Question

Place the following gases in order of increasing average molecular speed at \(25^{\circ} \mathrm{C}: \mathrm{Ar}, \mathrm{CH}_{4}, \mathrm{N}_{2}, \mathrm{CH}_{2} \mathrm{F}_{2}\)

Short answer

Order: \( \mathrm{CH}_4 > \mathrm{N}_2 > \mathrm{Ar} > \mathrm{CH}_2 \mathrm{F}_2 \).

Step-by-step solution

Understand the Relationship

The average molecular speed of a gas is related to its molar mass and temperature according to the equation \( v = \sqrt{\frac{3RT}{M}} \), where \( v \) is the average molecular speed, \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas.

Convert Temperature to Kelvin

Convert \(25^{\circ} \mathrm{C}\) to Kelvin by adding 273.15. So, \( T = 25 + 273.15 = 298.15 \) K.

Determine Molar Masses

Calculate the molar masses of the gases: \( \mathrm{Ar} = 39.948 \ \mathrm{g/mol} \), \( \mathrm{CH}_4 = 16.04 \ \mathrm{g/mol} \), \( \mathrm{N}_2 = 28.02 \ \mathrm{g/mol} \), \( \mathrm{CH}_2 \mathrm{F}_2 = 52.02 \ \mathrm{g/mol} \).

Calculate Molecular Speed Rankings

Since molecular speed is inversely proportional to the square root of the molar mass, the gas with the smallest molar mass will have the highest speed and vice versa. Order the gases from highest to lowest speed: \( \mathrm{CH}_4 > \mathrm{N}_2 > \mathrm{Ar} > \mathrm{CH}_2 \mathrm{F}_2 \).

Verify and Conclude

Double-check the order based on the equation and molar masses. The order holds as: \( \mathrm{CH}_4 \) \( (16.04) \), \( \mathrm{N}_2 \) \( (28.02) \), \( \mathrm{Ar} \) \( (39.948) \), \( \mathrm{CH}_2 \mathrm{F}_2 \) \( (52.02) \).

Key concepts

Molar Mass

Molar mass is a crucial property in chemistry that represents the mass of one mole of a chemical substance. It's measured in grams per mole (g/mol). Understanding molar mass is essential for calculations involving gases, as it directly affects properties such as density and average molecular speed.

To find the molar mass of a compound, sum the atomic masses of the individual elements in the compound, multiplied by their respective quantities:

• For \({\rm Ar}\): It is a monoatomic element, so its molar mass is simply the atomic weight, which is 39.948 g/mol.
• For \({\rm CH}_4\): Add the atomic masses of one carbon atom (12.01 g/mol) and four hydrogen atoms (1.01 g/mol each), resulting in approximately 16.04 g/mol.
• For \({\rm N}_2\): With two nitrogen atoms, each having an atomic mass of 14.01 g/mol, the total molar mass is 28.02 g/mol.
• For \({\rm CH}_2{\rm F}_2\): Combine the molar masses of one carbon, two hydrogens, and two fluorines (19.00 g/mol each) to get 52.02 g/mol.

Knowing the molar mass allows us to calculate and compare the average speed of gas molecules under similar conditions.

Temperature Conversion

Temperature conversion is essential when dealing with scientific equations that require temperature in Kelvin. The Kelvin scale is the standard unit of temperature used in thermodynamics and provides an absolute measure of temperature.

To convert Celsius to Kelvin, use the conversion formula: \[ T(K) = T(°C) + 273.15 \]For example, to convert \(25^{\circ} \text{C}\) to Kelvin, simply add 273.15, resulting in 298.15 K. This step ensures your calculations involving gases are consistent with scientific standards, as many formulas, including the ideal gas law and molecular speed equations, require temperature in Kelvin.

Remember:

• Celsius is relative to the freezing and boiling points of water.
• Kelvin is an absolute temperature scale, starting at absolute zero where molecular motion ceases.

With this conversion, comparisons of molecular speeds use accurate temperature values.

Ideal Gas Constant

The ideal gas constant, represented by \( R \), plays a significant role in the behavior of gases and in thermodynamic equations. It provides the proportionality factor that transforms the units of gas properties in equations.

The value of \( R \) varies depending on the units used in a given context:

• \( R = 8.314 \) J/(mol·K) for energy calculations
• \( R = 0.0821 \) L·atm/(mol·K) for volume-based equations

The choice of \( R \) should match the units of other parameters like pressure or volume in your calculation.

In calculating the average molecular speed, we often use \( R = 8.314 \; \text{J/(mol·K)} \) since speed involves kinetic energy, linking energy units with temperature and molar mass of the gas. This ensures the integration of these various units into cohesive analyses.

Gas Properties

Gas properties help us understand and predict the behavior of gases. They can be influenced by factors like temperature, pressure, and molar mass. Key properties to consider include:

• Average Molecular Speed: Calculated using the formula \( v = \sqrt{\frac{3RT}{M}} \), where \( v \) is speed, \( R \) is the ideal gas constant, \( T \) is temperature in Kelvin, and \( M \) is molar mass.
• Pressure: The force exerted by gas molecules when they collide with the walls of their container, typically measured in atmospheres (atm) or Pascals (Pa).
• Volume: The space that a gas occupies, often measured in liters (L) or cubic meters (m³).
• Temperature: A measure of the average kinetic energy of gas molecules, impacting their speed and behavior.

These properties are interrelated, and understanding one can lead to insights about another. For example, knowing the molar mass and temperature of a gas can help determine its average molecular speed, allowing predictions about its behavior under different conditions.