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A Barometric Formula without the Hydrostatic Pressure Assumption

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Barometric formulas are important mathematical equations used to understand and predict the behavior of the atmosphere pressure at different altitudes. Since the first development by Pierre-Simon de Laplace in the 18th century, the fundamental assumptions leading to barometric formulas have been considering air as an ideal gas at steady-state, and considering atmospheric pressure as a hydrostatic pressure following Pascal’s law. Being rigorous however, gases do not follow Pascal’s law since the molecules are on average so far from each other that they cannot transmit the weight of their neighboring molecules in the vertical direction. For this reason, a new barometric formula has been derived without recurring to the hydrostatic pressure assumption. Instead of Pascal’s law, the conservation of momentum is used to describe the effect of gravity on the vertical molecular density profile. Then, after determining the temperature profile (which can be derived by solving the energy conservation equation, or can be empirically obtained), the molecular density profile can be solved, and the vertical pressure profile can be directly obtained from the ideal gas equation. The barometric formula obtained, which is almost equivalent to the current barometric formula used by the standard atmospheric model (the US Standard Atmosphere of 1976), was tested considering a set of experimental barometric measurements reported from different locations worldwide. Even though only a slight difference is obtained, the new expression no longer requires assuming atmospheric pressure as hydrostatic. The wide success of previous barometric formulas can be explained by the fact that the pressure drop predicted by the conservation of momentum deviates by less than 4% from Pascal’s law. Finally, a multicomponent model of air was considered, which allows the estimation of atmospheric composition changes with altitude.
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Vol. 5, 2020-14
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (1 / 22)
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A Barometric Formula without the Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research, 050030 Medellin, Colombia
hugo.hernandez@forschem.org
doi:
Abstract
Barometric formulas are important mathematical equations used to understand and predict
the behavior of the atmosphere pressure at different altitudes. Since the first development by
Pierre-Simon de Laplace in the 18th century, the fundamental assumptions leading to
barometric formulas have been considering air as an ideal gas at steady-state, and considering
atmospheric pressure as a hydrostatic pressure following Pascal’s law. Being rigorous however,
gases do not follow Pascal’s law since the molecules are on average so far from each other that
they cannot transmit the weight of their neighboring molecules in the vertical direction. For
this reason, a new barometric formula has been derived without recurring to the hydrostatic
pressure assumption. Instead of Pascal’s law, the conservation of momentum is used to
describe the effect of gravity on the vertical molecular density profile. Then, after determining
the temperature profile (which can be derived by solving the energy conservation equation, or
can be empirically obtained), the molecular density profile can be solved, and the vertical
pressure profile can be directly obtained from the ideal gas equation. The barometric formula
obtained, which is almost equivalent to the current barometric formula used by the standard
atmospheric model (the US Standard Atmosphere of 1976), was tested considering a set of
experimental barometric measurements reported from different locations worldwide. Even
though only a slight difference is obtained, the new expression no longer requires assuming
atmospheric pressure as hydrostatic. The wide success of previous barometric formulas can be
explained by the fact that the pressure drop predicted by the conservation of momentum
deviates by less than  from Pascal’s law. Finally, a multicomponent model of air was
considered, which allows the estimation of atmospheric composition changes with altitude.
Keywords
Atmosphere, Barometric formula, Energy conservation, Hydrostatic pressure, Lapse rate,
Molecular distributions, Momentum conservation, Pascal’s law, Troposphere
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (2 / 22)
www.forschem.org
1. Introduction
In a recent publication, Lente and Ősz [1] explained in great detail different lines of thinking
resulting in the most common barometric formulas used to describe changes in atmospheric
pressure with altitude. Interestingly, all the approaches considered provide fairly good
approximations for describing the pressure profile, even though some of them were based on
different principles.
They classified the models according to the temperature profile at different altitudes into:
Constant temperature models and variable temperature models. While the constant
temperature assumption of the whole atmosphere is clearly unrealistic, it is the starting point
of the derivation of the formulas. Once the temperature profile is known, the extension as a
variable temperature model is relatively straightforward. Among the constant temperature
models considered, only the model based on the statistical thermodynamics approach cannot
be extended as a variable temperature model.
Excluding the statistical thermodynamics approach, all other approaches presented for
deriving the barometric formula rely on the hydrostatic pressure assumption, or equivalently,
on the validity of Pascal’s law for atmospheric air. In other words, these approaches assume
the validity of the following expression:


