Pressure vs. altitude in the Earth's atmosphere. In plotting quantities about the atmosphere, we usually plot altitude on the vertical axis, even though it is the independent variable.The scale height is indicated by H.
where we have used the fact that ln(P/P0) = ln(P) - ln(P0). If we raise e to the left-hand side, it should equal e raised to the right-hand side.
The quantity kT0/mg has dimensions of length, and is the distance over which the pressure falls to 1/e of its original value. We call this quantity the scale height, H, where
Example 23.1 Scale heights Compute the scale height for the Earth's atmosphere as well as that for an atmosphere of pure oxygen (O2) and pure hydrogen (H2).
For a gas of molecular mass Amp (where mp is the proton mass), the scale height is
The variation of pressure with altitude is shown in Fig. 23.11.
(300 K)(1.38 X 10-16 erg/K) (1.67 X 10-24 g)(980 cm/s)A
For A = 29, H = 8.7 km. For oxygen, A = 32, so H = 7.9 km; for hydrogen, A = 2, so H = 125 km.
The drop off shown in Fig. 23.11 explains several things. If we go up to an altitude of 2 km, the atmospheric pressure is already down to 80% of what it is at sea level. This explains why it is
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