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

Option (c) \(1.2 \times 10^{23}\) molecules.

Step 1: Understand the Problem

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

Step 2: Use Avogadro's Law

Avogadro's Law states that equal volumes of gases at the same temperature and pressure contain the same number of molecules. Thus, the number of molecules in CO2 is directly proportional to its volume.

Step 3: Calculate the Proportion

Since 500 mL of NH3 has \(6.0 \times 10^{23}\) molecules, the proportion for 100 mL is calculated as follows:Proportion factor = \(\frac{100 \text{ mL}}{500 \text{ mL}} = \frac{1}{5}\).

Step 4: Calculate Molecules in 100 mL

Using the proportion factor calculated, find the number of molecules in 100 mL of CO2:\[\text{Number of molecules in 100 mL CO2} = \frac{1}{5} \times 6.0 \times 10^{23} = 1.2 \times 10^{23}\].

Step 5: Select the Correct Answer

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

Key Concepts

STP (Standard Temperature and Pressure)

Standard Temperature and Pressure, often abbreviated as STP, is a reference point used in chemistry to provide a common benchmark for comparing gas volumes. At STP, the temperature is set at 0 degrees Celsius (273.15 Kelvin), and the pressure is set at 1 atmosphere (atm). These conditions are typically used because they simplify calculations involving gases. When gases are measured under STP, it allows chemists to assume ideal behavior and simplifies the use of Avogadro's Law, which directly applies under these conditions.
Understanding STP is crucial in solving problems involving gas volumes and molecules, such as determining how many gas molecules are present in a given volume. Always remember that gases at STP have predictable behavior, which makes calculations more straightforward.

proportionality of gas volumes

One of the foundational principles in chemistry involving gases is the proportionality of gas volumes under constant temperature and pressure. Avogadro's Law states that equal volumes of gases, at the same temperature and pressure, contain the same number of molecules. This means if you compare any two gases under identical conditions, there is a direct proportion between the gas's volume and the number of molecules it contains.
This principle was used in the original exercise to calculate the number of molecules in a different volume of the gas, specifically determining the quantity of molecules present in 100 mL of CO2 given a known quantity in 500 mL of NH3. The straightforward proportional relationship simplifies calculations and helps predict how changes in volume will affect the number of gas molecules.

molecular calculations

Calculating the number of molecules in a given volume of gas is facilitated by understanding the proportionality aspect explained by Avogadro's Law. Since volumes of gases at STP are directly proportional to the number of molecules, a proportion can be set up to easily find out how many molecules are in a sample.
In the exercise, we found that 500 mL of NH3 contains \(6.0 \times 10^{23}\) molecules. By establishing a proportion using a smaller sample, like 100 mL, the solution easily computed the number of molecules as \(1.2 \times 10^{23}\) through proportion factors. This simplification relies on understanding that gas volume ratios equate to molecule number ratios, making it a vital concept in molecular calculations.