Question

A nitrogen gas molecule at surface of the earth happens to have the rms speed for that has at exactly 0°C. If it were to go straight up without colliding with other molecules, how high will it rise

Answer 1

1) Determine the speed:

$$v_\text{rms}=\sqrt{3RT/M}=493\text{ m/s}.$$

2) How high will it rise assuming the acceleration due to gravity does not change with height?

$$h=\frac{v_\text{rms}^2}{2g}=12.4\text{ km}.$$

As we see, the height under this assumption is not that much to take the change of g with height into consideration. It's just a bit higher than the height at which aircraft fly.