(1.1)
where is the atmospheric pressure at altitude ,  is the air molecular density at altitude ,
 is the average molecular mass of air, and is the gravitational acceleration (also at
altitude ). The negative sign indicates that the atmospheric pressure decreases with altitude.
Eq. (1.1) basically expresses that the atmospheric pressure is caused by the weight of the mass
of air above a surface.
Pascal’s law was originally proposed for describing pressure differences in liquids [2], and has
been shown unsuitable for gases [3,4]. In gases, the increase in pressure as the altitude
decreases is not directly caused by the weight of the column of gas, as indicated by Pascal’s
law. In reality, the additional pressure is caused by an increase in mass density and in vertical
molecular velocities as altitude is reduced. Such effect is the result of the gravitational
acceleration, but not of weight transmission along the gas body in the vertical direction. Even
more, the increase in atmospheric pressure close to the surface is even larger than the pressure
increase expected by the increased weight of air. However, under certain conditions gases can
behave approximately as a Pascal fluid [3].
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (3 / 22)
www.forschem.org
Thus, even though common barometric formulas are good approximations for describing the
vertical atmospheric pressure profile, they are based on a wrong assumption: The validity of
the hydrostatic assumption in gases.
The purpose of this report is presenting an alternative derivation of a barometric formula
without assuming hydrostatic pressure, but using conservation equations instead.
2. Molecular Description
The common derivations of the barometric law assume atmospheric air to be an ideal gas. For
typical environmental conditions, and for practical purposes, this assumption is reasonable. An
ideal gas is a low-density molecular system where each molecule, on average, experiences a
net intermolecular force of zero. However, in the case of atmospheric air the average net force
is not zero because of the gravitational force acting on all molecules. In addition,
intermolecular forces acting between the ideal gas molecules and the solid surface play an
important role in the macroscopic mechanical stabilization of the atmosphere.
On the other hand, the fact that the average intermolecular force (between ideal gas
molecules) is zero does not imply that molecules do not (or cannot) collide. On the contrary,
collisions between ideal gas molecules, while relatively infrequent compared to other systems,
are very important for chemical kinetics and transport phenomena [5].
A schematic representation of an arbitrary fraction of the atmosphere is presented in Figure 1.
Air molecules move randomly in the horizontal direction, with an average zero velocity. This
also implies that there is a homogeneous molecular distribution in the horizontal direction.
However, in the vertical direction, the velocities are greatly influenced by gravity, which result
in a steady vertical distribution profile. Of course, no molecular system is static. However, an
average steady-state can be obtained, which will be considered in the development of the
model.
The steady-state assumption implies that the molecular concentration profile remains
constant, despite the fact that molecules are in permanent motion. Under this assumption, the
net flux of molecules crossing an imaginary horizontal boundary is zero, but the individual
upward and downward fluxes are never zero. This is illustrated in Figure 2.
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (4 / 22)
www.forschem.org
Figure 1. Molecular representation of atmospheric air. Black dots represent air molecules. A
molecular concentration profile develops vertically as a result of gravity.
Figure 2. Molecules crossing an imaginary horizontal boundary located at height in a short
period of time. Green arrows represent the vertical velocity of molecules crossing the boundary
in the upward direction. Blue arrows represent the vertical velocity of molecules crossing the
boundary in the downward direction.
The distribution of vertical molecular velocities at each particular height will be influenced by
gravity but also by molecular collisions. Considering the large number of collisions taking place
over short periods of time, and following the ideas of Maxwell [6] and Boltzmann [7], a
reasonable assumption is that the vertical molecular velocities at each height are normally
distributed, with a mean value of zero. At the surface, by assuming perfectly elastic collisions,
the steady-state requirement is still satisfied.
Considering air as a multicomponent system, the vertical molecular speed (the magnitude of
the molecular velocity) of species at height can then be described by the following
probability distribution function:




(2.1)
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (5 / 22)
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corresponding to the probability distribution of the absolute value of a normal random
variable, where  is the standard deviation in vertical molecular speed at height .
The vertical dynamics of a single non-interacting molecule is presented in Figure 3, when solar
radiation absorption by the atmosphere is neglected. The molecular vertical speed at the
surface is , being the same in both upward and downward directions. As the molecule rises,
its speed decreases by the decelerating effect of gravity , until the molecule completely
stops at a height . Then, the molecule falls increasing its speed as a result of gravity, until it
reaches the ground again. The same cycle is repeated at steady-state.
Figure 3. Vertical dynamic behavior of a single molecule. Molecular ground speed is . The
maximum height reached by the molecule is .
The vertical speed of the molecule can be described mathematically as a function of height
() using the following expression:
 
(2.2)
where 

(2.3)
3. Vertical Atmospheric Profiles
At steady-state, the number of molecules of a certain species crossing any horizontal plane at
altitude in the upward direction must be exactly the same as the number of molecules of
species crossing the plane in the downward direction. Mathematically, this can be expressed
as:
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (6 / 22)
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
(3.1)
where represents the molecular flux of species , and the superscripts and represent the
upward and downward direction, respectively.
On the other hand, a balance of momentum for each component in a horizontal atmospheric
section of volume  at steady-state yields:






(3.2)
which can be transformed into :





(3.3)
Assuming a symmetrical velocity distribution (particularly the normal distribution described in
Section 2) then: 

(3.4)
and considering positive the upward direction Eq. (3.3) becomes:

(3.5)
where  is the mean molecular speed in a single direction. Thus, for an infinitesimally small
height :


(3.6)
Please notice that Eq. (3.6), obtained from the conservation of momentum, is conceptually the
correct expression to be used instead of the hydrostatic assumption (Eq. 1.1). Of course, the
equivalent pressure acting on the horizontal plane is related to , but it is not exactly
identical [8].
In addition, a molecular flux corresponds to the product between the molecular density and
the average molecular speed  [8]:

(3.7)
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (7 / 22)
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Therefore: 
 

 

(3.8)
or equivalently, 





(3.9)
The average molecular speed can be obtained from the vertical speed distribution (Eq. 2.1):
 
 


 

(3.10)
However, due to spatial and temporal limitations, infinite molecular velocities are not possible.
Thus, we would expect that the mean molecular velocity should be lower than this ideal value.
Using a correction factor incorporating these limitations we obtain:


(3.11)
Now, assuming that at steady-state the thermal energy of the molecules at a certain height is
homogeneously distributed in all directions, then:

(3.12)
Therefore,


(3.13)
and 



(3.14)
From which Eq. (3.9) becomes:
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (8 / 22)
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
 



(3.15)
Separating variables results in: 




(3.16)
which can be integrated into:

 




(3.17)
where  is the molecular density of species at a reference altitude (e.g. sea level).
Clearly, the molecular density profile can only be obtained if the atmospheric temperature
profile is known. Let us recall that the temperature profile is greatly influenced by solar
radiation absorption. However, if this effect is neglected (which is a common assumption for
tropospheric models), then temperature changes are only attributed to gravitational effects.
Considering the non-interacting model previously described, and assuming ideal gas molecules,
then the average kinetic energy of a molecule of species as a function of height will be:

(3.18)
where represents the degrees of freedom of motion of the molecular species .
Using Eq. (2.2), we obtain:
 

(3.19)
Now, since the temperature of air at a certain height can be estimated as a function of the
average kinetic energy of the diatomic molecules, then:


 


(3.20)
where for
, and  is the molar fraction of species at height ,
given by:
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (9 / 22)
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(3.21)
and

(3.22)
where is the number of components in atmospheric air.
Therefore, the vertical temperature profile becomes:


(3.23)
where the temperature contribution becomes zero when
.
Assuming air as a pure system composed of molecules with an average molecular mass with
an average degrees of freedom of motion , then the temperature profile simply becomes:


(3.24)
where is the constant lapse rate, estimated in this case as:


