Problem 42 - Chemistry and Chemical Reactivity

Question

Equal masses of gaseous \(\mathrm{N}_{2}\) and \(\mathrm{Ar}\) are placed in separate flasks of equal volume at the same temperature. Tell whether each of the following statements is true or false. Briefly explain your answer in each case. (a) There are more molecules of \(\mathrm{N}_{2}\) present than atoms of Ar. (b) The pressure is greater in the Ar flask. (c) The Ar atoms have a greater rms speed than the \(\mathrm{N}_{2}\) molecules. (d) The \(\mathrm{N}_{2}\) molecules collide more frequently with the walls of the flask than do the Ar atoms.

Step-by-Step Solution

Answer

(a) True. (b) False. (c) True. (d) True.

Step 1: Calculate Moles of Each Gas

The molar mass of nitrogen, \( \mathrm{N}_{2} \), is 28 g/mol (14 g/mol per nitrogen atom) and for argon (\( \mathrm{Ar} \)) it is 40 g/mol. If equal masses are present, the moles are calculated as follows: \( n = \frac{\text{mass}}{\text{molar mass}} \). Since \( \mathrm{N}_{2} \) has a lower molar mass, there will be more moles of \( \mathrm{N}_{2} \) than of \( \mathrm{Ar} \).

Step 2: Examine Statement (a)

Statement: There are more molecules of \( \mathrm{N}_{2} \) present than atoms of \( \mathrm{Ar} \). Since moles are directly proportional to the number of molecules (Avogadro's number), more moles of \( \mathrm{N}_{2} \) mean more molecules than atoms of \( \mathrm{Ar} \). This statement is true.

Step 3: Analyze Pressure in Each Flask

According to the ideal gas law, \( PV = nRT \), pressure \( P \) is proportional to \( n \) when volume \( V \) and temperature \( T \) are constant. More moles (\( \mathrm{N}_{2} \)) result in a slightly higher pressure in the \( \mathrm{N}_{2} \) flask rather than the \( \mathrm{Ar} \) flask. Statement (b) is false.

Step 4: Compare RMS Speeds

The root mean square speed \( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \), where \( M \) is molar mass. Since \( \mathrm{Ar} \) has a lower molar mass than \( \mathrm{N}_{2} \), the \( \mathrm{Ar} \) atoms will have a greater rms speed. Statement (c) is true.

Step 5: Frequency of Collision with Walls

The frequency of collisions is linked to both the number of particles and their speeds. More \( \mathrm{N}_{2} \) molecules and similar speed as \( \mathrm{Ar} \) means \( \mathrm{N}_{2} \) will collide more frequently. Statement (d) is true.

Key Concepts

Gaseous Nitrogen

Nitrogen gas, denoted as \( \mathrm{N}_2 \), is a diatomic molecule, meaning it consists of two nitrogen atoms bonded together. It is an essential component of our atmosphere and plays a crucial role in chemical reactions. When studying or working with gaseous nitrogen, especially in ideal gas contexts, key factors include its molar mass and behavior under different conditions.
Nitrogen gas is lighter compared to many other gases, with a molar mass of approximately 28 g/mol. This lower molar mass influences how nitrogen behaves in gas mixtures, parameters like pressure, and even the speed of its molecules in a given environment. Understanding these properties is fundamental when comparing nitrogen to other gases like argon.

Molar Mass Comparison

Molar mass is a critical concept in understanding the behavior of gases. It is the mass of one mole of a substance, typically expressed in grams/mole. For nitrogen, the molar mass is 28 g/mol, while for argon it is 40 g/mol.

• This difference in molar mass means that for equal masses of nitrogen and argon, there will be more moles of nitrogen than argon.
• More moles of a substance imply more molecules, since moles are directly related to the number of molecules via Avogadro's number.

This principle is essential when applying the ideal gas law, as it affects calculations involving pressure, volume, and the number of particles present in a given space. By comparing molar masses, one can infer which gas would exhibit a greater effect under similar conditions.

Root Mean Square Speed

Root mean square speed (rms speed) is a concept used to describe the average speed of gas molecules. It is given by the formula \( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molar mass.

• Since molar mass \( M \) appears in the denominator, a lower molar mass results in a higher rms speed. This means lighter gas molecules move faster, on average, than heavier ones.
• In the given scenario, argon, having a lower molar mass than nitrogen \( \mathrm{N}_2 \), will have a greater rms speed.

This concept helps in understanding and predicting how gas molecules move and interact, affecting pressure and collision rates within a gas container.

Gas Molecule Collisions

Understanding gas molecule collisions is vital for comprehending gas pressure and behavior in a container. These collisions are influenced by factors such as the number of gas particles and their speed.
Frequent collisions mean higher pressure and are affected by both the volume of the container and the speed of the molecules:

• More molecules, such as in the case of nitrogen \( \mathrm{N}_2 \) due to its lower molar mass, result in more frequent collisions.
• Even if \( \mathrm{Ar} \) atoms are faster due to their higher rms speed, the sheer number of nitrogen molecules leads to more collisions with the walls of the container.

This frequency of collisions relates directly to the principles of the ideal gas law, explaining why \( \mathrm{N}_2 \) would collide more frequently despite the high speed of argon atoms.