In the present section three generalization of the barometric formula are presented that allow to amplify the range of applicability of the formula.


Temperatures throughout the troposphere, the lowest layer of Earth’s atmosphere, cover a wide range of values. This layer is heated from below, and it is wannest at the bottom near Earth’s surface (average temperature = 15°C), and coldest at its top (-57°C), and, typically, the temperature drops about 6.5°C per km (= p) increase in altitude. The troposphere extends from Earth’s surface up to a height of 6 km (in polar regions in winter) to 17 km (middle latitudes) above sea level, the average altitude being 13 km. Most of the mass (about 75%-80%) of the atmosphere is in this region, and almost all weather occurs within it. The sunlight heats the Earth’s surface that radiates the heat back into the adjacent atmosphere. Temperatures in the troposphere, both at the surface and at various altitudes, do vaiy based on latitude, season, time of day or night, regional weather conditions, and in some circumstances, the temperature at the top of the troposphere can be as low as -80°C. The phenomenon known as “temperature inversion” means that the temperature in some part of the troposphere gets wanner with increasing altitude, contrary to the normal situation.32 Consider the case of uniform gravitational field, g, with a vertical temperature gradient, where the temperature changes linearly with height, as in the following eq (2.14), which is a good approximation for the troposphere.

Here p [K/km] is a positive constant. Now, by the aid of eq (2.3) or (2.5) we obtain,

Equation (2.15) is a good model for the pressure dependence on altitude up to 11 km with: g0 = 9.8 m s-2,/>0 = 103 Pa, TQ = 288 К (15°C), and p = 6.5 K/km.


This constant varies with latitude, longitude, and elevation. These variations are rather small and lead to a minor error. The general case for an isothermal atmosphere where the acceleration g depends on altitude z, is given by the following equation.

According to the law of gravitation, and noting that the mass of the atmosphere is quite small compared to Earth’s mass we have,

Here, M = mass of the Earth, and at the Earth’s surface: g0 = GM/R2. Now, inserting eq (2.17) into eq (2.16), we have eq (2.18) that integrates into eq (2.19),

Now, with 1 IH=mgJkT we obtain,

Calculations show that a noticeable difference between data obtained from eqs (2.1) and (2.20) shows up only for г > 0.017? ~ 64 km (~ 8%, TQ = 288K), that is, well above the stratosphere. Following eq (2.20) pressure approaches a nonzero value for г —» oo, which is physically questionable. This aspect shows that a static and isothermal atmosphere is intrinsically unstable. A deeper analysis of this problem is discussed in Ref. 1.


Suppose that Earth atmosphere and solid Earth rotate with a unique angular velocity со (for the troposphere this is quite appropriate), this means that due to this rotation, the weight of gas is not the same at the pole than at the equator. At the pole, acceleration of gravity is described by eq (2.17), that is, g(z) = g0(+:/R)-2. At the equator, instead, the weight is decreased by the centrifugal force ?nco2(R +c), that is, the effective acceleration is smaller than the acceleration of gravity and obeys eq (2.21), where Ф(г) is given by eq (2.22). Assuming, со = 7.27-105 rad/s, gg = 9.8 m s2, and R = 6.4-106 m, we have co2R/ g0 ~ 3.45-10'3. It is clear from eq (2.21) that the centrifugal force becomes important (change in gravity acceleration > 1%) for (:) >0.01, that is, for :/R > 0.43, which leads to a large г = 2700 km.