(3.25)
For this linear temperature profile, 

(3.26)
And therefore Eq. (3.17) becomes:




(3.27)
Integrating with respect to the altitude results in:



(3.28)
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (10 / 22)
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And solving out for the molecular density we get:



(3.29)
Finally, the barometric formula of atmospheric air, under the steady-state assumption, can
simply be obtained from the ideal gas equation:
(3.30)
where represents the equivalent pressure of the atmosphere at altitude .
Thus,


(3.31)
When a multicomponent model is considered, the molecular density of species is determined
by: 
 


(3.32)
where the temperature is given by Eq. (3.23).
These equations cannot be solved analytically, unless a certain assumption regarding the
temperature profile is done. For example, assuming a linear temperature profile:


(3.33)
where is the overall lapse rate approximately given by:



(3.34)
The previous equation is only an approximation because the atmospheric composition is
expected to change with altitude, and thus, the lapse rate is not truly constant.
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (11 / 22)
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The most general barometric expression considering multicomponent air and an arbitrary
temperature profile valid for an arbitrary altitude interval would then be:



(3.35)
where the atmospheric conditions ( and ) at a certain altitude are known and used
as reference values.
In addition, the gravitational acceleration in this expression can be replaced by a suitable
altitude-dependent model.
4. Model Validation
Lente and Ősz [1] compared the different barometric formulas with the semi-empirical US
standard atmosphere (USSA) model [9]. In the USSA model, the temperature profiles are
represented by empirical piecewise models, whereas the pressure profile is obtained by
applying Pascal’s law in gases (Eq. 4 in [9]) using the empirical temperature, assuming ideal gas
behavior and standard sea level conditions 
[10].
The profiles described by the USSA model are compared in Figure 4 to the predictions of the
barometric formula derived in the previous Section (Eq. 3.31), using an estimated lapse rate
value (from Eq. 3.25) of , and the empirical lapse rate used by the USSA model
of . The standard deviation between the USSA model and the barometric formula
considering the theoretical estimation of the lapse rate and assuming was , with
an average relative absolute deviation of . The standard deviation between the USSA
model and the barometric formula using the empirical lapse rate and was  and
the average relative absolute deviation of . Interestingly, when the theoretical lapse rate
is used with the present model the results are closer to the USSA model than when the
empirical lapse rate is considered.
The USSA model was used as a reference model. However, a model validation using
experimental data is also possible. For this purpose, a set of barometric pressure
measurements in the troposphere reported almost simultaneously from different locations
worldwide was collected from the following website: https://www.timeanddate.com/weather/.
The information was collected on June 9, 2020 between 18:00 and 19:00 GMT. The current
barometric pressures at 182 different locations with altitudes over 500 m above sea level are
presented in the Appendix 1. In addition, the standard sea level pressure and the barometric
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (12 / 22)
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pressure reported at Mt. Everest [11] were included in the dataset. The comparison of the
models with the collected set of experimental data is presented in Figure 5.
Figure 4. Barometric models comparison in the troposphere. Red line: US Standard Atmosphere
(USSA) model [9]. Blue line: New barometric model using Eq. (3.25)  and
. Green line: New barometric model using the USSA empirical lapse rate 
and .
Figure 5. Barometric models validation using reported barometric data (gray circles) in the
troposphere. Red line: US Standard Atmosphere (USSA) model, . Blue line: New
barometric model using Eq. (3.25) , . Green line: New
barometric model using the USSA empirical lapse rate , .
We can observe that the experimental observations are somehow contained within the USSA
model  and the completely theoretical barometric model without the
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (13 / 22)
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hydrostatic pressure assumption . Considering the empirical lapse rate of
 in the new model causes a deviation from the experimental observations particularly
at higher altitudes. This deviation does not penalize much the goodness of fit 
because only a few experimental observations are available at high altitudes.
Although the theoretical ideal gas model neglecting solar radiation does not accurately predict
the vertical pressure profile, it qualitatively explains why temperature decreases with altitude
in the troposphere. However, from a quantitative point of view, the effect of solar radiation is
of utmost importance. On one hand, the surface temperature increases, and energy can be
transferred as heat to the atmosphere; but on the other hand, the atmosphere itself also
absorbs solar radiation. The complexity of this situation increases by considering that solar
radiation intensity changes along the day, along the year, and is different at each location.
Furthermore, radiation absorption is greatly influenced by the local atmospheric composition.
However, as a result of the temporal and spatial averages in solar radiation and absorption, the
cooling effect caused by the gravitational force is partially compensated. That is why the
observed temperature change is smaller than the expected theoretical lapse rate when solar
radiation is neglected.
Figure 6. Vertical tropospheric temperature profiles. Blue diamonds: Experimental
observations. Red line: Theoretical ideal gas lapse rate without solar radiation
. Green line: USSA model empirical lapse rate . Blue line: Best
quadratic fit of experimental observations.
These differences might be illustrated by considering the vertical temperature profile models,
and comparing them with the experimental results. This comparison is presented in Figure 6.
Clearly, neither of the linear profiles correctly describes the experimental observations.
Particularly, a quadratic regression model provides a very good qualitative description of the
A Barometric Formula without the
Hydrostatic Pressure Assumption
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ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (14 / 22)
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temperature profile. The empirical model obtained is the following (considering temperature in
and altitude in ): 
(4.1)
The empirical model presented in Eq. (4.1) can be approximated also as follows:

(4.2)
Using this temperature model (Eq. 4.2), the vertical density and pressure profiles become (from
Eq. 3.32 and 3.35 assuming a single component):



(4.3)


(4.4)
Figure 7. Barometric models validation using reported barometric data (gray circles) in the
troposphere. Red line: US Standard Atmosphere (USSA) model, . Green line: New
barometric model using Eq. (4.4) , . Green line: New barometric model
using Eq. (4.4) , .
The vertical pressure profile given by Eq. (4.4) is comparatively shown in Figure 7. In addition,
the experimental data was used to fit the value using model (4.4). The result obtained
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
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26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (15 / 22)
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 represents approximately an equivalent maximum molecular velocity of 
.
The new barometric model presented in this work provides an improved description of the
barometric data , while relying on valid first principles (mass,
momentum and energy balances) and not on the misconception of using Pascal’s law in gases.
The rate of change in pressure with respect to altitude can then be obtained for a linear
temperature profile from Eq. (3.31):




