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At what temperature does the rms speed of (a) H2
Chapter , Problem 39(choose chapter or problem)
At what temperature does the rms speed of
(a) \(\mathrm{H}_{2}\) (molecular hydrogen) and
(b) \(\mathrm{O}_{2}\) (molecular oxygen) equal the escape speed from Earth (Table 13-2)? At what temperature does the rms speed of
(c) \(\mathrm{H}_{2}\) and
(d) \(\mathrm{O}_{2}\) equal the escape speed from the Moon (where the gravitational acceleration at the surface has magnitude 0.16g)? Considering the answers to parts (a) and (b), should there be much
(e) hydrogen and
(f) oxygen high in Earth’s upper atmosphere, where the temperature is about 1000 K?
Questions & Answers
QUESTION:
At what temperature does the rms speed of
(a) \(\mathrm{H}_{2}\) (molecular hydrogen) and
(b) \(\mathrm{O}_{2}\) (molecular oxygen) equal the escape speed from Earth (Table 13-2)? At what temperature does the rms speed of
(c) \(\mathrm{H}_{2}\) and
(d) \(\mathrm{O}_{2}\) equal the escape speed from the Moon (where the gravitational acceleration at the surface has magnitude 0.16g)? Considering the answers to parts (a) and (b), should there be much
(e) hydrogen and
(f) oxygen high in Earth’s upper atmosphere, where the temperature is about 1000 K?
ANSWER:Step 1 of 7
The rms speed equals the escape speed from Earth.
The rms speed equals the escape speed from the moon.
The minimum speed needed to escape from Earth is when the gravitational force on the particle equals the centripetal force of that particle.
\(\begin{array}{r} F_{g}=F_{c} \\ G \frac{m M_{E}}{R_{E}^{2}}=m \frac{v_{\text {esc }}^{2}}{R_{E}} \end{array}\)
Where \(M_{E}\) is Earth's mass and \(R_{E}\) is the radius of Earth. This yields
\(v_{e s c}=\sqrt{\frac{2 G M_{E}}{R_{E}}}\)
The root mean square speed is given by:
\(v_{r m s}=\sqrt{\frac{3 R T}{M}}\)
Where R is the gas constant, T is the temperature and M is the molecular mass. If the the rms speed equals the escape speed from Earth; the temperature will be given as:
\(T=\frac{2 G M_{E} M}{3 R R_{E}}\)