Atmospheric concentration decrease for ideal isothermal conditions

Range ~9 km
Organism Biosphere
Reference Hans Kuhn, Horst-Dieter Försterling, David H. Waldeck, Principles of Physical Chemistry, 2009 2nd edition Wiley & sons, pp.421-422
Method "In the above consideration the influence of gravity on the molecules has been neglected. Is this reasonable? If the molecules are under a gravitational field, for example, the atmosphere on the Earth, then a concentration gradient will be established. How significant is this concentration gradient? In a gravitational field g each molecule is pulled down by a force f=mg (m=mass of a molecule g=9.81m/s^2 on Earth), but diffuses up as a result of thermal motion. At a certain height d the potential energy in the gravitational field f d equals the thermal energy kT: fd=mgd=kT. [Investigators] can use this relationship to estimate the characteristic length of the gravitational field’s influence. Solving for d, [Investigators] find d=kT/mg. For bromine gas (m=26.4e-26kg), g=9.81m/s^2, and T=300K [they] obtain d=1.6 km. Hence it is reasonable to ignore these effects for the 10-cm-long cylinder discussed above. In box 14.5 [ link note-link is from 1st edition of ref, so page numbers are 382-3 rather than 421-2 box 11.4 rather than 14.5], [they] show that the concentration profile in the Earth’s gravitational field is c(h)=c(0)×c^(-mgh/kT)=c(0)×c^(-h/d). Thus d can be identified with the height h at which the concentration has dropped to c(0)/e. For the Earth’s atmosphere d is about 9 km. This effect explains why the air gets ‘thinner’ as [one climbs] a mountain. In an artificial gravity field of an ultracentrifuge (e.g., g[centrifuge]=10^6m/s^2), d can be brought to laboratory dimensions (d˜10cm) and the dependence of d on the mass of the molecules can be used to separate isotopes."
Comments For a value of ~8.4km, see Uni of Texas,The isothermal atmosphere link :"As a first approximation, let [investigators] assume that the temperature of the atmosphere is uniform. In such an isothermal atmosphere, [they] can directly integrate the previous equation to give p=p[0]exp(-z/z[0]) (equation 327). Here, p[0] is the pressure at ground level (z=0), which is generally about 1 bar, or 1 atmosphere (10^5N/m^2 in SI units). The quantity z[0]=RT/µg (equation 328) is called the isothermal scale-height of the atmosphere. At ground level, the temperature is on average about 15° centigrade, which is 288° kelvin on the absolute scale. The mean molecular weight of air at sea level is 29 (i.e., the molecular weight of a gas made up of 78% Nitrogen, 21% Oxygen, and 1% Argon). The acceleration due to gravity is 9.81 m/s^2 at ground level. Also, the ideal gas constant is 8.314 joules/mole/degree. Putting all of this information together, the isothermal scale-height of the atmosphere comes out to be about 8.4 kilometers."
Entered by Uri M
ID 111406