The exosphere is the outermost layer of the Earth's atmosphere

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Kinetic theory and transport in gases · Kinetic theory and transport in gases

Thermal Physics, Ch Temperature, pressure and the kinetic theory of gases

Problem

The exosphere is the outermost layer of the Earth's atmosphere, and above roughly 600 km600\,\mathrm{km} it sits at about 1500 K1500\,\mathrm{K}. At that height the escape speed is about 10.7 km s−110.7\,\mathrm{km\,s^{-1}} (a little below the 11.2 km s−111.2\,\mathrm{km\,s^{-1}} at the surface), and an atom there that moves upwards faster than that leaves the Earth for good. Treat the helium in the exosphere as an ideal gas at T=1500 KT = 1500\,\mathrm{K} with the Maxwell-Boltzmann speed distribution. The molar mass of helium is 4.00 g mol−14.00\,\mathrm{g\,mol^{-1}}. Ignore the effect of gravity on the shape of the distribution and any collisions suffered on the way out. Use kB=1.381×10−23 J K−1k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}} and NA=6.022×1023 mol−1N_A = 6.022\times10^{23}\,\mathrm{mol^{-1}}. ((a)) Estimate the most probable, mean and root-mean-square speeds of a helium atom there. ((b)) Estimate the fraction of the helium atoms whose speed exceeds the escape speed, whatever their direction. This is a fraction of speeds, not the fraction that escapes, which would also depend on direction. For X≫1X\gg 1 the fraction of molecules with v>Xvpv > X v_p is closely 2π(X+12X)e−X2,\frac{2}{\sqrt{\pi}}\left(X+\frac{1}{2X}\right)e^{-X^{2}}, the large-XX form of the tail integral of f(v)f(v).

Browse the problem library