We have to compute (a) the temperature at which the rms speeds of molecular hydrogen and molecular oxygen to be equal to the speed of escape from the Earth’s surface. (b) We have to repeat the calculations for the Moon, assuming the gravitational acceleration on its surface to be 0.16 g. If the temperature high in the Earth’s upper atmosphere is about 1000 K, we have to answer whether we expect to find much hydrogen there and much oxygen there.
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Solution: Click For PDF Version The escape speed from the surface of a spherical object of mass M and radius R is given by the requirement that the magnitude of the kinetic energy and the gravitational potential energy of a particle to escape to infinity from the surface of the object should be equal. That is
or
For the Earth
Mass of a hydrogen molecule is
(a) Therefore, the temperature at which the rms speed of a hydrogen molecule will equal the escape speed from the surface of the Earth will be
As mass of an oxygen molecule is 16 times that of a hydrogen molecule, we have
Therefore, in the Earth’s upper atmosphere where the temperature is about 1000 K, the fraction of the hydrogen molecules with speed greater than the escape from the Earth will be more than the fraction of oxygen molecules with speed more than the escape speed from the Earth. (b) We repeat the above calculation for the Moon. Data of the moon are
radius of the
moon
acceleration due to gravity on the Moon’s surface
Therefore, the speed of escape from the surface of the Moon
At the Moon the temperature of the hydrogen gas for which the rms speed will be equal to the escape speed from the surface of the Moon will be
And,
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,and
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