Question

Compute the temperature at which the rms speed is equal to the speed of escape from the surface of the earth for hydrogen and for oxygen. The temperature of the upper atmosphere is about 1000 K. Would you expect to find a lot of hydrogen there or a lot of oxygen.

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$v_{escape}=\sqrt{\frac{2GM}{R}}=\sqrt{2gR}=\sqrt{2\times 9.8\times 6400\times {10}^3}=1.12\times {10}^{4}m{s}^{-1}{v}_{rms}=\sqrt{\frac{3RT}{M}}orT=\frac{M{v}_{rms}^2}{3R}\therefore \text{For hydrogen}T=\frac{2\times {10}^{-3}\times (1.12\times {10}^4{)}^2}{3\times 8.3}=1.01\times {10}^{4}K\text{For oxygen}T=\frac{32\times {10}^{-3}\times (1.12\times {10}^4{)}^2}{3\times 8.3}=16.2\times {10}^{4}K$
Since the actual temperature which is 1000 K is closed to $1.01\times {10}^4$, we expect less hydrogen and more oxygen in the upper atmosphere.