Question

Consider the following gases, all at STP: \(\mathrm{Ne}, \mathrm{SF}_{6}, \mathrm{~N}_{2}\) \(\mathrm{CH}_{4}\). (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

In summary, SF6 deviates most from assumption 3 and has the highest molecular volume relative to the space occupied. Ne is closest to an ideal gas, has the highest root-mean-square molecular speed, and would effuse more rapidly than N2. All gases at STP have equal average kinetic molecular energies.

Step-by-step solution

SF6 is likely to deviate the most from assumption 3, as it is a large and complex molecule with a higher molecular volume, making its individual molecular volume more significant compared to the others. Consequently, the volume of individual SF6 molecules cannot be considered negligible compared to the total gas volume. #b) Closest to an Ideal Gas#

Ne (a noble gas) is closest to an ideal gas in terms of behavior. This is because it is monatomic and has weak dispersion forces due to the absence of permanent electric dipoles and its small atomic size, making its intermolecular forces less significant. #c) Highest Root-Mean-Square Molecular Speed#

Based on the formula \(v_{rms} = \sqrt{\frac{3RT}{M}}\), we can see that the root-mean-square speed is inversely proportional to the square root of the molar mass of the gas. Therefore, among the given gases, Ne, having the lowest molar mass, would possess the highest root-mean-square molecular speed. #d) Highest Molecular Volume Relative to the Space Occupied#

SF6 has the highest molecular volume relative to the space occupied by the gas. This is due to the larger molecular size and complexity, resulting in a greater individual molecular volume compared to the other gases. #e) Highest Average Kinetic Molecular Energy#

The average kinetic molecular energy is given by \(\frac{3}{2}k_BT\), which does not depend on the mass or the properties of the gas molecules. Since all the gases are at STP (i.e., the same temperature), their average kinetic molecular energies will be equal. #f) Effusion More Rapidly than \(\mathrm{N}_{2}\)#

According to Graham's law of effusion, the rate of effusion for a gas is inversely proportional to the square root of its molar mass. Thus, among the given gases, the one with the lowest molar mass will effuse more rapidly. In this case, Ne has the lowest molar mass and would effuse more rapidly than N2.

Key concepts

Ideal Gas Behavior

Ideal gas behavior is a hypothetical concept that simplifies the complex interactions within a gas for easy analysis. According to this concept, a few basic assumptions are made which include: the gas particles are in constant, random motion; there are no intermolecular forces apart from during collisions; the collisions are perfectly elastic, meaning no energy is lost; and the volume of the gas particles themselves is negligible compared to the space the gas occupies.

Real gases can approximate this behavior under certain conditions, typically at high temperatures and low pressures where the particles are farther apart and the effects of intermolecular forces are minimized. Noble gases, such as Neon (Ne), come closest to this ideal as they are monatomic and exhibit minimal attraction or repulsion between particles. This is why, in the exercise, Ne is regarded as the closest to an ideal gas among the given examples.

Root-Mean-Square Molecular Speed

The root-mean-square (rms) molecular speed is a measure of the speed of particles within a gas. It represents an average speed that takes into account the different velocities of all the particles. The rms speed is important because it helps us understand how fast the particles of a gas are moving on average, which affects properties like diffusion, effusion, and reaction rates.

The formula for the rms speed is \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where \(R\) is the gas constant, \(T\) is the temperature (in Kelvin), and \(M\) is the molar mass of the gas. Since \(T\) and \(R\) are constants in this scenario, the determining factor for rms speed is the molar mass: the lower the molar mass, the higher the rms speed. Therefore, amongst the gases mentioned, Ne with the lowest molar mass possesses the highest rms speed.

Molecular Volume

Molecular volume refers to the space that a single molecule of a substance occupies, an important factor when comparing gases. While ideal gas behavior assumes that molecular volume is negligible relative to the container volume, in reality, the size and shape of gas molecules do affect how a gas behaves, especially under high pressures or low temperatures.

Larger and more complex molecules like SF6 have a significant individual molecular volume, influencing properties like flow and compression. In the mentioned exercise, SF6 has the largest molecular volume, which accounts for its deviation from ideal gas behavior since it makes assumption 3 (the point about negligible volume) of the kinetic molecular theory less valid.

Average Kinetic Energy

The average kinetic energy of gas molecules is directly related to their temperature. According to the kinetic molecular theory, it's expressed by the equation \(\text{average kinetic energy} = \frac{3}{2}k_BT\), with \(k_B\) representing the Boltzmann constant and \(T\) the temperature. This equation indicates that, at a given temperature, all gas molecules have the same average kinetic energy, regardless of mass or size.

Since the exercise specifies that all gases are at the same temperature (STP - standard temperature and pressure), they all share equal average kinetic energy. This concept underlies the principle that temperature is a measure of the average energy of molecular motion in a gas.

Graham's Law of Effusion

Graham's law of effusion states that the rate at which a gas effuses (escapes through a tiny opening) is inversely proportional to the square root of its molar mass. Mathematically, it's expressed as \(\text{rate of effusion} \propto \frac{1}{\sqrt{M}}\). This implies that lighter gases effuse more quickly than heavier ones, due to their higher average speeds.

In comparison to \(\mathrm{N}_2\), Ne would effuse more rapidly because it has a lower molar mass. Knowing this law helps us predict and compare rates of effusion for gases, which can be crucial in industrial processes and in understanding the behavior of gases in different environments.