According to the kinetic molecular theory, how does the mean molecular speed of water molecules change as the temperature increases?

• A. The mean molecular speed remains constant as temperature increases.
• B. The mean molecular speed increases, but not as rapidly as the temperature; it is proportional to the square root of temperature.
• C. The mean molecular speed increases at the same rate as the temperature; it is directly proportional to temperature.
• D. The mean molecular speed decreases as temperature increases.

Answer: (B)

Verified step by step guidance

1. Recall that the kinetic molecular theory relates the temperature of a gas to the average kinetic energy of its molecules. The average kinetic energy is proportional to the absolute temperature (in Kelvin).
2. Understand that the mean molecular speed (often denoted as \( v \)) is related to the kinetic energy by the equation \( \frac{1}{2} m v^2 = \frac{3}{2} k_B T \), where \( m \) is the mass of a molecule, \( k_B \) is Boltzmann's constant, and \( T \) is the temperature.
3. Rearrange the equation to solve for the mean molecular speed \( v \): \[ v = \sqrt{\frac{3 k_B T}{m}} \]. This shows that the speed depends on the square root of the temperature.
4. Interpret this relationship: as temperature \( T \) increases, the mean molecular speed \( v \) increases proportionally to \( \sqrt{T} \), meaning it increases but not as fast as the temperature itself.
5. Conclude that the correct understanding is that the mean molecular speed increases with temperature, but it is proportional to the square root of the temperature, not directly proportional or constant.