Question

The “escape velocity” necessary for objects to leave the gravitational field of the Earth is 11.2 km s-1 . Calculate the ratio of the escape velocity to the root-mean-square speed of helium, argon, and xenon atoms at 2000 K. Does your result help explain the low abundance of the light gas helium in the atmosphere? Explain.

Answer 1

To calculate the rms speeds of helium, argon, and xenon atoms at 2000 K , we substitute the molar masses in kg mol^-1 of the three gases successively into the expression,

taking T = 2000 K and R = 8.315 J mol^-1 K^-1

Molar mass for Helium = 4.0026 x 10^?3 kg mol^-1
Molar mass for Argon = 39.948 x 10^?3 kg mol^-1
Molar mass for Xenon = 131.29 x 10^?3 kg mol^-1

We get the rms speed for the given gases as followed

Helium = 3.53 km s^-1
Argon = 1.12 km s^-1
Xenon = 0.62 km s^-1

The ratio of the escape velocity to the root-mean-square speed of the given gases

The ratio of helium = 31.5 %
The ratio of argon = 10.0 %
The ratio of xenon = 5.5 %

Yes, the fraction (or ratio) of the speed exceeding the escape velocity (31.5%) is larger in helium, which makes low abundancy of the helium in the atmosphere.