If it is assumed that atmosphere and Earth rotate as a whole, irrespective of height, the upper limit of the atmosphere at the equator can be obtained from eq (2.21) by setting g(z) = 0. Then, Ф(:) = 0 —*■ :/R = 5.6, and the result is a meaningless z = 36,000 km. In fact, the density of outer space is attained for altitudes lower than 1000 km. Escape of molecules, atoms, and ions from the upper atmosphere occurs by thermal and photochemical mechanisms still in the presence of a significant inward force. Combining eqs (2.21) and (2.16), and reminding that Я1 = mgJkT we obtain for the barometric equation in the case of an isothermal atmosphere,

Integrating this expression we finally obtain,

Calculations show a noticeable difference (~ 2%) between data obtained from eqs (2.24), and (2.20) for r > 0.017? « 64 km, that is, well above the stratosphere. The major problem in applying the barometric formula to the real atmosphere, however, derives from the fact that the atmosphere is not in equilibrium.


Suppose a shaft is drilled down to the center of the Earth, notwithstanding the technical impossibility of this feat, namely, owing to the iimnense pressures and temperatures that exist inside the Earth, and to the physical state of its inner layers, it is interesting to imagine what would be the depth dependence of air pressure within this imaginary shaft. Let us first have a short excursus throughout history about this topic (a more detailed excursus with appropriate references see Ref. 2).


The motion of an object (neglecting drag) dropped in a bottomless shaft was again considered by Hooke in 1679. The main point under discussion was the effect of Earth’s rotation on the trajectory. Hooke obtained the correct result qualitatively: The object should oscillate like a pendulum, describing an ellipse, hi fact, an object dropped in a shaft connecting the poles of a homogeneous and spherical Earth behaves as a one-dimensional harmonic oscillator and strictly obeys Hooke’s “law,” although this is not the present standard pedagogical example. In 1882 the respected French civil engineer and applied mathematician Ё. Collignon (1831-1897) speculated on the possibility of travel between cities by means of long linear tunnels inside the Earth, in a kind of partial free-fall planetary subway, for which the transit time in the absence of drag is 42 min, independently of the location of the two cities. An account of his ideas, published on a semi-humorous tone in the scientific periodical La Nature, is suggestively entitled “From Paris to Rio de Janeiro in 42 minutes and 11 seconds” In it, the effect of pressure is discussed, and it is considered an insurmountable problem. Numerical estimates of the enormous pressures at several depths (but provided with no computational details) are given, but differ from the calculations given below by several orders of magnitude.


Assuming for simplicity that air temperature and Earth’s density are both uniform, eq (2.18) applies, where the acceleration of gravity now is,

Equation (2.18), with -R < : < 0, and with eq (2.25) now, after integration, becomes (reminding that, g0 = GM/R:, and 1 /Н = mg0/kT0),

As the reader can notice the dependence is similar to eq (2.1), apart from the multiplicative factor (<1) in the argument of the exponential that slightly reduces the variation, owing to the decrease of g with depth. The deepest gold mines in South Africa attain a depth of 3.9 km, ca., for which one obtains p = 1.6 atm, in good agreement with the observations. While eq (2.26) predicts a pressure of 1000 atm for : = -58 km, for г = -R, instead, the calculated pressure becomes: p(-R) = p0exp (R/2H) = 10165p0, clearly, a meaningless value, as the air for pressures of few tons of atm ceases to behave as an ideal gas. A more detailed calculation with the van der Waals equation is discussed in Ref. 2.


Whenever we take a formula that is valid only for an ideal gas under conditions of equilibrium and apply it to real cases, it is not a surprise if it does not fit the data perfectly. Nevertheless, pressure seldom departs from the average value by more than a few percent, and within this restriction the barometric equation given by eq (2.1) does its job. The nonideality of a gas has instead a much more dramatic influence on the barometric equation at negative heights.

It should be remarked that the atmosphere as a whole is never in a state of equilibrium, as it continuously exchanges mass and energy with its surroundings and this is why there is weather. To better understand this last topic, we should know something about the Bernoulli’s principle of fluid dynamics33 that is though valid only for ideal fluids. It states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid’s potential energy, it can be applied to various types of fluid flow, and there are different forms of Bernoulli’s equation for different types of flow. In Ref. 2 and 5 a full discussion of this principle is given.


