Problem 79 - General Chemistry: Principles and Modern Applications

Question

Calculate the average kinetic energy, \(\bar{e}_{k},\) for \(\mathrm{O}_{2}(\mathrm{g})\) at \(298 \mathrm{K}\) and \(1.00 \mathrm{atm}\)

Step-by-Step Solution

Answer

The average kinetic energy per oxygen molecule, \( \bar{e}_k \), at 298K and 1.00atm is approximately \(6.2 \times 10^{-21} \; \text{J}\).

Step 1: Identify the Given Variables.

The given temperature (T) is 298K. The Boltzmann constant (k) is a known constant that equals \(1.38 \times 10^{-23} \; \text{J/K}\).

Step 2: Insert the Given Values into the Formula.

Put the values of \(k\) and \(T\) into the formula \( \bar{e}_k = \frac{3}{2} \times k \times T \).

Step 3: Calculate the Average Kinetic Energy.

Now perform the multiplication to calculate the average kinetic energy \( \bar{e}_k = \frac{3}{2} \times (1.38 \times 10^{-23} \; \text{J/K}) \times 298\; \text{K}\).

Step 4: Simplify the Expression.

The result will be the numerical value for \( \bar{e}_k \), which is the average kinetic energy per molecule.

Key Concepts

Oxygen Gas

Oxygen gas, represented by the chemical formula \( \mathrm{O}_2 \), is a diatomic molecule composed of two oxygen atoms bonded together. It is a vital component of the Earth's atmosphere and is essential for the survival of aerobic organisms, including humans.

Some key characteristics of oxygen gas include:

• Colorless and odorless in its natural state.
• Approximately 21% of the Earth's atmosphere, making it the third most abundant element.
• Supports combustion but is not flammable itself.
• Used in various industrial processes, such as steel-making and water treatment.

Oxygen gas at a temperature of 298 K (approximately 25°C) and a pressure of 1 atm is considered to be in standard conditions, which are often used for scientific calculations to ensure consistency and accuracy.

Temperature

Temperature is a measure of the average kinetic energy of the particles in a substance. It is an essential factor in determining the energy state of a system and influences the behavior of gases like oxygen.

Some key points about temperature include:

• Measured in units such as Kelvin (K), Celsius (°C), and Fahrenheit (°F).
• The Kelvin scale is used in scientific calculations because it starts from absolute zero, the theoretical point where molecular motion stops.
• It is directly related to kinetic energy, meaning that as the temperature increases, so does the kinetic energy of the gas molecules.

In the context of the problem, the temperature of 298 K is crucial as it is the condition under which the average kinetic energy of the oxygen gas molecules is calculated.

Boltzmann Constant

The Boltzmann constant \( (k) \) serves as a bridge between macroscopic and microscopic physics. It is used to relate the kinetic energy of particles to temperature. The constant appears frequently in statistical mechanics and thermodynamics.

Here are some important details about the Boltzmann constant:

• It has a value of \( 1.38 \times 10^{-23} \; \text{J/K} \).
• Essential in calculations involving gases and energy, such as determining the average kinetic energy per molecule in a gas.
• Named after the Austrian physicist Ludwig Boltzmann, a pioneer in the field of statistical mechanics.

In the calculation provided, the Boltzmann constant is used along with the temperature (298 K) to find the average kinetic energy of the oxygen gas molecules, illustrating how fundamental constants facilitate understanding complex physical phenomena.