(4.5)
and for Pascal’s law is: 

(4.6)
Clearly, the results obtained are very similar, but they are derived from different principles. This
explains why the hydrostatic pressure assumption, while being invalid for gases, can
satisfactorily describe the behavior of pressure in the atmosphere. Thus, it is not the purpose
to change the mathematical model but only to change its interpretation. Furthermore, the
larger rate of change in pressure compared to Pascal’s law is consistent with previous
theoretical developments [3,13] as well as experimental results [4].
5. Composition Profile
In addition to the pressure profile, the barometric model developed also allows estimating the
vertical composition profile of atmospheric air, as long as the temperature profile and the
ground-level composition are already known. For this purpose, the empirical temperature
profile considered by the USSA model [9] will be used. In addition, a representative
composition of dry unpolluted air at ground level is given in Table 1 [12].
Table 2 shows the composition of atmospheric air at different altitudes, as predicted by the
multicomponent model using the empirical lapse rate. Figure 8 illustrates the changes in air
composition with altitude, and Figure 9 shows the average molecular mass profile.
The multicomponent model predicts a composition decrease in all molecules heavier than the
average molecular mass (i.e. Oxygen, Argon, carbon dioxide, Krypton, nitrous oxide and
Xenon) and an increase in all molecules lighter than the average molecular mass of air (i.e.
Nitrogen, Neon, Helium, methane and Hydrogen). The average molecular mass of air is
A Barometric Formula without the
Hydrostatic Pressure Assumption
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26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (16 / 22)
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observed in the troposphere to almost linearly decrease with altitude. The complete
atmospheric profiles obtained considering the empirical temperature profiles reported by the
USSA model are summarized in Appendix 2.
Table 1. Representative ground-level air composition [12] and additional air components
properties
Air component
Molecular
mass (g/mol)
Degrees of
freedom
%vol. / %mol
Nitrogen
28.01
5
78.084
Oxygen
32.00
5
20.946
Argon
39.95
3
0.934
Carbon Dioxide
44.01
9
0.036
Neon
20.20
3
1.82E-3%
Helium
4.00
3
5.24E-4%
Methane
16.04
15
1.60E-4%
Krypton
83.80
3
1.14E-4%
Hydrogen
2.02
5
5.00E-5%
Nitrous Oxide
44.01
9
3.00E-5%
Xenon
131.29
3
8.70E-6%
Average / Total
28.97
5
100
Table 2. Air composition at different altitudes obtained from the multicomponent model with
.
Altitude (km)
Composition
0
2
3
4
6
7
8
11
Nitrogen
78.08%
78.56%
78.80%
79.05%
79.55%
79.81%
80.07%
80.87%
Oxygen
20.95%
20.54%
20.33%
20.11%
19.67%
19.45%
19.22%
18.50%
Argon
9.34E-1%
8.70E-1%
8.38E-1%
8.07E-1%
7.46E-1%
7.16E-1%
6.86E-1%
6.00E-1%
Carbon
Dioxide
3.60E-2%
3.27E-2%
3.10E-2%
2.95E-2%
2.65E-2%
2.50E-2%
2.36E-2%
1.97E-2%
Neon
1.82E-3%
1.92E-3%
1.98E-3%
2.04E-3%
2.17E-3%
2.24E-3%
2.32E-3%
2.57E-3%
Helium
5.24E-4%
6.15E-4%
6.68E-4%
7.27E-4%
8.67E-4%
9.50E-4%
1.04E-3%
1.40E-3%
Methane
1.60E-4%
1.74E-4%
1.82E-4%
1.90E-4%
2.08E-4%
2.18E-4%
2.29E-4%
2.67E-4%
Krypton
1.14E-4%
7.98E-5%
6.64E-5%
5.49E-5%
3.71E-5%
3.02E-5%
2.45E-5%
1.26E-5%
Hydrogen
5.00E-5%
5.94E-5%
6.50E-5%
7.12E-5%
8.60E-5%
9.49E-5%
1.05E-4%
1.45E-4%
Nitrous Oxide
3.00E-5%
2.72E-5%
2.59E-5%
2.46E-5%
2.20E-5%
2.08E-5%
1.97E-5%
1.64E-5%
Xenon
8.70E-6%
4.47E-6%
3.16E-6%
2.22E-6%
1.07E-6%
7.28E-7%
4.92E-7%
1.42E-7%
Avg. Mol.
mass (g/mol)
28.97
28.94
28.93
28.92
28.89
28.88
28.86
28.83
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (17 / 22)
www.forschem.org
Figure 8. Atmospheric air composition changes with altitude predicted by the multicomponent
model with . Left plot: Major air components. Right plot: Minor air components.
Figure 9. Vertical profile of the average molecular mass of atmospheric air, predicted by the
multicomponent model with .
6. Conclusion
In this report, a barometric formula was derived assuming a steady-state condition, and a
normal distribution of vertical molecular velocities at each altitude, by using only conservation
equations. In this derivation, Pascal’s law (Eq. 1.1), which is a valid law for dense fluids but
invalid for gases, was not used. Instead, Pascal’s law was substituted by an equation obtained
from the conservation of momentum considering a thin, horizontal section of atmospheric air
(Eq. 3.6). The solution of this equation requires knowledge of the vertical temperature profile,
and thus, the energy conservation equation needs to be solved first. While a simple
temperature model can be obtained by neglecting the effect of solar radiation absorption, such
model is far from reality. For this reason, and due to the complexity of the conservation of
energy considering solar radiation, an empirical vertical temperature profile is used instead.
Particularly, the empirical profiles reported in the US Standard Atmospheric model [9] are
used. A correction factor accounting for the limitations in molecular speeds was fitted using a
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
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26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (18 / 22)
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set of barometric pressure measurements reported almost simultaneously at different
locations worldwide. The correction factor thus obtained () fitted very well
 the experimental data. Furthermore, using this numerical value for the
correction factor, the expression obtained closely resembles the formula obtained using
Pascal’s law, with an estimated pressure drop with altitude compared to the drop predicted by
Pascal’s law in the troposphere only . This may also explain why barometric models based
on the hydrostatic pressure assumption have been successfully used to describe the vertical
atmospheric pressure profile, despite the fact that Pascal’s law does not apply to gases.
Additional calculations using the same empirical temperature profile allow predicting the
changes in air composition with altitude. It is observed that the composition of light
components (below the molecular mass average) increase with increasing altitude, whereas
the composition of heavy components (above the molecular mass average) decrease with
altitude. For this reason, the average molecular mass of air decreases with altitude; particularly
in the troposphere, such decrease is almost linear.
By considering the particular temperature profile in any other atmospheric layer, Eq. (3.35) can
be used as the most general barometric formula to obtain the corresponding vertical pressure
profile.
Acknowledgments
