At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47° C ?

Solution

Physics sir · 4 steps

Equal rms speed means equal T/M, so the light hydrogen needs a much lower temperature.

1. Rms speed formula. For a gas of molar mass M at absolute temperature T, vᵣₘₛ = √(3RT/M). Two gases have the same vᵣₘₛ only when T/M is the same for both.

   T_(H₂)/M_(H₂) = T_(O₂)/M_(O₂)

2. Convert to kelvin. The formula needs absolute temperature, so T_(O₂) = 47 + 273 = 320 K. The molar masses are M_(H₂) = 2 and M_(O₂) = 32 g mol⁻¹.
3. Put in numbers. T_(H₂) = 320 × 2/32 = 320/16 = 20 K.
4. So. Hydrogen at 20 K moves as fast on average as oxygen at 47°C. That is option (a).

The idea to keep. Whenever two gases share the same rms speed, set T/M equal. Always put the temperature in kelvin first, since vᵣₘₛ depends on absolute temperature.

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