3. Distribution of Molecular Speeds

Concept Explanation:
In a gas, not all molecules move at the same speed. The distribution of molecular speeds can be described by the Maxwell-Boltzmann distribution, which shows that most molecules have speeds near the average value, but some move much faster or slower.

Key Points:
- Most Probable Speed: The speed at which the greatest number of molecules is moving.
- Average Speed: The mean speed of all the molecules in the gas.
- Root Mean Square Speed: A measure of the speed that gives the same kinetic energy as the average kinetic energy of the molecules in the gas.

Example Problem:

Calculate the root mean square speed of oxygen molecules (O₂) at room temperature (300 K). The molar mass of oxygen is 32 g/mol, and the gas constant R = 8.314 J/mol·K.

Solution:

The root mean square speed is given by the formula:
v_rms = √(3RT / M)
where R is the gas constant, T is the temperature in Kelvin, and M is the molar mass in kg/mol.

Convert the molar mass from grams to kilograms:
M = 32 g/mol = 0.032 kg/mol

Now, calculate the speed:
v_rms = √(3 × 8.314 × 300 / 0.032) = √23332.5 ≈ 483 m/s

So, the root mean square speed of oxygen molecules at 300 K is approximately 483 m/s.