The author gratefully acknowledges Prof. Jaime Aguirre (Universidad Nacional de Colombia)
for the several helpful discussions on the topic, and for reading the manuscript.
This research did not receive any specific grant from funding agencies in the public,
commercial, or not-for-profit sectors.
References
[1] Lente, G., & Ősz, K. (2020). Barometric formulas: various derivations and comparisons to
environmentally relevant observations. ChemTexts, 6, 1-14.
[2] Pascal, B. (1663). Traites de l'equilibre des liqueurs, et de la pesanteur de la masse de l'air.
Chez Guillaume Desprez, Paris.
[3] Hernandez, H. (2020). Pascal’s Law in Gases. ForsChem Research Reports, 5, 2020-09. doi:
10.13140/RG.2.2.36166.09285.
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (19 / 22)
www.forschem.org
[4] Hernandez, H. (2020). Testing Pascal's Law in Gases using Free Fall Experiments. ForsChem
Research Reports, 5, 2020-12. doi: 10.13140/RG.2.2.35747.89120.
[5] Jordan, P. C. (1979). Chemical Kinetics and Transport. Plenum Press, New York.
[6] Maxwell, J. C. (1860). Illustrations of the dynamical theory of gases. - Part I. On the motions
and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical
Magazine and Journal of Science, 19(124), 19-32.
[7] Boltzmann, L. (1872). Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen.
K. Acad. Wiss.(Wein) Sitzb., II Abt, 66.
[8] Hernandez, H. (2019). Calculation of Molecular Fluxes and Equivalent Pressure in Ideal
Gases. ForsChem Research Reports, 4, 2019-03. doi: 10.13140/RG.2.2.35898.44483.
[9] NOAA, NASA & USAF (1976). US standard atmosphere, 1976. Washington.
[10] Corda, S. (2017). Introduction to aerospace engineering with a flight test perspective. John
Wiley & Sons. p. 186.
[11] West, J. B. (1999). Barometric pressures on Mt. Everest: new data and physiological
significance. Journal of Applied Physiology, 86(3), 1062-1066.
[12] Brimblecombe, P. (1996). Air Composition and Chemistry. 2nd Ed. Cambridge University
Press, New York. p. 2.
[13] Hernandez, H. (2020). Effect of External Forces on the Macroscopic Properties of Ideal
Gases. ForsChem Research Reports, 5, 2020-07. doi: 10.13140/RG.2.2.33193.21608.
Appendix 1. Worldwide Barometric Pressure Dataset
The data presented here corresponds to a set of barometric pressure measurements in the
lower layer of the atmosphere (troposphere), reported almost simultaneously from different
locations worldwide. The data was obtained from the following weather website:
https://www.timeanddate.com/weather/, the 9th of June 2020 between 18:00 and 19:00 GMT. A
total of 180 different locations with altitudes over 500 m above sea level were collected. In
addition, the standard sea level pressure and the barometric pressure reported at Mt. Everest
[11] were included as extreme values in the dataset.
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (20 / 22)
www.forschem.org
Location
Altitude (m)
Pressure (Pa)
Location
Altitude (m)
Pressure (Pa)
Location
Altitude (m)
Pressure (Pa)
Abéché
540
95000
Edmonton
692
93400
Pedro Juan Caballero
662
94300
Abilene
524
95000
El Paso
1142
88900
Petrópolis
838
92200
Adama
1621
84700
Entebbe
1160
88900
Pirassununga
632
94800
Addis Ababa
2354
77800
Eskişehir
792
92300
Poprad
670
93600
Agadez
504
95000
Ferizaj
582
94300
Prince George
573
94500
Aguascalientes
1870
82700
Francistown
989
90900
Pristina
599
94100
Alajuela
955
91200
Freistadt
565
94600
Provo
1386
87000
Albuquerque
1510
85100
Fresnillo
2187
79700
Puebla
2155
79100
Ali Sabieh
713
92400
Ghardaïa
522
95700
Pune
567
94300
Alice Springs
581
95400
Gitarama
1872
82000
Puyo
950
92100
Almaty
790
92600
Granada
698
93800
Pyatigorsk
533
95200
Amarillo
1117
89000
Guadalajara
1543
84400
Querétaro
1817
83200
Amman
816
92300
Guarulhos
759
93400
Quito
2826
73800
Ankara
871
91600
Guatemala City
1533
86100
Rapid City
977
90400
Antananarivo
1287
87800
Guiyang
1067
89500
Regina
577
94700
Arequipa
2345
78300
Helena
1239
88100
Riyadh
590
94400
Arusha
1366
87000
Hesperia
971
91300
Rustenburg
1166
89200
Aurora
1647
83300
Hovd
1396
85600
Rwamagana
1543
85200
Bandung
709
93700
Hyderabad
507
95100
Sahuarita
824
92400
Bangalore
876
91600
Ibarra
2225
79400
Saint Thomas
760
93000
Baotou
1069
89000
Indore
554
93900
Salamanca
802
92700
Barquisimeto
569
95100
Innsbruck
570
94500
Salt Lake City
1312
87700
Béchar
779
92900
Islamabad
554
94400
Salta
1184
88300
Belo Horizonte
860
92500
Jaén
565
95300
San José
1161
89000
Bern
533
95100
Jerusalem
769
92500
San Luis Potosí
1876
81200
Bethlehem
753
92600
Kaduna
607
94600
San Marino
666
93400
Billings
946
91100
Kampala
1230
88200
San Salvador
667
94200
Bishkek
760
92900
Kathmandu
1305
87200
Santa Ana
650
94400
Bismarck
513
95200
Kempten
673
93400
Santa Fe
2132
79000
Bitola
615
93800
Kigali
1555
85100
Santiago
576
95000
Bloemfontein
1397
86500
Kiruna
532
95700
Sao Paulo
765
93300
Bogotá
2618
75700
Köniz
579
94600
Sarajevo
526
95000
Boise
822
92900
Kunming
1909
81600
Schwyz
515
95300
Brasilia
1091
90200
Lakewood
1683
83100
Sea Level [9]
0
101325
Bucaramanga
956
91100
Lanchow
1531
85000
Sion
514
95100
Butare
1750
83200
Las Vegas
629
95000
Sofia
551
94600
Calgary
1063
89500
León
1808
82000
Stavropol
580
94600
Cali
955
91200
Lhasa
3654
64300
Tabora
1207
88600
Campinas
686
94100
Lilongwe
1031
90500
Taiyuan
790
92400
Canberra
577
96000
Machu Picchu
2074
81100
Tegucigalpa
1054
90300
Caracas
875
91700
Madrid
658
94300
Tehran
1179
88900
Carson City
1425
86500
Malatya
960
90700
Tepic
936
90900
Cheyenne
1855
81200
Manizales
2139
79800
Texcoco
2250
77800
Chihuahua
1432
85500
Maun
943
91400
Thimphu
2307
75800
Chita
666
92800
Medellín
1479
86400
Tskhinvali
877
91700
Chur
592
94200
Medina
606
94500
Tucson
758
93100
Ciudad Juárez
1130
88700
Mendoza
767
92700
Tuxtla Gutierrez
536
95200
Cochabamba
2584
76200
Mexico City
2241
77900
Ulaanbaatar
1298
86100
Constantine
557
95300
Midland
847
92200
Ürümqi
832
91600
Cranbrook
933
91300
Molepolole
1145
89500
Valladolid
712
93700
Cuenca
2554
76200
Monterrey
545
95000
Valverde
571
95400
Cuernavaca
1522
85700
Mt. Everest [11]
8848
33600
Victorville
830
92800
Curitiba
915
91500
Munich
526
95200
Whistler
674
93800
Damascus
689
93400
Mwanza
1146
89000
Whitehorse
640
93400
Denver
1593
83800
Nairobi
1680
83700
Windhoek
1680
84000
Dire Dawa
1204
87800
Nekemte
2110
79700
Yaoundé
717
93500
Diyarbakir
671
93500
Nikšić
635
93600
Yeghegnadzor
1223
87400
Dodoma
1125
89400
Oaxaca
1566
84400
Yerevan
997
89800
Durango
1880
81500
Ogden
1310
87700
Zacatecas
2440
77400
Dushanbe
832
92100
Ouarzazate
1151
89200
Zaria
659
94100
Ebebiyín
564
94800
Oyem
664
93800
Ecatepec
2250
77800
Paradise
643
94800
A Barometric Formula without the
Hydrostatic Pressure Assumption
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ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (21 / 22)
