Problem 108 - Chemistry and Chemical Reactivity

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, explain why it is false. (a) The pressure in the flask containing \(\mathrm{N}_{2}\) is greater than that in the flask containing \(\mathrm{O}_{2}\) (b) There are more molecules in the flask containing \(\mathrm{O}_{2}\) than in the flask containing \(\mathrm{N}_{2}\).

Step-by-Step Solution

Answer

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

Step 1: Understand the Relationship between Pressure, Volume, and Temperature

According to the ideal gas law \( PV = nRT \), the pressure \( P \) of a gas is directly proportional to the number of moles \( n \), given that the volume \( V \) and temperature \( T \) are constant. This means that for equal volumes and temperatures, the pressure is dependent on the quantity of gas in moles.

Step 2: Compare Molar Masses of O2 and N2

Calculate the molar mass of both gases. The molar mass of \( \mathrm{N}_2 \) (Nitrogen) is \( 28.02 \text{ g/mol} \), while the molar mass of \( \mathrm{O}_2 \) (Oxygen) is \( 32.00 \text{ g/mol} \).

Step 3: Determine Number of Moles

For equal masses of the two gases, calculate the number of moles of each. Since moles \( n = \frac{\text{mass}}{\text{molar mass}} \), fewer moles of \( \mathrm{O}_2 \) will be present compared to \( \mathrm{N}_2 \), because \( \mathrm{O}_2 \) has a higher molar mass.

Step 4: Analyze Pressure Differences

Since \( \mathrm{N}_2 \) will have more moles than \( \mathrm{O}_2 \) in equal mass conditions, the pressure in the container with \( \mathrm{N}_2 \) will be higher based on the ideal gas law, assuming volumes and temperatures are equal.

Step 5: Fact-Check Both Statements

(a) True: The pressure in the flask with \( \mathrm{N}_2 \) is greater than that in the flask with \( \mathrm{O}_2 \) because there are more moles of \( \mathrm{N}_2 \). (b) False: There are more molecules in the \( \mathrm{N}_2 \) flask because it has more moles, not the \( \mathrm{O}_2 \) flask.

Key Concepts

Pressure and Moles

The ideal gas law is an important principle in chemistry expressed as \( PV = nRT \), where \( P \) stands for pressure, \( V \) is the volume, \( n \) represents the number of moles of the gas, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. When comparing the pressures of gases in containers of equal volume and at the same temperature, the key factor influencing the pressure is the number of moles \( n \). This is because pressure is directly proportional to the amount of gas, assuming temperature and volume are held constant.

If two gases are measured by mass, the gas with more moles will exert greater pressure. For example, let's compare equal masses of nitrogen \( \mathrm{N}_2 \) and oxygen \( \mathrm{O}_2 \). Assuming each gas has the same mass but different molar masses, they will contain different numbers of moles. This difference directly affects the pressure each exerts under identical conditions of volume and temperature.

Molar Mass Comparison

Understanding molar masses is crucial when dealing with gases and their behaviors. A molar mass is the mass of one mole of a substance, which tells us how much one mole of a particular type of particle weighs. For oxygen \( \mathrm{O}_2 \), the molar mass is 32.00 \( \text{g/mol} \), and for nitrogen \( \mathrm{N}_2 \), it's 28.02 \( \text{g/mol} \).

• Since nitrogen has a lower molar mass compared to oxygen, a given mass of nitrogen will contain more moles than the same mass of oxygen.
• Thus, with identical mass samples, nitrogen will have a larger number of molecules than oxygen due to its lighter molar mass.

This difference impacts the number of molecules present in a given volume, influencing pressure and behavior, as stated by the ideal gas law. When comparing gases, the molar mass allows us to properly convert mass to moles, leading us to more accurately predict their behavior in various conditions.

Gas Behavior

The behavior of gases can be effectively studied through the lens of the ideal gas law, focusing on how pressure, volume, and temperature interact. When examining gases like \( \mathrm{N}_2 \) and \( \mathrm{O}_2 \), it's important to refer to their molecular states and how they respond to specific conditions.

From the ideal gas law perspective:

• For containers of the same volume and temperature, the amount of gas, expressed in moles, determines the pressure exerted by the gas inside the container.
• For gases compared under equal conditions but different molar masses, the number of moles directly determines how many molecules are present, affecting the pressure substantially.

An example scenario: with the same mass of nitrogen and oxygen, more moles of nitrogen result in more molecules. Consequently, nitrogen gas will show higher pressure than oxygen if kept at the same volume and temperature. This understanding of gas behavior allows scientists and students alike to predict and explain Differences observed in real-world applications or experimental setups.