Barometric formula

The barometric formula relates the change of atmospheric pressure with altitude, pQi) = p(0)exp. (6.46) 

p(h) is the pressure at altitude h, p(0) is the pressure at reference level zero altitude, M the molar mass of the gas, g the earth acceleration, R the gas constant, and T the absolute temperature. In spite of its simplicity, it is valid for heights up to 6 km with an error of less than 5%. 

The history of the barometric formula is described in the literature [8]. Famous scientists laid the foundations to the barometric formula, among ihtmGalileC Torricelli Pascal, and Boyle. The explicit formula goes back to Laplace. For this reason, the barometric formula is sometimes called Laplace formula. 

There are connections to other fields of science. In 1909, Perrin showed that small particles suspended in water exhibit a similar relation, where the pressure is replaced with the concentration. 

There are several different approaches to derive the barometric formula, i.e., based on hydrostatic considerations, statistical considerations, and energetic considerations [8]. We show here certain methods to derive the barometric formula. 

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Note that this expression is equivalent to the barometric formula which gives the variation of atmospheric pressure ( c) with elevation (oc r). A first-order dependence on the distance variable holds in the barometric equation, since the acceleration is constant in this case. 

The barometric formula governs the spatial distribution of the molecules in a gravity field according to the equation... 

The bottomhole pressure of injection wells was determined according to the known barometric formula. However, for the gas wells a correction was made for the increase in gas density. The steam density was calculated to be 20-30% higher on account of the moisture content, and gas density was assumed to equal 0.7 of air density. Inasmuch as the pressure changes in the production-observation wells were not significant, the effects that the injection wells exerted mutually among themselves could be ignored. In other words, it was assumed that all injection wells operated independently of one another. Before the pressure field map was constructed, all pressures measured in the production-observation wells were brought up to the moment at which steam injection was discontinued. Visual extrapolation was used for that puipose. 

For the energetic derivation of the barometric formula we consider Fig. 6.9, right sketch. We inspect an exchange of a certain gas from a lower altitude to a higher altitude. This thought experiment is done by removing both portions of gas from the column. 

Question 6.3. In practice, in the atmosphere, from where should a gas take the thermal energy needed for isothermal expansion The answer is partly by radiation. On the other hand, even in elementary school it is taught that air will cool when it is forced to move upward, when the foehn is discussed. So the barometric formula has a restricted field of application. 

In the earth atmosphere, there are usually certain fluxes of heat, therefore both approaches that treat the gas column as a closed thermodynamic system are not correct. The adiabatic case is more suitable, if the respective process of movement of a gas to some height is fast in comparison to the rate of heat conduction, thus entropy exchange of the gas. It was pointed out by Schaefer that the isothermal assumption in the barometric formula is highly unrealistic, and the adiabatic equation should be used [13]. 

Further, we have introduced the subscript (w), in order not to confuse the vapor pressure with the osmotic pressure. Due to the reduction in the vapor pressure of the solution, pure solvent should condense into the column from the bath, system ("). This would be sound if the system would operate in the absence of gravity. However, due to gravity, the pressure is decreasing with increasing altitude. The barometric formula says that the pressure reduction as a function of height is... 

Recall that p ih ) in Eq. (6.80) is the vapor pressure of the solution at height h, and p" h ) in Eq. (6.81) is the vapor pressure of the pure solvent or the other osmotic half cell corrected to the same height fi by the barometric formula. In equilibrium, just these vapor pressures should be equal. Actually, by the relation... 

Beyond the basic applications of the Boltzmann distribution, partieularly in statistical thermodynamics or even in quantum applieations sueh as modeling populations of atoms in laser media, they are fimdamental also in Environmental Physics by the famous modeling of the pressure with the atmosphere known as the barometric formula. 

A relation known as the barometric formula is useful for estimating the change in atmospheric pressure with altitude. The formula is given hy P =... 

Isotope effects can also be observed in the concentration distribution of gaseous substances in mechanical fields. In the earth s gravitational field, the pressure decreases with increasing altitude, which can be described by the barometric formula... 

Derive the following barometric formula which describes the isothermal... 

Equation 8.1.13 is the barometric formula for a pure ideal gas. It shows that in the equilibrium state of a tall column of an ideal gas, the pressure decreases exponentially with increasing elevation. 

This derivation of the barometric formula has introduced a method that will be used in Sec. 9.8.1 for deahng with mixtures in a gravitational field. There is, however, a shorter derivation based on Newton s second law and not involving the chemical potential. Consider one of the thin slab-shaped phases of Fig. 8.1. Let the density of the phase be p, the area of each horizontal face be. 4s, and the thickness of the slab be %h. The mass of the phase is then m = pA ih. The pressure difference between the top and bottom of the phase is 8p. Three vertical forces act on the phase an upward force pA at its lower face, a downward force — p + 8p)As at its upper face, and a downward gravitational force —mg = -pAsghh. If the phase is at rest, the net vertical force is zero pA - p + 8p)v4s — pA ghh = 0, or 8p = —pghh. In the limit as the number of phases becomes infinite and 8/ and 8p become infinitesimal, this becomes... 

Equation 9.8.9 shows that each constituent of an ideal gas mixture individually obeys the barometric formula given by Eq. 8.1.13 on page 197. 

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