The root mean square speed of molecules of a given mass of a gas at 27°C and 1 atmosphere pressure is 200 ms⁻¹. The root mean square speed of molecules of the gas at 127°C and 2 atmosphere pressure is x/√3 ms⁻¹. The value of x will be _________.

Solution

Physics sir · 4 steps

RMS speed depends only on temperature, so ignore the pressure change and scale by the square root of T.

1. What vrms depends on. For a given gas, vᵣₘₛ = √(3RT/M). The molar mass M is fixed, so the speed depends only on the absolute temperature T. Pressure does not appear, so doubling it changes nothing here.
2. Convert to kelvin. T₁ = 27 + 273 = 300 K and T₂ = 127 + 273 = 400 K.

   v₂/v₁ = √(T₂/T₁) = √(400/300) = 2/√3

3. Put in numbers. v₂ = 200 × 2/√3 = 400/√3 m s⁻¹.
4. So. Comparing with x/√3 gives x = 400.

The common slip: 231. Reporting the speed itself, 400/√3 ≈ 231, gives 231. The question asks for x, the numerator, which is 400.

The idea to keep. For a fixed gas, vᵣₘₛ ∝ √T with T in kelvin. A pressure change in the question is a distractor unless the temperature also changes with it.

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