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

(c) \(1.2 \times 10^{23}\)

Step 1: Understand the Situation

The problem provides the number of molecules in a certain volume of NH3, and asks for molecules in a given volume of CO2, both at Standard Temperature and Pressure (STP). At STP, 1 mole of any gas occupies 22.4 liters.

Step 2: Calculate Moles of NH3

We know that 500 mL of NH3 contains Avogadro's number of molecules. Since 1 mole of gas at STP is 22.4 L, convert 500 mL to liters (500 mL = 0.5 L) to find the moles of NH3.\[ \text{Moles of } NH_3 = \frac{0.5}{22.4} \approx 0.0223 \text{ moles} \]

Step 3: Mole to Molecule Conversion for NH3

Convert the moles of NH3 to molecules using the fact that 1 mole is equivalent to \(6.022 \times 10^{23}\) molecules.\[ 0.0223 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mole} \approx 1.34 \times 10^{22} \text{ molecules} \]

Step 4: Compare with Given

The given states that 500 mL of NH3 has \(6 \times 10^{23}\) molecules, while our calculation gives \(1.34 \times 10^{22}\). The problem is using a simplified number rather than exact calculations.

Step 5: Calculate Molecules in 100 mL of CO2

Since at STP, volumes relate directly to moles and hence molecules, use proportions to find molecules in 100 mL of CO2.\[ \frac{100 \text{ mL}}{500 \text{ mL}} \times 6 \times 10^{23} = 1.2 \times 10^{23} \text{ molecules} \]

Step 6: Final Step: Determine the Correct Answer Choice

Based on the calculated number of molecules in 100 mL of CO2 being \(1.2 \times 10^{23}\), the correct choice is (c).

Key Concepts

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry, representing the number of particles, typically atoms or molecules, in one mole of a substance. This number is approximately \(6.022 \times 10^{23}\).
This immense quantity provides a bridge between the atomic scale and the macroscopic world. By relating molar quantities to a measurable number of particles, Avogadro's number allows chemists to perform calculations involving reactions and stoichiometry with precision.
When given the number of moles of a substance, you can multiply by Avogadro's number to find out how many molecules are present. Similarly, if you know the number of molecules, dividing by Avogadro's number gives the number of moles.

• Essential for conversions between moles and molecular counts.
• Facilitates stoichiometric calculations in chemical reactions.

Mastering the use of Avogadro's number helps students understand the scale at which molecular interactions occur.

Volume to Mole Conversion

Converting volume to moles is a critical skill in chemistry, particularly under conditions known as Standard Temperature and Pressure (STP), where calculations simplify significantly. At STP, 1 mole of any ideal gas occupies 22.4 liters. This equivalence is pivotal for transferring between physical and chemical quantities.
To convert a volume of gas at STP into moles, divide the volume in liters by 22.4 liters per mole. This calculation allows you to determine how many moles of a gas are present in a given volume. For instance, to find the moles of a gas in 500 mL, first convert milliliters to liters, then use the STP condition:

• 1 liter = 1000 milliliters, so 500 mL is 0.5 L.
• For 0.5 L of gas at STP: \[ \text{Moles} = \frac{0.5}{22.4} \approx 0.0223 \text{ moles} \]

This type of conversion is common in problems that involve gaseous reactants or products at known conditions.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) refers to a set of conditions used as a common reference point in chemistry to ensure consistent data comparison. STP is defined as a temperature of 273.15 Kelvin (0 degrees Celsius) and a pressure of 1 atmosphere. At these conditions, gases behave predictably, as described by the Ideal Gas Law, making calculations more straightforward.
STP allows chemists to work with gases under standardized conditions ensuring that density, volume, and moles of gases can be compared across different experiments. A key property of gases at STP is that 1 mole of any ideal gas occupies a volume of 22.4 liters.

• Provides uniformity in scientific measurements.
• Simplifies calculations involving gases, as seen in our example problem.
• Essential for calculating molar volumes in introductory chemistry.

Understanding STP is foundational for students dealing with gas laws and properties, as it provides consistency and clarity in calculations.