  • • barometric formula
  • • pressure
  • • mathematics
  • • generalization
  • • ideal gas


1. Berberan-Santos, M. N.; Bodunov, E. N.; Pogliani, L. On the Barometric Formula. Am. J. Phys. 1997, 65, 404-412.

  • 2. Berberan-Santos. M. N.; Bodunov, E. N.; Pogliani, L. On the Barometric Formula Inside the Earth. J. Math. Chew. 2010, 47, 991-1004.
  • 3. Hall, D. S.; Standish. T. E.: Behazin, M; Keech. P. G. Corrosion of Copper-Coated Used Nuclear Fuel Containers Due to Oxygen Trapped in a Canadian Deep Geological Repository. Coiros. Eng. Sci. Techn. 2018, 53, 309-315.
  • 4. Eymuller, C.; Wanninger, C.; Hoffmann. A.: Reif. W. Semantic Plug and Play - Self- Descriptive Modular Hardware for Robotic Applications. Int. J. Semant. Comput. 2018, 12, 559-577.
  • 5. Alonso-Gonzalez, E.; et al. Daily Gridded Datasets of Snow Depth and Snow Water Equivalent for the Iberian Peninsula from 1980 to 2014. Earth Syst. Sci. Data 2018,10, 303-315.
  • 6. Hall, D. S.; Standish, T. E.; Behazin, M.; Keech. P. G. Corrosion of Copper-Coated Used Nuclear Fuel Containers Due to Oxygen Trapped in a Canadian Deep Geological Repository. Int. J. Coiros. Process. Coiros. Conti: 2018, 53, 309-315.
  • 7. Zhao, F.; Luo. H.; Zhao, X.; Pang. Z.; Park, H. HYFI: Hybrid Floor Identification Based on Wireless Fingerprinting and Barometric Pressure IEEE T. hid. Info. 2017,13, 330-341.
  • 8. Akinnubi, R. T.; Adeniyi, M. O. Modeling of Diurnal Pattern of Air Temperature in a Tropical Environment: Ile-Ife and Ibadan, Nigeria. Model Earth Syst. Environ. 2017, 3, 1421-1439.
  • 9. Masse. F.; Gonzenbach, R.; Paraschiv-Ionescu, A.; Luft, A.R.; Aminian, K. Wearable Barometric Pressure Sensor to Improve Postural Transition Recognition of Mobility- Impaired Stroke Patients. IEEE Pans. Neural Syst. Rehabil Eng. 2016,24,1210-1217.
  • 10. Wua, Z.; Zhou, X.; Liu, X.; Ni, Y.; Zhao, K.; Peng, F.: Yang, L. Investigation on the Dependence of Flash Point of Diesel on the Reduced Pressure at High Altitudes. Fuel 2016,181, 836-842.
  • 11. Garcia-Diez, R.; Gollwitzer, C.; Rrumrey, M. Nanoparticle Characterization by Continuous Contrast Variation in Small-Angle X-Ray Scattering with a Solvent Density Gradient. J. Appl. Ciyst. 2015, 48, 20-28.
  • 12. Fajardo, S.; Frankel. G. S. Gravimetric Method for Hydrogen Evolution Measurements on Dissolving Magnesium. J. Electrochem. Soc. 2015,162, C693-C701.
  • 13. Cleasby. I. R.; Wakefield. E. D.; Bearhop, S.: Bodey, T. W.; Votier, S. C.; Hamer, К. C. Three-Dimensional Tracking of a Wide-Ranging Marine Predator: Flight Heights and Vulnerability to Offshore Wind Farms. J. Appl Ecol. 2015, 52, 1474-1482.
  • 14. Xia. H.; Wang, X. ; Qiao, Y.; Jian, J.; Chang, Y. Using Multiple Barometers to Detect the Floor Location of Smart Phones with Built-in Barometric Sensors for Indoor Positioning. Sensors 2015, 15, 7857-7877.
