Question

If equal masses of \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) are placed in separate containers of equal volume at the same temperature, which of the following statements is true? If false, tell why it is false. (a) The pressure in the flask containing \(\mathrm{N}_{2}\) is greater than that in the llask containing \(\mathbf{O}_{2}\) (b) There are more molecules in the flask containing \(\mathrm{O}_{2}\) than in the flask containing \(\mathbf{N}_{2}\)

Short answer

Statement (a) is true; (b) is false.

Step-by-step solution

Determine Molar Mass

First, calculate the molar mass of both gases. The molar mass of \(\mathrm{O}_{2}\) is approximately 32 g/mol (16 g/mol per oxygen atom) and the molar mass of \(\mathrm{N}_{2}\) is approximately 28 g/mol (14 g/mol per nitrogen atom).

Calculate Moles Given Equal Masses

Since the masses of \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) are equal, let's say \(m\) grams each, the moles \(n\) can be calculated using \(n = \frac{m}{\text{Molar Mass}}\). Therefore, the moles of \(\mathrm{O}_{2}\) are \(\frac{m}{32}\) and the moles of \(\mathrm{N}_{2}\) are \(\frac{m}{28}\).

Compare Number of Molecules

Using Avogadro's law, the number of molecules is directly proportional to the number of moles. Therefore, since \(\frac{m}{28} \gt \frac{m}{32}\), there are more moles, and thus more molecules, of \(\mathrm{N}_{2}\) than \(\mathrm{O}_{2}\).

Analyze Pressure with Ideal Gas Law

According to the ideal gas law \(PV = nRT\), at constant volume \(V\) and temperature \(T\), pressure \(P\) is directly proportional to moles \(n\). Since \(\mathrm{N}_{2}\) has more moles in the same conditions, the pressure is higher in the \(\mathrm{N}_{2}\) flask.

Validate or Refute Statements

(a) True. The pressure in the flask containing \(\mathrm{N}_{2}\) is greater because it has more moles.\(\mathrm{N}_{2}\). (b) False. There are more molecules in the flask containing \(\mathrm{N}_{2}\), not \(\mathrm{O}_{2}\).

Key concepts

Molar Mass Explained

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is a crucial concept in chemistry because it allows us to convert between the mass of a substance and the number of particles or molecules it contains. For diatomic gases like oxygen (\(\mathrm{O}_2\)) and nitrogen (\(\mathrm{N}_2\)), the molar mass is the combined mass of the respective atoms. Oxygen has a molar mass of approximately 32 g/mol, and nitrogen has a molar mass of about 28 g/mol.
This means that for a given mass, nitrogen has more moles than oxygen because it is lighter on a per-mole basis. Understanding molar mass helps us predict how different gases behave under the same conditions.

The Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as \(PV = nRT\), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume.
• \(n\) is the number of moles.
• \(R\) is the gas constant.
• \(T\) is the temperature in Kelvin.

This equation shows that if the volume and temperature of a gas are constant, the pressure is directly proportional to the number of moles. In our scenario, nitrogen will exert a higher pressure than oxygen since it has more moles, given the same mass due to its lower molar mass.

Understanding Avogadro's Law

Avogadro's law states that equal volumes of gases at the same temperature and pressure contain the same number of molecules. It implies that the number of moles of a gas is directly related to the number of molecules. Since nitrogen has more moles than oxygen when equal masses are measured, it also has more molecules.
This relationship is crucial for predicting gas behavior in chemical reactions and processes. It helps us understand why, in this problem, nitrogen, having more moles, consequently has a larger number of molecules under the same conditions.

Molecular Comparison

When comparing molecules of different gases, it’s important to consider both the number of moles and the type of molecules. In this exercise, analyzing equal masses of \(\mathrm{O}_2\) and \(\mathrm{N}_2\) highlights how molecular weight affects the number of particles.
Despite having an equal mass, nitrogen molecules outnumber oxygen molecules because nitrogen has a lower molar mass. This contrasts with intuitive thinking that equal masses would mean an equal number of molecules. Such insights enable us to tackle real-world problems where molecular interactions and counts are crucial.