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

Short answer

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

Step-by-step solution

Determine the relation between volume and moles at STP

Since we're dealing with gases at Standard Temperature and Pressure (STP), we can use the property that one mole of any gas occupies 22.4 liters (or 22400 mL). This means that the volume is directly proportional to the number of moles of gas.

Calculate moles of NH3 in 500 mL

We are given that 500 mL of NH3 contains \(6.0 \times 10^{23}\) molecules. Using the relation between moles and volume at STP, we can find the number of moles in 500 mL by first understanding the proportionality: \(\frac{6.0 \times 10^{23}}{500 \text{ mL}}\) relates molecules to mL for NH3.

Calculate moles of CO2 in 100 mL

Since we need to find the number of molecules in 100 mL of CO2, and volumes are directly proportional to moles at STP, we proportionately scale the number of molecules for 100 mL. Using the information for NH3: \(\frac{6.0 \times 10^{23}}{500 \text{ mL}} \times 100 \text{ mL} = 1.2 \times 10^{23}\) molecules for 100 mL of CO2.

Conclusion

Thus, 100 mL of CO2 at STP contains \(1.2 \times 10^{23}\) molecules, which matches option (c).

Key concepts

Avogadro's Law

Avogadro's Law is a fundamental principle in chemistry, named after the Italian scientist Amedeo Avogadro. This law simplifies the study of gases and makes predictions about gas behavior more accessible. It states that equal volumes of any gases, at the same temperature and pressure, contain an equal number of molecules.

The formula representing this law is:\[ V \propto n \]where \( V \) is the volume of the gas and \( n \) is the number of moles. This means if you know the volume of one gas at a given condition, you can predict the volume of another identical number of moles of a different gas under the same conditions.

Avogadro's Law allows chemists to switch between volumes and particles easily. It links macroscopic phenomena (like volume) with microscopic properties (like number of molecules), making it easier to understand and predict the behavior of gases.

STP Conditions

Standard Temperature and Pressure (STP) are the conditions where temperature is at 273.15 K (0°C) and pressure is at 1 atm (atmosphere). These conditions are crucial for simplifying gas calculations.

Under STP, any ideal gas occupies a standard volume of 22.4 liters per mole. This uniformity provides a convenient way to compare different gases or perform calculations without confusion.

• Temperature: 273.15 K or 0°C
• Pressure: 1 atm
• Volume of one mole of gas: 22.4 L (or 22400 mL)

These conditions are widely used in chemistry to ensure consistency and are essential when applying Avogadro's Law. By working under STP conditions, students can directly equate volumes to number of moles using the standard volume of any ideal gas. This makes calculations straightforward and reduces complexity in problem-solving.

Molecular Calculations

Molecular calculations in gas chemistry involve converting between volumes, masses, and number of molecules. These calculations benefit greatly from Avogadro's Law and the determined conditions of STP.

In the example problem, we used the fact that at STP, one mole of any gas contains Avogadro's number of molecules, which is approximately \( 6.022 \times 10^{23} \). By determining how many moles corresponded to a given volume, we could convert this information into the number of molecules.

The steps to perform molecular calculations typically involve:

• Identifying the conditions (STP in this case) and appropriate gas laws to apply.
• Using proportional relationships between the known quantities (like volume) and desired quantities (like moles or molecules).
• Applying Avogadro's number for converting moles into molecules.

By following a strict methodical approach, molecular calculations provide accurate results determining how many molecules occupy certain volumes under defined conditions.