Problem 113 - Chemistry The Central Science In Si Units

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 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?

Step-by-Step Solution

Answer

(a) SF6, (b) Ne, (c) CH4, (d) SF6, (e) All equal, (f) CH4, (g) SF6.

Step 1: Identify real gas properties

The kinetic-molecular theory assumes no attractive or repulsive forces between molecules, which is more valid for lighter and less polarizable molecules. (a) Because \( \mathrm{SF}_6 \) is a larger molecule with more electrons, it has stronger London dispersion forces and thus is most likely to deviate from ideal behavior.

Step 2: Evaluate ideal gas characteristics

An ideal gas is one that follows the kinetic-molecular theory under all conditions. Lighter and non-polar molecules are closer to ideal behavior.(b) Among the gases listed, Neon (\( \mathrm{Ne} \)) with its small, non-polar, and monoatomic nature is closest to an ideal gas.

Step 3: Calculate root-mean-square speed

The root-mean-square speed \( u_{rms} \) is given by the formula \( u_{rms} = \sqrt{\frac{3RT}{M}} \), where \( M \) is the molar mass of the gas. The gas with the smallest molar mass will have the highest \( u_{rms} \).(c) Methane (\( \mathrm{CH_4} \)) has the lowest molar mass (16 g/mol) among the options, giving it the highest root-mean-square speed.

Step 4: Total molecular volume

The molar volume of the gas can be influenced by the size of the molecules.(d) Sulfur hexafluoride (\( \mathrm{SF_6} \)) has the largest and the most complex structure, resulting in the highest total molecular volume relative to the space it occupies.

Step 5: Average kinetic energy

At a given temperature, the average kinetic energy of gas molecules is primarily determined by temperature, not molecular weight. Hence all gases at the same temperature have the same average kinetic energy. (e) All gases listed have the same average kinetic energy at a given temperature.

Step 6: Determine rate of effusion

According to Graham's law, the rate of effusion is inversely proportional to the square root of its molar mass. Therefore, lighter gases effuse more rapidly.(f) Methane (\( \mathrm{CH_4} \)), being lighter than \( \mathrm{N_2} \), will effuse more rapidly.

Step 7: Identify largest van der Waals parameter

The van der Waals \( b \) parameter accounts for the volume occupied by gas molecules, and larger molecules have more significant \( b \) values.(g) Sulfur hexafluoride (\( \mathrm{SF_6} \)), due to its large molecular size, has the largest van der Waals \( b \) parameter among the listed gases.

Key Concepts

Real Gas Behavior

In the realm of gases, not all follow the same rules under every condition. Real gases sometimes deviate from the idealized behavior described by the kinetic-molecular theory. This theory suggests that molecules have no forces of attraction or repulsion between them. These assumptions work well for small, light molecules under low pressure and high temperature.
For instance, gases like sulfur hexafluoride (\( \mathrm{SF_6} \)), which have larger molecules and more electrons, experience stronger London dispersion forces. As a result, they deviate more from ideal gas laws compared to small, non-polar molecules.

Ideal Gas Laws

Ideal gases are hypothetical gases that perfectly obey kinetic-molecular theory under all conditions. The ideal gas law formula is usually written as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.
Among common gaseous elements, Neon (\( \mathrm{Ne} \)) is closest to ideal behavior. It is monoatomic, small, light, and non-polar, conditions favorable for ignoring intermolecular forces.

Root-Mean-Square Speed

The root-mean-square speed (\( u_{rms} \)) is a measure of the average speed of molecules in a gas. At a given temperature, gases with lighter molecules typically have higher speeds. The \( u_{rms} \) is calculated as \( u_{rms} = \sqrt{\frac{3RT}{M}} \), where \( M \) is the molar mass.
Methane (\( \mathrm{CH_4} \)), with its relatively low molar mass, exhibits the highest root-mean-square speed among gases like \( \mathrm{SF_6} \), \( \mathrm{N_2} \), and \( \mathrm{Ne} \), at the same temperature.

Graham's Law

Graham's Law explains the effusion and diffusion of gases. It states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass: \( \text{Rate} \propto \frac{1}{\sqrt{M}} \).
In simpler terms, lighter gases effuse faster than heavier ones. For example, methane (\( \mathrm{CH_4} \)) will effuse more quickly than nitrogen (\( \mathrm{N_2} \)) because it has a lower molar mass.

Van der Waals Equation

The van der Waals equation is an adjusted version of the ideal gas law, accounting for the volume of gas particles and the attractive forces between them: \[ \left(P + \frac{an^2}{V^2}\right)(V-nb) = nRT \]
Here, \( a \) corrects for intermolecular attractions and \( b \) accounts for the volume occupied by the gas molecules. Large gases like sulfur hexafluoride (\( \mathrm{SF_6} \)) have greater \( b \) values due to their significant molecular size. Thus, \( \mathrm{SF_6} \) commonly shows the largest deviation from ideal behavior as described by van der Waals parameters.