Question

Place the following gases in order from lowest to highest average molecular speed at \(25^{\circ} \mathrm{C}: \mathrm{He}, \mathrm{Ar}, \mathrm{O}_{2},\) \(\mathrm{I}_{2}\).

Short answer

Order: \( \text{I}_2 \), \( \text{Ar} \), \( \text{O}_2 \), \( \text{He} \)

Step-by-step solution

Understand the Concept

The average molecular speed of a gas is inversely related to its molar mass by the formula \( v = \sqrt{\frac{3RT}{M}} \), where \( v \) is the average speed, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( M \) is molar mass of the gas. Therefore, gases with lower molar masses have higher average speeds.

List the Molar Mass of Each Gas

Determine the molar mass of each gas:- \( \text{He} \): 4 g/mol- \( \text{Ar} \): 40 g/mol- \( \text{O}_2 \): 32 g/mol- \( \text{I}_2 \): 254 g/mol

Order the Gases by Molar Mass

Order the gases from lowest molar mass to highest, which gives:1. \( \text{He} \)2. \( \text{O}_2 \)3. \( \text{Ar} \)4. \( \text{I}_2 \)

Determine the Order of Average Molecular Speed

Since gases with lower molar masses have higher speeds, the order of increasing average molecular speed is the reverse of the order of increasing molar mass. Thus, the gases in order of increasing average molecular speed are:1. \( \text{I}_2 \)2. \( \text{Ar} \)3. \( \text{O}_2 \)4. \( \text{He} \)

Key concepts

Average Molecular Speed

The concept of average molecular speed is a central part of the Kinetic Molecular Theory, which explains how particles move in gases. Average molecular speed gives us an idea of how fast gas molecules are moving. It depends on factors like temperature and the gas's molar mass. A key formula illustrating this is \[ v = \sqrt{\frac{3RT}{M}} \] where:

• \( v \) is the average speed
• \( R \) is the universal gas constant
• \( T \) is the temperature measured in Kelvin
• \( M \) is the molar mass of the gas

This formula tells us that gases with a lower molar mass will have a higher average speed at a given temperature. For example, helium atoms, having a relatively low molar mass, move faster on average compared to heavier gases like iodine.

Molar Mass

Molar mass is an essential property of substances that influences their behavior in various chemical reactions. It's the mass of one mole of a substance and is written in grams per mole (g/mol). For gases, molar mass significantly impacts their average speed, as detailed in the equation for average molecular speed. Knowing the molar mass helps us determine how different gases will behave under similar conditions, such as temperature and pressure.
For the gases in the exercise:

• Helium (He) has a molar mass of 4 g/mol
• Oxygen (O\(_2\)) has a molar mass of 32 g/mol
• Argon (Ar) has a molar mass of 40 g/mol
• Iodine (I\(_2\)) has a molar mass of 254 g/mol

Knowing the molar mass, we can predict and compare the speed at which different gas molecules will move. The lighter helium molecules zip around faster compared to the much weightier iodine molecules, which are slower due to their larger molar mass.

Gas Laws

Gas laws are a set of relationships describing the behavior of gases. These laws include Boyle's Law, Charles's Law, and the Ideal Gas Law, among others. They provide insight into how gases respond to changes in pressure, volume, and temperature.
In the context of average molecular speed, the Ideal Gas Law connects these variables with the equation \( PV = nRT \), where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the gas constant
• \( T \) is the temperature in Kelvin

This law, combined with the equation for molecular speed, helps us understand that while temperature increases linearly with molecular speed, molar mass inversely affects how fast these molecules move. This is why lighter gases like helium move more swiftly than heavier ones such as iodine, when compared under similar conditions.