Question

The “escape velocity” necessary for objects to leave the gravitational field of the Earth is 11.2...

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.

Homework Answers

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.

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