Problem 27 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Consider the following gases at SATP (Section 8.6): argon, krypton, nitrogen, methane, hydrogen chloride, chlorine, carbon dioxide, helium. (a) Which gas would be expected to most closely follow ideal behaviour? (b) Which gas would deviate most from ideal behaviour? (c) Which gas would have the highest and lowest root mean square speeds? (d) Which gas would effuse most slowty?

Step-by-Step Solution

Answer

(a) Helium; (b) Hydrogen chloride; (c) Highest: Helium, Lowest: Chlorine; (d) Chlorine.

Step 1: Understanding Ideal Gas Behavior

Ideal gases are those that perfectly abide by the ideal gas law: \(PV = nRT\). Gases behave ideally at high temperatures and low pressures as intermolecular forces become negligible. Gases with the smallest molecular size and weakest intermolecular forces follow ideal behavior closely. Argon (Ar) and helium (He) are noble gases with negligible intermolecular forces.

Step 2: Identifying Least Ideal Behavior

Gases with stronger intermolecular forces and larger molecular sizes deviate more from ideal behavior. Hydrogen chloride (HCl) and chlorine (Cl\(_2\)) have polar bonds and large molecular sizes, respectively, leading to stronger intermolecular forces compared to other gases in the list.

Step 3: Calculating Root Mean Square Speed

The root mean square speed \(\text{(rms)}\) is given by the formula \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where \(M\) is the molar mass. Gases with lower molar masses will have higher rms speeds. The lightest gas is helium (with an atomic mass of about 4), and the heaviest is chlorine (with a molar mass of about 71).

Step 4: Determining Effusion Speed

Effusion refers to the process by which gas molecules escape through a small hole. According to Graham's law, the rate of effusion is inversely proportional to the square root of the molar mass \(\left(\frac{1}{\sqrt{M}}\right)\). Heavier gases effuse slower. Chlorine, with the highest molar mass, will effuse most slowly.

Key Concepts

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion which act between neighboring particles. For gases, these forces are significant because they dictate how a gas will deviate from ideal gas behavior.

Ideal gases are assumed to have no intermolecular attractions, but real gases do experience these forces. There are several types of intermolecular forces, including:

• London dispersion forces: These are weak and are present in all molecules. They are more significant in heavier atoms.
• Dipole-dipole interactions: Occur between molecules that have permanent dipoles.
• Hydrogen bonds: A type of dipole-dipole interaction that happens when hydrogen is bonded to a highly electronegative atom.

Noble gases like helium and argon experience very weak intermolecular forces, as they are nonpolar and only have London dispersion forces. This makes them follow ideal behavior more closely.

Root Mean Square Speed

The root mean square speed is a way to express the average speed of particles in a gas. It's an important concept because it provides insight into how fast gas particles are moving at a given temperature. The formula is:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

Here, \( R \) is the universal gas constant, \( T \) is the temperature (in Kelvin), and \( M \) is the molar mass of the gas particles.

Lighter gases, such as helium, have a higher root mean square speed because their smaller mass requires more speed to have the same energy as heavier gases. Conversely, heavier gases like chlorine have a lower root mean square speed. This means that at the same temperature, a helium atom will move much faster than a chlorine molecule.

Effusion

Effusion is the process that describes how gas molecules escape through tiny openings. This process is governed by Graham's law, which shows the relationship between the rate of effusion and the mass of the particles involved.

According to Graham's law, the rate of effusion is inversely proportional to the square root of the molar mass:
\[ \text{Rate of effusion} \propto \frac{1}{\sqrt{M}} \]\

Consequently, gases with a lower molar mass will effuse more quickly than those with a higher molar mass. This is because lighter molecules travel faster and can escape more rapidly through an opening. In the given example, helium, having the lowest molar mass, would effuse the fastest, whereas chlorine, being the heaviest, would effuse the slowest.

Noble Gases

Noble gases belong to a group in the periodic table known for their lack of chemical reactivity. This is due to their full valence electron shells which make them stable and unlikely to form chemical bonds.

These gases include helium, neon, argon, krypton, xenon, and radon. Key characteristics include:

• Low Boiling and Melting Points: They exist as gases at room temperature.
• Nonpolarity: Since they do not have a permanent dipole, noble gases primarily exhibit weak London dispersion forces.
• Lack of Color, Odor, and Taste: They are often found naturally in their pure form in the Earth's atmosphere.

Owing to these properties, noble gases are often used in situations that require nonreactive atmospheres, such as in light bulbs and other electronics. Additionally, because of their weak intermolecular forces, they tend to behave closely to an ideal gas under standard conditions.