The root mean square velocity of molecules of gas is

Solution

Physics sir · 3 steps

Kinetic energy grows with T, so speed grows only with √T: that square root is the whole story.

1. Start from energy. Kinetic theory says the average kinetic energy of a molecule is proportional to temperature: 1/2mv̅²̅ = 3/2k_BT. So the mean of v², not v itself, goes as T.

   vᵣₘₛ = √v̅²̅ = √(3k_BT/m) = √(3RT/M)

2. Read the dependence. For a given gas, m (or M) is fixed, so vᵣₘₛ ∝ √T. For example, making T four times larger only doubles vᵣₘₛ.
3. So. The rms velocity is proportional to the square root of temperature, which is option (d).

The idea to keep. Temperature measures kinetic energy, and kinetic energy goes as v². So every molecular speed (rms, average, most probable) goes as √(T/M).

Answer from the NTA final answer key. Solution by PrepWiser.