Problem 61 - Chemistry The Molecular Science

Question

Place these gases in order of increasing average molecular speed at \(25^{\circ} \mathrm{C}: \mathrm{Kr}, \mathrm{CH}_{4}, \mathrm{~N}_{2},\) and \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\).

Step-by-Step Solution

Answer

Order: CH₂Cl₂ < Kr < N₂ < CH₄.

Step 1: Understanding Molecular Speed

The average molecular speed of gases is related to their molar mass. According to the kinetic molecular theory, at a given temperature, the average speed of gas molecules is inversely proportional to the square root of their molar mass.

Step 2: Determine the Molar Mass of Each Gas

Calculate the molar mass for each gas: - Krypton (Kr) = 83.80 g/mol - Methane (CH₄) = 16.04 g/mol - Nitrogen (N₂) = 28.02 g/mol - Dichloromethane (CH₂Cl₂) = 84.93 g/mol.

Step 3: Use the Formula for Average Speed

The formula for average molecular speed is given by \[ v = \sqrt{\frac{3RT}{M}} \] where \( v \) is the molecular speed, \( R \) is the constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass. Since all gases are at the same temperature, only the molar mass affects their relative speeds.

Step 4: Rank Based on Molar Mass

The gases should be ranked from highest to lowest molecular speed, which corresponds to lowest to highest molar mass. - CH₄: 16.04 g/mol - N₂: 28.02 g/mol - Kr: 83.80 g/mol - CH₂Cl₂: 84.93 g/mol.

Step 5: Write the Order of Increasing Molecular Speed

The order of increasing molecular speed, from slowest to fastest, is CH₂Cl₂, Kr, N₂, CH₄.

Key Concepts

Kinetic Molecular Theory

The kinetic molecular theory provides a framework for understanding the behavior of gas molecules. According to this theory, gas molecules are in continuous motion, and this motion contributes to observable properties such as pressure and temperature. One key aspect of this theory is that the average speed of gas molecules is related to their kinetic energy.
This means that at a constant temperature, the kinetic energy, and therefore speed, varies based on the molar mass of the gas molecules. More precisely, the average speed of gas molecules at a given temperature is inversely proportional to the square root of their molar mass. This means that lighter molecules will move faster than heavier ones, assuming the temperature is the same.
Understanding this concept helps us predict how different gases will behave when exposed to the same conditions, highlighting the role of molecular mass in dictating the physical properties of gases. This relationship is particularly useful when ranking molecules based on speed, as demonstrated in the exercise where the lighter methane molecules move faster than heavier dichloromethane molecules.

Molar Mass Calculation

Calculating the molar mass of a substance is a fundamental skill in chemistry, crucial for understanding various properties of gases. The molar mass of a substance is the mass of one mole of its particles, usually measured in grams per mole (g/mol). To find the molar mass:

• Add the atomic masses of all the atoms present in a molecule. These values can be found on the periodic table.
• For molecules like methane (CH₄), add the masses from one carbon atom (approximately 12.01 g/mol) and four hydrogen atoms (about 1.01 g/mol each), resulting in a total molar mass of about 16.04 g/mol.
• Each type of gas has a distinct molar mass. For example, krypton (Kr) has a molar mass of around 83.80 g/mol, whereas dichloromethane (CH₂Cl₂) has approximately 84.93 g/mol.

This method allows us to compare different gases and predict their physical behaviors, as in the exercise where knowing the molar mass was essential to ranking the gases by their molecular speeds. The molar mass directly influences properties like diffusion rates and speeds of molecules under constant temperatures.

Gas Molecules

Gas molecules exhibit unique behaviors that make them a central topic of study in chemistry. These molecules occupy space, have mass, and are in constant, random motion. In gases, the molecules are far apart compared to liquids and solids, making their density much lower.
The freedom of movement in gas molecules means they readily expand to fill containers. They also exert pressure on the walls of their containers as they collide with them. The kinetic molecular theory explains that the pressure is due to the movement and collisions of these molecules. The faster they move, the more frequent and forceful the collisions.
Several factors affect the behavior of gas molecules, including:

• Temperature: Higher temperatures increase molecular speeds and energy.
• Molar Mass: As we discussed, lighter molecules move more quickly than heavier ones.
• Volume: Gas molecules will spread out to fill the available volume.

Understanding these principles helps us comprehend and predict how gases will respond under various conditions, such as changes in temperature or pressure.