Question

Consider the following gases, all at STP: Ne, \(\mathrm{SF}_{6}, \mathrm{~N}_{2}\), \(\mathrm{CH}_{4}\). (a) Which gas is most likely to depart from assumption 3 of the kinetic molecular theory (Section 10.7)? (b) Which one is closest to an ideal gas in its behavior? (c) Which one has the highest root-mean-square molecular speed? (d) Which one has the highest total molecular volume relative to the space occupied by the gas? (e) Which has the highest average kinetic molecular energy? (f) Which one would effuse more rapidly than \(\mathrm{N}_{2}\) ?

Short answer

(a) SF6 is most likely to depart from assumption 3 of the Kinetic Molecular Theory as it contains larger and heavier molecules with stronger intermolecular forces. (b) Ne behaves closest to an ideal gas due to its weak intermolecular forces and small volume. (c) CH4 has the highest root-mean-square molecular speed as it has the smallest molar mass among the given gases. (d) SF6 has the highest total molecular volume relative to the space occupied by the gas. (e) All gases have the same average kinetic molecular energy as they are at STP. (f) CH4 would effuse more rapidly than N2 as it has a smaller molar mass.

Step-by-step solution

Understand the properties of gases according to the Kinetic Molecular Theory

To break down this exercise, it is important to recall the main assumptions of the Kinetic Molecular Theory: 1. Gases consist of a large number of small particles (atoms or molecules) in constant random motion. 2. The interactions between gas particles are negligible (no attractive or repulsive forces between them). 3. The volume occupied by gas particles themselves is negligible compared to the volume of the container. 4. The average kinetic energy of gas particles is proportional to the temperature in Kelvin. 5. The collisions between gas particles and between particles and the container walls are elastic (no loss of kinetic energy).

Understanding the Molecular Structures and Molar Masses of the Gases

In order to analyze each gas's properties at STP, we need to understand their molecular structures and molar masses. Here are the molar masses of the given gases: Ne: 20.18 g/mol (monoatomic) SF6: 146.06 g/mol (polar covalent compound) N2: 28.02 g/mol (diatomic) CH4: 16.04 g/mol (tetrahedral)

Answering Part (a) - Departure from Assumption 3

Since SF6 is a polar covalent compound containing larger and heavier molecules, it tends to have stronger intermolecular forces. Thus, its particles occupy a more significant volume than the other gases, making SF6 the gas most likely to depart from assumption 3 of the Kinetic Molecular Theory.

Answering Part (b) - Ideal Gas Behavior

Gases that exhibit ideal behavior have weak or no intermolecular forces and occupy a small volume. Since Ne is a monoatomic gas with weak intermolecular forces, it behaves closest to an ideal gas among the given gases.

Answering Part (c) - Highest Root-mean-square Molecular Speed

The root-mean-square molecular speed (u) is given by the equation \(u = \sqrt{\frac{3RT}{M}}\), where R is the gas constant, T is the temperature, and M is the molar mass of the gas. At constant temperature, the gas with the smallest molar mass will have the highest root-mean-square speed. Since CH4 has the smallest molar mass among the given gases (16.04 g/mol), it has the highest root-mean-square molecular speed.

Answering Part (d) - Highest Total Molecular Volume

As mentioned in Step 3, the gas with the strongest intermolecular forces and larger molecules will have the highest total molecular volume relative to the space they occupy. This gas is SF6, as it has the largest and heaviest molecules compared to Ne, N2, and CH4.

Answering Part (e) - Highest Average Kinetic Molecular Energy

According to the Kinetic Molecular Theory, the average kinetic energy of gas particles is proportional to the temperature. Given that all gases are at STP, they all have the same temperature, and thus, they all have the same average kinetic molecular energy.

Answering Part (f) - Gas with Higher Effusion Rate than N2

Effusion rate is inversely proportional to the square root of the molar mass of the gas. The gas with the smaller molar mass effuses more rapidly. Comparing the molar masses of the given gases, we can determine that CH4 (16.04 g/mol) is the gas that would effuse more rapidly than N2 (28.02 g/mol).

Key concepts

Ideal Gas Behavior

In order to understand ideal gas behavior, it's helpful to think about the conditions under which a gas behaves perfectly according to the assumptions of the Kinetic Molecular Theory. These assumptions include the lack of intermolecular forces between gas particles and that the volume occupied by these particles is negligible in relation to the volume of their container. This means that ideal gases don't experience attractions or repulsions between molecules and the space between them is vast compared to their own size.

Among the gases given, Ne (neon) closely aligns with ideal gas behavior characteristics. This is because Ne is a noble gas with no polar characteristics, leading to nearly nonexistent intermolecular forces. As a monoatomic gas, the individual particles also occupy a very small volume. Thus, under standard temperature and pressure (STP), Ne behaves almost ideally.

Molar Mass

Molar mass is a crucial factor in determining the behavior of gases. It is defined as the mass of one mole of a substance, typically expressed in grams per mole (g/mol). The molar mass can affect various properties like the speed of molecules and their effusion rates. For the gases discussed:

• Ne: 20.18 g/mol
• SF6: 146.06 g/mol
• N2: 28.02 g/mol
• CH4: 16.04 g/mol

The molar mass matters because gases with lower molar masses, like CH4 (methane), tend to have higher molecular speeds and effuse faster when compared to heavier gases like SF6. In contrast, larger molar mass gases like SF6 present more considerable deviations from ideal gas laws due to stronger intermolecular interactions and larger volume occupied by individual molecules.

Root-Mean-Square Speed

The root-mean-square (rms) speed of gas molecules is an important concept in understanding gas behavior. It is defined by the equation \[ u = \sqrt{\frac{3RT}{M}}\]where:

• "u" is the root-mean-square speed
• "R" is the universal gas constant
• "T" is the temperature in Kelvin
• "M" is the molar mass in kilograms per mole

From the formula, it can be observed that rms speed is inversely related to molar mass. Thus, at constant temperature, a gas with a lower molar mass will have a higher rms speed. For our gases, CH4, with its smallest molar mass of 16.04 g/mol, claims the highest rms speed, meaning its molecules move fastest among the lineup.

Intermolecular Forces

Intermolecular forces are forces that mediate the interaction between molecules. These forces are not only crucial for understanding why substances are in gas, liquid, or solid forms at different temperatures, but also influence how closely a gas aligns with ideal behavior.

In the context of the exercise, SF6 (sulfur hexafluoride) was found to deviate most from ideal gas behavior due to these forces. Since SF6 is a polar covalent compound, it experiences stronger intermolecular forces compared to the other gases like Ne and CH4. These forces cause SF6 to occupy more space as molecules are strongly attracted to each other, leading to a more realistic assessment of its volume when compared to ideal gases. As a consequence, gases like SF6 tend to show deviations from the assumptions of Kinetic Molecular Theory.