Barometric height formula

Consider a particular example the distribution of molecular concentration (or pressure) in a field of gravity not very high above the earth s surface. In this case, the gravitational field 

The barometric formnla can be revmtten in a different form. Remember that m/K = niNfjKTNf) = (MIR) (M is a mole mass and R is the universal gas constant). Therefore, 

If there was no thermal movement of molecules, they would all fall to the ground under the action of gravitational force, and all the air would concentrate as a layer, several meters in thickness, on the surface of the ground. If there were no gravitational force field, the molecules would scatter throughout open space. The atmosphere is 

Notice that the earth s gravitational field is completely unable to keep gas in a current balance. According to Boltzmann s formula, a nonzero (let it be very small) concentration of molecules is present at any height. In fact, it is possible that as a result of manifold casual collisions a molecule can gain a speed greater than the second space speed (u = 11.2 km/sec) and can leave the earth. Therefore, the earth s atmosphere continuously, but very slowly, loses its light component. The moon, in particular, has already lost its atmosphere as the gravitational field of the moon is much weaker than that of the earth s. 

In the cabin of an airborne plane, the barometer shows an identical pressure of p = 19 kPa all the time because of what the pilot counts as the flying-height constant. However, the temperature of the outside air changes from Tj = 5°C to T2 = 1°C. What uncertainty Ah in the definition of height has the pilot supposed Assume pressure Po at the ground to be normal. 

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For concentrated solutions and high colmnns of liquid the above calculation is not permissible, as the density of the vapour in the column of height h is not constant, but diminishes in accordance with the barometric height formula. Moreover, the solution at the membrane is under a considerable hydrostatic... 

Through integration the barometric height formula is obtained... 

The dimensionless constant Ag = amg/k T characterizes the strength of this potential. For large z, the density profile of an ideal gas of monomers at the lattice would follow the standard barometric height formula for the monomer density... 

This is the barometric height formula. It allows us to calculate the atmospheric pressure above the earth s surface, a graph of which (Figure 3.5b) is analogous to that presented in Figure 3.5a. In fact, it is assumed that the atmospheric temperature is constant and convection absent. 

Solution To find the solution we have to use the barometric height (bh) formula (refer to Section 3.2.2). The barometer can show constant pressure p at various temperatures to T2 onboard only in the case where the plane is not at height h (which the pilot counts constant), but at some other height Write the barometric height formula (3.2.7) for these two heights ... 

Particular attention should be paid to the fact that standards contain various formulas for computations and calculations, conversion factors, and correction factors. For example, a method of test for calorific value of gaseous fuels by the water-flow calorimeter (ASTM D 900-46T) has tables showing corrections for reduction of observed barometric heights to standard conditions. These corrections include those of temperature, latitude, and altitude. A standard method of testing and tolerances for woolen yarns (ASTM 403-44) has formulas for computing yarn numbers and a yarn number conversion table. 

Formula (4.1) is often called the barometer formula, since it gives the variation of barometric pressure with altitude. It indicates a gradual decrease of pressure with altitude, going exponentially to zero at infinite height. 

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... 

Figure 10.2 (a) The concept of a chemical potential that includes a field leads to the well-known barometric formula at thermal equilibrium in which T is uniform, (b) The actual state of the earth s atmosphere is not in thermal equilibrium the temperature varies with height as shown... 

Equation 1.7b, known as the barometric or hypsometric formula, gives a good description of the pressure distribution in the troposphere (i.e., up to a height of 11 km), provided the local temperature is used in tiie formula. That temperature is given by the empirical relation T(K) = 288.15 - 6.5 x IQ- (K/m)h. 

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