For the same gas, the ratio of vᵣₘₛ at temperatures T and 4T is:

Solution

Physics sir · 3 steps

Same gas, so only temperature changes, and vᵣₘₛ grows as the square root of T.

1. Start from the formula. For an ideal gas of molar mass M, the rms speed depends on temperature and molar mass.

   vᵣₘₛ = √(3RT/M)

2. Same gas. R and M do not change, so vᵣₘₛ ∝ √T. The temperature goes up by a factor of 4, so the speed goes up by a factor of √4 = 2.
3. So. vᵣₘₛ(T)/vᵣₘₛ(4T) = √(T/4T) = 1/2, so the ratio is 1 : 2, option (a).

The idea to keep. For a fixed gas, vᵣₘₛ ∝ √T. Multiply the temperature by n and the rms speed changes by √n, not by n.

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