Oxygen (O2) has a molar mass of 32.0 g>mol. What is (a) the average translational kinetic energy of an oxygen molecule at a temperature of 300 K;

Verified step by step guidance

1. Start by understanding the concept of translational kinetic energy for a gas molecule. The average translational kinetic energy of a molecule in an ideal gas is given by the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub> = \frac{3}{2}kT</math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is the Boltzmann constant and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is the temperature in Kelvin.
2. Identify the given values: The temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is 300 K. The Boltzmann constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is approximately <math xmlns="http://www.w3.org/1998/Math/MathML">1.38 \times 10^{-23} \text{ J/K}</math>.
3. Substitute the given values into the formula for average translational kinetic energy: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub> = \frac{3}{2} \times 1.38 \times 10^{-23} \text{ J/K} \times 300 \text{ K}</math>.
4. Perform the multiplication and division to find the average translational kinetic energy. This involves multiplying the constants and the temperature, then dividing by 2.
5. The result from the calculation will give you the average translational kinetic energy of an oxygen molecule at 300 K in joules. Remember, this energy is per molecule, not per mole.

Key Concepts

Kinetic Theory of Gases

The kinetic theory of gases explains the behavior of gases in terms of the motion of their molecules. It states that gas molecules are in constant random motion and that the temperature of a gas is directly proportional to the average kinetic energy of its molecules. This theory is crucial for understanding how temperature affects molecular motion and energy.

Translational Kinetic Energy

Translational kinetic energy refers to the energy due to the motion of a molecule as it moves through space. For a single molecule, it is given by the formula (3/2)kT, where k is the Boltzmann constant and T is the temperature in Kelvin. This concept is essential for calculating the average kinetic energy of gas molecules at a given temperature.

Boltzmann Constant

The Boltzmann constant (k) is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It has a value of approximately 1.38 x 10^-23 J/K. This constant is used in the formula for translational kinetic energy and is key to connecting macroscopic and microscopic physical quantities.