![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 .](https://www.researchgate.net/profile/Hugo-Hernandez-16/publication/349822552/figure/fig4/AS:998009680703488@1614955458442/Barometric-models-comparison-in-the-troposphere-Red-line-US-Standard-Atmosphere-USSA_Q320.jpg)
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Vol. 5, 2020-14
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (1 / 22)
www.forschem.org
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)
www.forschem.org
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)
www.forschem.org
(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)
www.forschem.org
(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)
www.forschem.org
(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)
www.forschem.org
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)
www.forschem.org
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)
www.forschem.org
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)
www.forschem.org
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
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (14 / 22)
www.forschem.org
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
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (15 / 22)
www.forschem.org
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
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (16 / 22)
www.forschem.org
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
1
2
3
4
5
6
7
8
9
10
11
Nitrogen
78.08%
78.32%
78.56%
78.80%
79.05%
79.30%
79.55%
79.81%
80.07%
80.33%
80.60%
80.87%
Oxygen
20.95%
20.74%
20.54%
20.33%
20.11%
19.90%
19.67%
19.45%
19.22%
18.98%
18.75%
18.50%
Argon
9.34E-1%
9.02E-1%
8.70E-1%
8.38E-1%
8.07E-1%
7.76E-1%
7.46E-1%
7.16E-1%
6.86E-1%
6.57E-1%
6.28E-1%
6.00E-1%
Carbon
Dioxide
3.60E-2%
3.43E-2%
3.27E-2%
3.10E-2%
2.95E-2%
2.79E-2%
2.65E-2%
2.50E-2%
2.36E-2%
2.23E-2%
2.09E-2%
1.97E-2%
Neon
1.82E-3%
1.87E-3%
1.92E-3%
1.98E-3%
2.04E-3%
2.10E-3%
2.17E-3%
2.24E-3%
2.32E-3%
2.40E-3%
2.48E-3%
2.57E-3%
Helium
5.24E-4%
5.67E-4%
6.15E-4%
6.68E-4%
7.27E-4%
7.93E-4%
8.67E-4%
9.50E-4%
1.04E-3%
1.15E-3%
1.27E-3%
1.40E-3%
Methane
1.60E-4%
1.67E-4%
1.74E-4%
1.82E-4%
1.90E-4%
1.99E-4%
2.08E-4%
2.18E-4%
2.29E-4%
2.41E-4%
2.53E-4%
2.67E-4%
Krypton
1.14E-4%
9.56E-5%
7.98E-5%
6.64E-5%
5.49E-5%
4.52E-5%
3.71E-5%
3.02E-5%
2.45E-5%
1.97E-5%
1.58E-5%
1.26E-5%
Hydrogen
5.00E-5%
5.45E-5%
5.94E-5%
6.50E-5%
7.12E-5%
7.82E-5%
8.60E-5%
9.49E-5%
1.05E-4%
1.16E-4%
1.30E-4%
1.45E-4%
Nitrous Oxide
3.00E-5%
2.86E-5%
2.72E-5%
2.59E-5%
2.46E-5%
2.33E-5%
2.20E-5%
2.08E-5%
1.97E-5%
1.85E-5%
1.74E-5%
1.64E-5%
Xenon
8.70E-6%
6.26E-6%
4.47E-6%
3.16E-6%
2.22E-6%
1.55E-6%
1.07E-6%
7.28E-7%
4.92E-7%
3.29E-7%
2.17E-7%
1.42E-7%
Avg. Mol.
mass (g/mol)
28.97
28.95
28.94
28.93
28.92
28.90
28.89
28.88
28.86
28.85
28.84
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
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (18 / 22)
www.forschem.org
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
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (21 / 22)
www.forschem.org
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
hugo.hernandez@forschem.org
26/08/2020 ForsChem Research Reports Vol. 5, 2020-14 (22 / 22)
www.forschem.org
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).