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Appendix 2. Complete Atmospheric Profiles
The following figures summarize the atmospheric vertical profiles (up to 86 km) obtained with
the barometric formula developed in this work. The starting point is the empirical temperature
profile presented in the 1976 U.S. Standard Atmosphere model [9].
Figure A1. Empirical vertical profile of the atmospheric air temperature, up to .
Figure A2. Vertical profile of the atmospheric air molecular density, up to , calculated
from Eq. (3.31). Left plot: Molecular density in original scale. Right plot: Molecular density in
logarithmic scale.
A Barometric Formula without the
Hydrostatic Pressure Assumption
Hugo Hernandez
ForsChem Research
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26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (22 / 22)
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Figure A3. Vertical profile of the atmospheric air pressure, up to , calculated from Eq.
(3.34). Left plot: Pressure in original scale. Right plot: Pressure in logarithmic scale.
Figure A4. Vertical profile of the atmospheric air pressure, up to , calculated from the
multicomponent barometric model. Left plot: Pressure in original scale. Right plot: Pressure in
logarithmic scale.
Figure A5. Vertical profile of the atmospheric air composition, up to , calculated from the
multicomponent barometric model. Left plot: Major components (composition in % mole).
Right plot: Minor components (composition in logarithmic scale of molar fraction).
... An alternative analytical expression describing the effect of mass density and external acceleration on pressure changes in presented [53]. This result, based on the conservation of momentum, is used to derive a new barometric formula without recurring to the hydrostatic pressure assumption based on Pascal's law [58]. The model obtained correctly fits experimental data of environmental pressure measured in different cities worldwide located at different altitudes. ...
... This model can be used as a mechanistic explanation of aerodynamic lift.  The new barometric formula obtained without the hydrostatic assumption [58] is improved by considering the effect of reversible chemical reactions [68]. The complex interaction between concentration and temperature at different altitudes do not have a simple analytical solution, unless no reaction is taking place. ...
... 4 th place was given to the mechanistic model of liquid evaporation having a single parameter (cohesion temperature), representing an alternative to the empirical Antoine equation (with three parameters) for describing vapor pressure [93,99]. Particularly, only the second report of § § This has enormous implications, not only on the barometric formula [58], but it also has raised serious questions on the author about the rotation of the Earth. Given that forces are not directly transmitted in gases, and considering the low viscosity of atmospheric air, we should be able to perceive the speed of rotation of Earth (about 1600 km/h) as a devastating wind! the series was considered in the list [99] as a personal appreciation to all the researchers that obtained and consolidating experimental information for more than 1400 chemical substances, used to obtain their corresponding cohesion temperature and energy barrier for evaporation. ...
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... Recently, an alternative derivation of the barometric formula was obtained without the hydrostatic assumption for the pressure of air [1], but taking into account that in reality Pascal's law is not valid for gases [2,3]. The barometric formula was obtained assuming air at steadystate, with a normal distribution of vertical molecular velocities at each altitude, and using only conservation equations (particularly, introducing the conservation of momentum instead of the hydrostatic pressure assumption). ...
... where is a constant lapse rate (negative temperature gradient), is the altitude with respect to the ground, ̃ is the molecular density of component (in a mixture of components), is the molecular mass of the -th component, is Boltzmann constant, is the gravitational acceleration, is a mean molecular velocity correction factor [1], and the subscript indicates an arbitrary reference point (usually the ground). ...
... Following the previous derivation of the barometric formula [1], the steady-state balance of momentum for each component in a horizontal atmospheric section of volume , assuming a symmetrical vertical velocity distribution, yields: ...
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The classical barometric formula used in atmospheric models is derived neglecting the presence of chemical reactions in the atmosphere. However, many chemical reactions are continuously taking place either promoted by sunlight or simply by the thermal motion of the molecules. In this report, the effect of chemical reactions on the barometric formula will be modeled and discussed. Such effect is not only related to individual molecular concentration profiles but also to thermal profiles when the heat of reaction is considered. The derivation of the model is based on a simple reversible chemical reaction, but it is also generalized for any arbitrary set of chemical reactions taking place in the system. Even under the steady-state assumption, the differential equations obtained do not provide a direct analytical solution and therefore, they must be numerically integrated. A particular example is presented for illustrating the model obtained but also the numerical solution method.
... For instance, it exerts a downwards vertical force on the ground, where is the pressure of atmospheric air, and is the total surface of the Earth in contact with atmospheric air. This force indirectly includes the effect of the gravitational force exerted by Earth on the mass of the atmosphere [4]. Thus, it will be the only interaction force considered between both systems. ...
... indicating that air at the bottom transfers to the body more energy than it receives back at the top. The ultimate source of energy used by air is gravity, which increases the speed of the molecules (increasing the temperature of air) [4], and is then transferred to the balloon (with a subsequent decrease in temperature of air at the bottom of the body). ...
