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.

Short answer

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

Step-by-step solution

Analyze Molar Masses

First, calculate the molar masses of nitrogen (\( \mathrm{N}_{2} \)) and argon (\( \mathrm{Ar} \)) to understand the number of moles present. The molar mass of \( \mathrm{N}_{2} \) is approximately 28 g/mol, and for \( \mathrm{Ar} \), it is approximately 40 g/mol. Since the masses are equal, this means that there will be more moles of \( \mathrm{N}_{2} \) because \( \mathrm{N}_{2} \) has a lower molar mass.

Evaluate Statement (a)

Since \( \mathrm{N}_{2} \) molecules have a smaller molar mass than Ar atoms, there are more moles of \( \mathrm{N}_{2} \) in the flask. More moles imply more molecules, so statement (a) is true.

Assess Pressures in Flasks

According to the ideal gas law, \( PV = nRT \), pressure depends on the number of moles, volume, and temperature. Both flasks have the same volume and temperature, but the \( \mathrm{N}_{2} \) flask has more moles. Therefore, the pressure in the \( \mathrm{N}_{2} \) flask is greater. Hence, statement (b) is false.

rms Speed Calculation

Root mean square speed (\( v_{\text{rms}} \)) is given by \( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \). Since \( \mathrm{Ar} \) has a larger molar mass than \( \mathrm{N}_{2} \), it will have a lower rms speed at the same temperature. Therefore, statement (c) is false.

Frequency of Collisions

Higher pressure and more molecules in the \( \mathrm{N}_{2} \) flask mean molecules collide more frequently with the walls. Collision frequency is related to the number of molecules and particle speed. Even though Ar atoms are heavier and slower, the greater number of \( \mathrm{N}_{2} \) molecules leads to more collisions overall. Thus, statement (d) is true.

Key concepts

Molar Mass

Molar mass is an important concept when discussing gases because it helps us determine the number of moles from a given mass. A mole is a unit used to express amounts of a chemical substance, equivalent to the molecular weight of the substance. For example, nitrogen \( \mathrm{N}_{2} \) has a molar mass of about 28 g/mol while argon \( \mathrm{Ar} \) has a molar mass of 40 g/mol.
When you have equal masses of two gases, the one with the lower molar mass will have more moles.

• This is because more molecules are needed to make up the same mass if each individual molecule is lighter.
• In this exercise, \( \mathrm{N}_{2} \) has more moles than \( \mathrm{Ar} \).

Gas Pressure

Gas pressure is influenced by the number of moles of a gas, as explained by the ideal gas law, given by the formula \( PV = nRT \).

• Here, \( P \) represents pressure, \( V \) is volume, \( n \) is number of moles, \( R \) is the gas constant, and \( T \) is temperature measured in Kelvin.
• By this principle, if two containers have equal volume and temperature, the gas with a higher number of moles will exert greater pressure.
• In this exercise, because \( \mathrm{N}_{2} \) has more moles than \( \mathrm{Ar} \), the pressure in the \( \mathrm{N}_{2} \) flask is actually higher, which is why statement (b) in the original exercise is false.

Root Mean Square Speed

The concept of root mean square (rms) speed is crucial to understanding the motion of gas molecules. The rms speed can be calculated with the equation \( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \).

• In the equation, \( M \) represents the molar mass of the gas in kg/mol, and \( T \) is the absolute temperature in Kelvin.
• The rms speed indicates how fast the molecules of a gas are moving on average.
• Since \( \mathrm{Ar} \) is heavier (higher molar mass) than \( \mathrm{N}_{2} \), at the same temperature, \( \mathrm{Ar} \) atoms move slower than \( \mathrm{N}_{2} \) molecules, making statement (c) false in the original exercise.

Collision Frequency

Collision frequency relates to how often gas molecules hit the walls of their container, playing a crucial role in defining the pressure.

• With more molecules, as found in the \( \mathrm{N}_{2} \) flask, there are naturally more collisions occurring with the walls, contributing to higher pressure.
• Collision frequency is also affected by speed: faster moving molecules tend to collide more frequently.
• Despite the fact that \( \mathrm{Ar} \) atoms are slower due to their larger mass, the greater number of \( \mathrm{N}_{2} \) molecules makes their collision frequency higher.
• This explains why statement (d) in the original exercise is accurate, as the \( \mathrm{N}_{2} \) molecules collide more often than \( \mathrm{Ar} \) atoms.