  • 15. Sphioni, J.; et al. Climate of the Carpathian Region in the Period 1961-2010: Climatologies and Trends of 10 Variables. Int. J. Climatol 2015, 35, 1322-1341.
  • 16. Fabien Masse, F.; Gonzenbach, R. R.; Ararni, A.; Paraschiv-Ionescu, A.; Luft, A. R.; Aminian, K. Improving Activity Recognition Using a Wearable Barometric Pressure Sensor in Mobility-Impaired Stroke Patients. J. NeuroEng. Rehab. 2015,12, 1-15.
  • 17. Ignaccolo, R.: Franco-Villoria. M.; Fasso, A. Modelling Collocation Uncertainty of 3D Atmospheric Profiles. Stoch. Environ. Res. Risk. Assess. 2015, 29, 417-429.
  • 18. Masse. F.; Вошке, A. K.: Chardonnens, J. ; Paraschiv-Ionescu, A.: Aminian, K. Suitability of Commercial Barometric Pressure Sensors to Distinguish Sitting and Standing Activities for Wearable Monitoring. Med. Eng. Phys. 2014, 36, 739-744.
  • 19. Igoe, D. P.; Parisi, A.; Carter, B. Smartphone-Based Android App for Determining ЦЛА Aerosol Optical Depth and Direct Solar Irradiances. Photochem. Photobiol 2014, 90, 233-237.
  • 20. Marthews, T. R.; Mallii, Y.; Iwata, H. Calculating Downward Longwave Radiation Under Clear and Cloudy Conditions Over a Tropical Lowland Forest Site: An Evaluation of Model Schemes for Hourly Data. Theor. Appl. Climatol. 2012,107, 461-477.
  • 21. Dubinova. A. A: Exact Explicit Barometric Formula for a Warm Isothermal Fermi Gas. Tech. Phys., 2009, 54, 210-213.
  • 22. Monti. D.; Ariano, P: Distasi, C.: Zamburlin, P: Bemascone. S.; Ferraro. M. Entropy Measures of Cellular Aggr egation. PhysicaA 2009, 388, 2762-2770.
  • 23. Bottecchia, O. L. The Barometric Formula as Resource for Teaching Chemistry. Quint. Nova 2009, 32, 1965-1970.
  • 24. Fink, J. K. Equilibrium, hi Physical Chemistiy in Depth. Springer: Berlin. Heidelberg, 2009.
  • 25. Lopez-Moreno, J. I.: Goyette, S.: Beniston, M. Impact of Climate Change on Snowpack in the Pyrenees: Horizontal Spatial Variability and Vertical Gradients. J. Hydro!. 2008, 374, 384-396.
  • 26. Bohm, T. D.: Griffin, S. L.; DeLuca Jr., P. M.; DeWerd, L. A. The Effect of Ambient Pressure on Well Chamber Response: Monte Carlo Calculated Results for the HDR 1000 Plus. Med. Phys. 2005. 32, 1103-114.
  • 27. Aim, S. R.; Johnson T. C. Carbon Cycling in Large Lakes of the World: A Synthesis of Production, Burial, and Lake-Atmosphere Exchange Estimates. Global Biogeochem. Cy., 2007, 21, 1-12, doi: 10.1029/2006GB002881.
  • 28. Pantellini. F. G. E. A Simple Numerical Model to Simulate a Gas in a Constant Gravitational Field. Am. J. Phys. 2000, 68, 61-68.
  • 29. Chandrasekhar, S. Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys. 1943,15, 1-91.
  • 30. Van Kamperi, N. G. Stochastic Processes in Physics and Chemistiy. 2nd ed; Springer: Berlin, 1985; pp. 195-210.
  • 31. Reif, F. Fundamentals of Statistical and Thermal Physics. McGraw-Hill: New York, 1965.
  • 32. html&edu=high.
  • 33.
< Prev   CONTENTS   Source   Next >