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The First Law of Thermodynamics represents the principle of energy conservation applied to the interaction between different macroscopic systems. The traditional mathematical description of the First Law (e.g.) is rather simplistic and lack universal validity, as it is only valid when several implicit assumptions are met. For example, it only considers mechanical work done associated with a change in volume of a system, but completely neglects other types of work. On the other hand, it employs the concept of entropy which is not only ambiguous but also implies only heat associated with a temperature difference, neglecting other types of heat transfer that may take place at mesoscopic and/or microscopic levels. In addition, it does not consider mass transfer effects. In the previous report of this series, a more general representation of the First Law is obtained considering different conditions and different types of interactions between the systems. In this report, the expression previously obtained is applied to different representative examples, involving macroscopic systems with no volume change, gas systems with volume change, and even a case where mass transfer between the systems takes place.
... Then, the molecular density profile (assuming air as a single-component system) becomes [4]: ...
... Table 1 summarizes the wing loading ( ) and the cruise velocity ( ) for different commercial aircrafts [6]. It also includes the vertical terminal velocity at ( ) estimated assuming an air temperature of , and density (corresponding to an altitude of [4]), and the corresponding relative cruise velocity . The relative cruise velocities obtained correspond to theoretical inclination angles between and . ...
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In this report, a simplified model of flotation is presented based on the determination of the forces arising from the collisions between air molecules and a solid, rigid, thin flat body. The net force emerging during the motion of a thin body in a fluid medium is explained by the differences in molecular collision frequencies at both faces of the body. Under certain conditions, the motion of the body in the medium is capable of counteracting (or even surpassing) the weight of the body, resulting in flotation (or elevation). The model obtained is discussed under three different scenarios are considered: 1) When the flat body is placed horizontally (e.g. skydiving), 2) when the flat body is inclined (e.g. aircraft wings), and 3) when the flat body is touching the ground (e.g. take-off and landing). A good qualitative explanation of flotation and flight is obtained from this simplified molecular model.
... Let us now consider a general barometric model obtained after combining the structure of different barometric models previously reported [15,16] The optimal parameter values will be obtained considering the set of experimental observations summarized in Table 4. ...
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Nonlinear regression consists in finding the best possible model parameter values of a given homoscedastic mathematical structure with nonlinear functions of the model parameters. In this report, the second part of the series, the mathematical structure of models with nonlinear functions of their parameters is optimized, resulting in the minimum estimation of model error variance. The uncertainty in the estimation of model parameters is evaluated using a linear approximation of the model about the optimal model parameter values found. The homoscedasticity of model residuals must be evaluated to validate this important assumption. The model structure identification procedure is implemented in R language and shown in the Appendix. Several examples are considered for illustrating the optimization procedure. In many practical situations, the optimal model obtained has heteroscedastic residuals. If the purpose of the model is only describing the experimental observations, the violation of the homoscedastic assumption may not be critical. However, for explanatory or extrapolating models, the presence of heteroscedastic residuals may lead to flawed conclusions.
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Pascal's law or Pascal's principle, enunciated almost 400 years ago, has been of utmost importance in a wide variety of scientific and engineering disciplines. Currently physics textbooks describe Pascal's law as follows: "A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container." Such phenomenon can be explained based on the propagation of external forces at a molecular level via intermolecular repulsion (following Newton's third law of motion). In the case of gases, such propagation of forces does not take place directly, although external forces do influence the pressure of a gas by changing the momentum of the molecules hitting the walls. There is, therefore, a relationship not necessarily identical to that stated in Pascal's law. However, under certain conditions, Pascal's law remains as a fairly good approximation for gases. For example, the International Standard Atmosphere model has assumed that the atmosphere is an ideal gas following Pascal's law, with satisfactory results.
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Conventional thermodynamic and transport properties of ideal gases have been widely investigated in the absence of external forces (or in the presence of external forces with a negligible effect). However, when the potential energy provided by external forces acting on the molecules of an ideal gas overcomes their thermal kinetic energy, relevant changes in the properties of the system are expected. In that case, non-constant local temperature, density and pressure profiles in the direction of the force are obtained. Local differences with respect to such forced equilibrium profiles will drive the transport of mass and energy. A net heat flux from a colder to a warmer region is therefore possible, which would not be observed under "normal" conditions without significant external forces. The external force is also capable of increasing the overall temperature of the system, since the potential energy provided by the force becomes kinetic energy at the molecular level. In this report, a mathematical description of the molecular position and velocity distribution in the presence of external forces is derived from the fundamental equations of motion, and assuming an initial Maxwell-Boltzmann distribution of molecular velocities and a uniform distribution of molecular positions. Particularly, approximate mathematical expressions of these distributions are proposed for practical purposes. These distributions are then used to obtain the local temperature, density and pressure profiles in the direction of the force. The results obtained, while being approximate, might provide an alternative for modeling different phenomena involving gases under the effect of relevant external forces (e.g. atmospheric models).
