Concept explainers
(a)
The escape velocity for an object on Earth.
(a)
Answer to Problem 112P
The escape velocity is
Explanation of Solution
Write the expression for conservation of energy
Here,
Write the expression for kinetic energy of a particle
Here,
Write the expression for gravitational potential
Here,
At infinite distance both kinetic and potential energies are considered zero.
Substitute
Rearrange for
Substitute
Thus, the escape velocity of an object is
(b)
The average speed of hydrogen atom at
(b)
Answer to Problem 112P
The average speed of hydrogen atom is
Explanation of Solution
Write the expression for average speed of an atom
Here,
Substitute
Thus, the average speed of hydrogen is
(c)
The average speed of oxygen molecule at
(c)
Answer to Problem 112P
The average speed of oxygen molecule is
Explanation of Solution
Substitute
Thus, the average speed of oxygen molecule is
(d)
The reason for more oxygen than hydrogen on the Earth’s atmosphere.
(d)
Answer to Problem 112P
The average speed of hydrogen is greater for hydrogen.
Explanation of Solution
Particles with velocity lesser than that of the escape velocity cannot escape Earth’s atmosphere.
From the above results, the escape velocity is about six times the average speed of the hydrogen atom. Thus, a small fraction of hydrogen atoms might possess velocity greater than the required escape velocity to cross the Earth’s atmosphere.
The average speed of oxygen molecules is lesser than that of hydrogen and hence only a smaller fraction will possess the required speed to escape Earth’s gravitational pull. Thus, oxygen is found more than hydrogen in the Earth’s atmosphere.
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