Oxygen molecules \(\left(\mathrm{O}_{2}\right)\) are 16 times as massive as hydrogen molecules \(\left(\mathrm{H}_{2}\right)\). Carbon dioxide molecules \(\left(\mathrm{CO}_{2}\right)\) are 22 times as massive as \(\mathrm{H}_{2}\) a. Compare the average speed of \(\mathrm{O}_{2}\) and \(\mathrm{CO}_{2}\) molecules in a volume of air. b. Does this ratio of the speeds in part (a) depend on air temperature?

Gas molecules move at different speeds depending on their masses. The average speed of a gas molecule is inversely proportional to the square root of its molar mass (\text{M}). This relationship is shown by the equation: \(v \backsim \frac{1}{\root{2}{\text{M}}}\). The lighter the molecule, the faster it moves on average. For instance, hydrogen molecules (\text{H}_2) are very light compared to oxygen (\text{O}_2) and carbon dioxide (\text{CO}_2). As a result, hydrogen molecules will have a higher average speed. In the exercise, oxygen is 16 times heavier than hydrogen and carbon dioxide is 22 times heavier. So, to compare the speeds, you would use the mass in this inverse relationship.

The speed of gas molecules is also affected by temperature. As the temperature increases, molecules move faster. This relationship is given by the equation: \(v \backsim \root{2}{\text{T}}\), where \(T\) represents the temperature. When comparing gas molecules at the same temperature, the temperature factor cancels out because both gases are exposed to the same thermal energy. Therefore, while the actual speed of the molecules will change with temperature, the ratio of their speeds stays constant if the mass ratio is considered the same. In our exercise, the speed ratio between \(\text{CO}_2\) and \(\text{O}_2\) will remain unaffected by temperature differences.

The kinetic theory of gases explains how gas particles move and interact. It assumes that gas molecules are in constant, random motion and that the pressure and temperature of a gas are a result of collisions between molecules and the walls of the container. According to this theory:

• Molecules have negligible volume compared to the distance between them.
• No intermolecular forces act between them, except during collisions.
• Collisions are perfectly elastic, meaning no kinetic energy is lost.
• The average kinetic energy of gas molecules is directly proportional to the temperature of the gas in Kelvin.

These principles explain why lighter molecules move faster and how temperature affects their speed, aligning with our observations in the given exercise. Understanding this theory helps to predict and explain behavior such as diffusion and pressure changes in gases.