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Three different lines of thinking (mechanical, mixed thermodynamical-mechanical, statistical thermodynamic) are presented to derive the noted barometric formula, which gives the altitude dependence of the pressure of a gas in a gravity field. It is shown that the first two methods can be extended to non-isothermal cases, whereas statistical thermodynamics relies on the concept of thermal equilibrium and its usefulness is limited to the isothermal barometric formula. The temperature changes in the gravity field are taken into account by two different methods: simple conservation of energy, and a more refined line of thought based on the adiabatic expansion of an ideal gas. The changes in gravitational acceleration are also considered in further refinements. Overall, six different formulas are derived and their usefulness is tested on the atmosphere of the Earth. It is found that none of the formulas is particularly useful above an altitude of 20 km because radiation effects make the temperature changes in the atmosphere difficult to predict by simple theories. Finally, the different components of air are also considered separately in the context of the barometric formula, and it is shown that the known composition changes of the atmosphere are primarily caused by photochemical processes and not by the gravity field. Graphical abstract
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Molecules are in permanent motion and interaction. Even though a system might be at rest from a macroscopic point of view, it is restless at the molecular scale. Such continuous molecular motion results in the flow of molecules through space. The number of molecules crossing a boundary per unit time per unit area is denoted as a molecular flux. Three different types of boundaries are considered: Completely permeable, semi-permeable rigid and impermeable rigid boundaries. When the system is contained by rigid (impermeable or semi-permeable) boundaries such molecular flux against the walls produce a net force per unit area that is denoted as pressure. The purpose of this report is deriving mathematical expressions for estimating the molecular flux and pressure on rigid boundaries in ideal gases, using only the classical equations of motion and the distribution of molecular positions and velocities in the system. The pressure at the interior of an ideal gas system is zero because there are no forces acting on the molecules. Thus, the concept of equivalent pressure, as the pressure that would be obtained in the presence of an impermeable rigid boundary, is introduced. The mathematical results presented are based on the following general assumptions: i) Molecular collisions in the gas are negligible (ideal gas assumption). ii) There are no external forces acting on the system. iii) There is no net macroscopic motion of the gas parallel to the surface. iv) Molecular positions are uniformly distributed in the system volume. v) The orthogonal component of molecular velocity is normally distributed. vi) The system considered is large compared to the size of the molecules. The results obtained are consistent with the continuity equation. Furthermore, when the ideal gas system is at macroscopic rest, the classical ideal gas thermodynamic equation of state is obtained. In addition, three different regimes of molecular flux and pressure are observed: The thermal motion regime, the macroscopic regime and the transition regime between the first two.
Book
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Introduction to Aerospace Engineering with a Flight Test Perspective is an introductory level text in aerospace engineering with a unique flight test perspective. Flight test, where dreams of aircraft and space vehicles actually take to the sky, is the bottom line in the application of aerospace engineering theories and principles. Designing and flying the real machines is often the reason that these theories and principles were developed. This book provides a solid foundation in many of the fundamentals of aerospace engineering, while illuminating many aspects of real-world flight. Fundamental aerospace engineering subjects that are covered include aerodynamics, propulsion, performance, and stability and control.
Article
This book is about the atmosphere and humanity's influence on it. For this new edition, Brimblecombe has rewritten and updated much of the book. In the early chapters, he discusses the geochemical, biological and maritime sources of the trace gases. Next, he examines the chemistry of atmospheric gases, suspended particles, and rainfall. After dealing with the natural atmosphere, he examines the sources of air pollution and its effects, with all scenarios updated from the last edition. Scenarios include decline in health, damage to plants and animals, indoor pollution, and acid rain. The final chapters, also revised, are concerned with the chemistry and evolution of the atmospheres of the planets of the solar system. Students with an interest in chemistry and the environmental sciences will find this book highly valuable.
Article
Barometric pressures (PB) near the summit of Mt. Everest (altitude 8, 848 m) are of great physiological interest because the partial pressure of oxygen is very near the limit for human survival. Until recently, the only direct measurement on the summit was 253 Torr, which was obtained in October 1981, but, despite being only one data point, this value has been used by several investigators. Recently, two new studies were carried out. In May 1997, another direct measurement on the summit was within approximately 1 Torr of 253 Torr, and meteorologic data recorded at the same time from weather balloons also agreed closely. In the summer of 1998, over 2,000 measurements were transmitted from a barometer placed on the South Col (altitude 7,986 m). The mean PB values during May, June, July, and August were 284, 285, 286, and 287 Torr, respectively, and there was close agreement with the PB-altitude (h) relationship determined from the 1981 data. The PB values are well predicted from the equation PB = exp (6.63268 - 0.1112 h - 0.00149 h2), where h is in kilometers. The conclusion is that on days when the mountain is usually climbed, during May and October, the summit pressure is 251-253 Torr.
Traites de l'equilibre des liqueurs, et de la pesanteur de la masse de l'air
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Pascal, B. (1663). Traites de l'equilibre des liqueurs, et de la pesanteur de la masse de l'air. Chez Guillaume Desprez, Paris.
Illustrations of the dynamical theory of gases. -Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine
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Maxwell, J. C. (1860). Illustrations of the dynamical theory of gases. -Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 19(124), 19-32.