas volume, pressure, and temperature, with Avogadro's Law being key in understanding gas behavior:

• Avogadro's Law: It states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules.
• Pressure Influence: The volume of a gas decreases as pressure increases, assuming constant temperature (Boyle's Law).
• Temperature Influence: The volume of a gas increases with rising temperature if pressure is constant (Charles's Law).

In our exercise, we used the relationship between volume and the number of molecules at constant STP to determine the count of molecules, illustrating how proportionality underlies gas laws and calculations.