Question

A sealed container holds the gases \(\mathrm{H}_{2}, \mathrm{~N}_{2}\), and \(\mathrm{O}_{2}\). Which of these gases corresponds to each of the following descriptions? (a) highest kinetic energy (b) lowest kinetic energy (c) fastest molecules (d) slowest molecules

Short answer

(a) All gases are the same, (b) All gases are the same, (c) \(\mathrm{H}_2\), (d) \(\mathrm{O}_2\).

Step-by-step solution

Understand the properties of kinetic energy in gases

Kinetic energy in gases is influenced by the temperature and the mass of the molecules. All gases at the same temperature have the same average kinetic energy. Hence, the kinetic energy does not depend on the type of gas but rather the temperature.

Kinetic energy at a constant temperature

Since all gases in the container are at the same temperature, they all have the same average kinetic energy due to the equation \( KE = \frac{3}{2} kT \), where \( k \) is the Boltzmann constant and \( T \) is the temperature. Therefore, all gases have the same kinetic energy, meaning (a) and (b) cannot be differentiated at the same temperature.

Examine molecular speed of the gases

The molecular speed of a gas is given by the formula \( v = \sqrt{\frac{3kT}{m}} \), where \( m \) is the molar mass. Lighter gases move faster at the same temperature. Compare the molar masses: \( \mathrm{H}_2 = 2 \text{ g/mol} \), \( \mathrm{O}_2 = 32 \text{ g/mol} \), \( \mathrm{N}_2 = 28 \text{ g/mol} \).

Determine the fastest and slowest molecules

Since \( \mathrm{H}_2 \) has the lowest molar mass, it will have the highest speed and hence is the fastest. \( \mathrm{O}_2 \) has the highest molar mass and will have the lowest speed, making it the slowest.

Key concepts

Molecular Speed

Molecular speed refers to how fast the molecules of a gas move. This speed is primarily determined by the temperature and the molar mass of the gas. At a constant temperature, every molecule in a gas will have the same average kinetic energy. However, the speed of individual gas molecules can vary based on their mass.
For gases at the same temperature, lighter molecules will generally move faster than heavier ones. This relationship is captured in the formula:

• \( v = \sqrt{\frac{3kT}{m}} \), where
  • \( v \) is the molecular speed
  • \( k \) is the Boltzmann constant
  • \( T \) is the absolute temperature
  • \( m \) is the molar mass of the gas

Using this formula, we see that hydrogen molecules (\( \mathrm{H}_2 \)) move faster than oxygen (\( \mathrm{O}_2 \)) and nitrogen (\( \mathrm{N}_2 \)) molecules, owing to their lower molar mass.

Molar Mass

Molar mass is a key concept that helps determine the characteristics of a gas, such as how fast its molecules will move at a given temperature. It is defined as the mass of one mole of a substance, typically measured in grams per mole (g/mol).
In the context of this exercise, understanding the molar mass of each gas type helps explain their relative molecular speeds:

• Hydrogen (\( \mathrm{H}_2 \)): 2 g/mol
• Nitrogen (\( \mathrm{N}_2 \)): 28 g/mol
• Oxygen (\( \mathrm{O}_2 \)): 32 g/mol

A larger molar mass means the gas molecules are heavier, making them move more slowly compared to lighter gas molecules at the same temperature. This is why hydrogen, with the smallest molar mass, moves fastest, while oxygen, with the largest, moves slowest.

Gas Properties

Gas properties like pressure, volume, temperature, and kinetic energy are fundamental topics in chemistry. The behavior of gases, encapsulated by the ideal gas law, is essential in understanding these properties.

• Kinetic energy, for example, is the energy a gas molecule possesses due to its motion and is described by the equation
  \( KE = \frac{3}{2}kT \).
• Temperature directly relates to kinetic energy: higher temperatures typically lead to higher energy and faster-moving molecules.
• Pressure is generated by collisions of gas molecules against the walls of their container. At an equal temperature, gases exert the same pressure given equal volumes.

While all gases at the same temperature will have the same kinetic energy, due to their different molar masses, their speed and behaviors can vary greatly, influencing how they interact with their environment. Understanding these properties offers invaluable insight into gas behavior.