Question

Consider the following gases, all at STP: Ne, SF \(_{6}, \mathrm{N}_{2}, \mathrm{CH}_{4}\) . (a) Which gas is most likely to depart from the assumption of the kinetic-molecular theory that says there are no attractive or repulsive forces between molecules? (b) Which one is closest to an ideal gas in its behavior? (c) Which one has the highest root-mean-square molecular speed at a given temperature? (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 \(\mathrm{N}_{2} ?\) (g) Which one would have the largest van der Waals \(b\) parameter?

Short answer

(a) SF\(_{6}\) is most likely to depart from the assumption of no attractive or repulsive forces between molecules. (b) Ne is closest to an ideal gas in its behavior. (c) Ne has the highest root-mean-square molecular speed at a given temperature. (d) SF\(_{6}\) has the highest total molecular volume relative to the space occupied by the gas. (e) All gases have the same average kinetic energy at the same temperature. (f) Ne would effuse more rapidly than N\(_{2}\). (g) SF\(_{6}\) would have the largest van der Waals b parameter.

Step-by-step solution

a. Analysis of Departure from Kinetic-Molecular Theory

Kinetic-molecular theory assumes that there are no attractive or repulsive forces between molecules. The gas that is most likely to depart from this assumption will be the one with the highest molecular weight and larger electronic cloud. In this case, that is SF\(_{6}\).

b. Analysis of Ideal Gas Behavior

Ideal gases are assumed to have no forces between molecules and occupy no volume. The gas closest to this behavior would be the one with the smallest molecular size, and that interacts less with other molecules. In this case, the noble gas Ne would be the closest to an ideal gas.

c. Analysis of Root-Mean-Square Molecular Speed

Root-mean-square molecular speed is given by the formula \(v_{rms}=\sqrt{\frac{3RT}{M}}\), where R is the ideal gas constant, T is the temperature, and M is the molar mass. At a given temperature, the gas with the smallest molecular weight will have the highest root-mean-square molecular speed. Thus, Ne has the highest root-mean-square molecular speed.

d. Analysis of Total Molecular Volume

The total molecular volume relative to the space occupied by the gas is directly related to the size of the molecules. In this case, SF\(_{6}\) has the largest molecular size, so it would have the highest total molecular volume relative to the space occupied by the gas.

e. Analysis of Average Kinetic Energy

The average kinetic energy of a gas is given by the formula \(\bar{E}_K = \frac{3}{2}RT\), where R is the ideal gas constant and T is the temperature. Note that average kinetic energy does not depend on the molecular mass of the gas. So, at the same temperature, all these gases will have the same average kinetic energy.

f. Analysis of Effusion Rate

The effusion rate of a gas is inversely proportional to the square root of its molecular weight. Therefore, the gas with the smallest molecular weight will effuse more rapidly than N\(_{2}\). In this case, the lighter gas is Ne, so it would effuse more rapidly than N\(_{2}\).

g. Analysis of van der Waals Parameter b

The van der Waals parameter b is a measure of the size of the gas molecules and their repulsive forces. The gas with the largest molecules and the strongest repulsive forces should have the highest van der Waals b parameter. Considering the options, SF\(_{6}\) has the largest molecular size, and therefore it would have the largest van der Waals b parameter.

Key concepts

Effusion Rate

Effusion refers to the process by which gas particles escape through a tiny opening into a vacuum. It's a fascinating aspect of the kinetic-molecular theory, where the effusion rate depends heavily on the molecular weight of the gas. According to Graham's Law of Effusion, the effusion rate ( \[ R \]) is inversely proportional to the square root of the molar mass ( \[ M \]) of the gas:

• Effusion Rate Formula: \[ R \propto \frac{1}{\sqrt{M}} \]

This means that lighter gases effuse faster than heavier ones. Let's consider the examples: among Ne (Neon), SF _{6} (Sulfur Hexafluoride), \( \mathrm{N}_{2} \) (Nitrogen), and \( \mathrm{CH}_{4} \) (Methane), Neon is the lightest. This gives it the edge in effusion rate, making it effuse more rapidly than \( \mathrm{N}_{2} \). Understanding the effusion rate is essential, especially in industrial processes where gas separation is required.

Ideal Gas Behavior

The concept of an ideal gas is central to understanding gas behavior and thermodynamics. An ideal gas perfectly follows the assumptions of the kinetic-molecular theory:

• No volume is occupied by gas particles.
• No intermolecular forces exist among particles.
• The collisions between particles are perfectly elastic.

Real gases, however, deviate from these conditions at high pressure and low temperature. Among the given gases (Ne , \(SF _{6} \), \( \mathrm{N}_{2} \), \( \mathrm{CH}_{4} \)), Neon, a noble gas, is nearest to exhibiting ideal behavior. Its monatomic structure and small size mean minimal interaction and volume, thus adhering closest to ideal gas laws. The simplicity of the Ne atom allows for modeling real-world scenarios, offering practical understanding while simplifying calculations for scientists and engineers.

Kinetic Energy

In the realm of gases, kinetic energy remains a cornerstone concept of kinetic-molecular theory. The average kinetic energy ( \(\bar{E}_K \)) in a gas is defined by the equation:

• Average Kinetic Energy: \[ \bar{E}_K = \frac{3}{2}RT \]

Here, \(R\) is the ideal gas constant, and \(T\) is the temperature. A critical insight is that, at a given temperature, the average kinetic energy is the same for all gases, regardless of molar mass. This principle is apparent when considering gases like Ne , \( \mathrm{SF}_{6} \), \( \mathrm{N}_{2} \), and \( \mathrm{CH}_{4} \) at Standard Temperature and Pressure (STP). Despite molecular differences, the average kinetic energy remains unchanged, reflecting how temperature solely influences the kinetic energy in gaseous matter. This uniformity in kinetic energy underpins much of thermodynamic behavior in gases.