The barometric distribution law

If p is the density of the fluid, then dm = pA dz using this in the expression for dp yields 

If the density of a fluid is independent of pressure, as is the case for liquids, then Eq. (2.34) may be integrated immediately. Since p and g are constants, they are removed from the integral and we obtain 

The integration constant C is evaluated in terms of the pressure at ground level when z = 0, p = Pq. Using these values in Eq. (2.37), we find that In p = C. Substituting this value for C and rearranging reduces Eq. (2.37) to 

Since the density is proportional to the pressure, and the number of moles per cubic metre is proportional to the pressure, Eq. (2.39) can be written in two other equivalent forms  

It is advisable to look more closely at this exponential type of distribution law, since it occurs so frequently in physics and physical chemistry in a more general form as the Boltzmann distribution. Equation (2.36) is most informative in discussing the exponential distribution it can be written 

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The barometric distribution law gives the distribution of atmospheric gas molecules in terms of their molar mass M, height h, absolute temperature T, the acceleration due to gravity g, and the ideal gas constant R. The pressure p at height h is given in terms of that at zero height po as... 

Conversion of height into relative vapor pressure for wetting films. Inserting the parameters for He in the barometric distribution law ... 

The tendency for particles to settle is opposed by their Brownian diffusion. The number density distribution of particles as a function of height z will tend to an equilibrium distribution. At low concentration, where van t Hoff s law applies, the barometric height distribution is given by... 

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