By what factor will the rms speed of gas molecules increase if the temperature is increased from 30 ∘C to 130 ∘C?

Question

By what factor will the rms speed of gas molecules increase if the temperature is increased from 30 ∘C∘C to 130 ∘C∘C?

Answer 1

The rms speed (v_rms) of gas molecules is related to the temperature (T, in ^oK) of the gas by:

v_rms = √((3 × R × T)/M )

...where R is the ideal gas constant, and M is the molar mass of the gas.

For the initial temperature T₀, v₀ = √((3 × R × T₀)/M ), and for the final temperature T₁, v₁ = √((3 × R × T₁)/M )

The factor of the rms speeds increases by the ratio of the temperatures in ^oK:

v₁/v₀ = √(T₁/T₀)

... as the common factor 3 × R/M cancels out.

The temperatures are given in ^oC, so after converting to ⁰K:

v₁/v₀ = √((130 + 273 )/( 30 +273 )) = 1.15