Question

(c) (i) On the surface of a planet of mass \(\mathrm{M}\) and radius \(\mathrm{R}\), the gravitational potential energy of a molecule of mass \(\mathrm{m}\) is \(-\frac{G M m}{R}\). Show that the escape speed of a molecule from the surface is \(\sqrt{\frac{2 G M}{R}}\).

(ii) The rms thermal speed of a molecule of mass \(m\) is given by \(v_{\text {th }}=\left(\frac{3 k T}{m}\right)^{1 / 2}\) where \(k\) is Boltzmann's constant . Using the appropriate temperature value from part (b) calculate the \(\mathrm{rms}\) thermal speed of atoms/molecules of \(\mathrm{H}_{2}, \mathrm{He}, \mathrm{O}_{2}\) and \(\mathrm{CO}_{2}\) at the surface of Mercury and discuss why Mercury has very little atmosphere. (Atomic/molecular masses: \(\mathrm{H}-2 ; \mathrm{He}-4 ; \mathrm{O}_{2}-32 ; \mathrm{CO}_{2}-44 ; 1\) atomic mass unit (amu) \(=1.66 \times 10^{-27} \mathrm{~kg}\)

Stefan-Boltzmann Constant, \(\sigma=5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}\)

Boltzmann's Constant, \(k=1.38 \times 10^{-23} \mathrm{JK}^{-1}\) )

Answer 1

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