Question

Which of the following pairs of gases contain equal number of molecules? (a) \(\mathrm{CO}_{2}\) and \(\mathrm{NO}_{2}\) (b) \(\mathrm{CO}\) and \((\mathrm{CN})_{2}\) (c) \(\mathrm{NO}\) and \(\mathrm{CO}\) (d) \(\mathrm{N}_{2} \mathrm{O}\) and \(\mathrm{CO}_{2}\)

Short answer

Pair (d), \(\mathrm{N}_{2} \mathrm{O}\) and \(\mathrm{CO}_{2}\), contain an equal number of molecules.

Step-by-step solution

Understand Avogadro's Law

According to Avogadro's Law, equal volumes of gases at the same temperature and pressure contain an equal number of molecules.

Identify Gas Molar Masses

Calculate the molar mass for each gas in the given pairs. This will help determine if they occupy the same volume under identical conditions.- Molar Mass of \( \mathrm{CO}_{2} = 44 \text{ g/mol} \)- Molar Mass of \( \mathrm{NO}_{2} = 46 \text{ g/mol} \)- Molar Mass of \( \mathrm{CO} = 28 \text{ g/mol} \)- Molar Mass of \( (\mathrm{CN})_{2} = 52 \text{ g/mol} \)- Molar Mass of \( \mathrm{NO} = 30 \text{ g/mol} \)- Molar Mass of \( \mathrm{N}_{2} \mathrm{O} = 44 \text{ g/mol} \)

Compare Each Pair of Gases

Examine each pair to see if their molar masses match, as matching molar masses imply the same number of molecules if their conditions are identical.(a) \( \mathrm{CO}_{2} \; (44 \text{ g/mol}) \) vs \( \mathrm{NO}_{2} \; (46 \text{ g/mol}) \): Different(b) \( \mathrm{CO} \; (28 \text{ g/mol}) \) vs \( (\mathrm{CN})_{2} \; (52 \text{ g/mol}) \): Different(c) \( \mathrm{NO} \; (30 \text{ g/mol}) \) vs \( \mathrm{CO} \; (28 \text{ g/mol}) \): Different(d) \( \mathrm{N}_{2} \mathrm{O} \; (44 \text{ g/mol}) \) vs \( \mathrm{CO}_{2} \; (44 \text{ g/mol}) \): Same

Conclusion

The pair \((\mathrm{d}) \; \mathrm{N}_{2} \mathrm{O} \text{ and } \mathrm{CO}_{2}\) have the same molar mass, indicating they contain an equal number of molecules under the same conditions.

Key concepts

Molar Mass Calculation

Calculating molar mass is an essential skill when working with gases. It's the first step in understanding how gases compare in terms of their molecular content. Molar mass refers to the mass of one mole of a molecule and is usually expressed in grams per mole (g/mol). To find it, simply add up the atomic masses of all the atoms in a molecule. For example, the molar mass of carbon dioxide ( CO_2 ) is 44 g/mol, because it comprises one carbon atom (12 g/mol) and two oxygen atoms (16 g/mol each). When comparing gases, molar mass helps us determine whether two gases could contain the same number of molecules when assessed under identical conditions, as per Avogadro's Law. It's a routine but crucial calculation in chemistry that ensures you understand how substances behave in different scenarios.

Equal Volumes of Gases

According to Avogadro's Law, equal volumes of gases, at the same temperature and pressure, hold the same number of molecules. This principle is fundamental when comparing different gases and judging their behavior and composition. Whether you have a liter of oxygen or a liter of helium, they both contain the same number of molecules. This concept facilitates comparisons between gases without needing to delve into the complexities of individual molecular structures or weights. While their chemical identities and molar masses may differ, understanding this principle shows they behave similarly in terms of quantity when volume, temperature, and pressure are held constant. This understanding is pivotal for predicting how gases will act in various chemical processes or reactions.

Equal Number of Molecules

To determine if two gas samples contain an equal number of molecules, one must first ensure they share the same volume, temperature, and pressure conditions, as per Avogadro's Law. Then, verify that their molar masses are identical. When these criteria are met, the gases, despite their different chemical compositions, will house an equal number of molecules. Often, the inquiry involves calculating and comparing molar masses, as mentioned earlier. In the case of our exercise, pairs such as N_2O and CO_2 have identical molar masses (44 g/mol), which means they possess the same number of molecules under identical conditions. Learning to recognize these conditions prepares you for further chemical calculations and enriches your understanding of gas behavior.