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On the Barometric Formula Inside the Earth J Math Chem (2010) 47:990–1004 DOI 10.1007/s10910-009-9620-7 ORIGINAL PAPER On the barometric formula inside the Earth Mario N. Berberan-Santos · Evgeny N. Bodunov · Lionello Pogliani Received: 27 June 2009 / Accepted: 16 October 2009 / Published online: 30 October 2009 © Springer Science+Business Media, LLC 2009 Abstract The assumptions leading to the barometric formula are discussed, with remarks on the influence of temperature, gravitational field, Earth rotation, and non- equilibrium conditions. A generalization of the barometric formula for negative heights, e.g., for pressure inside shafts and deep tunnels, is also presented, and some surprising conclusions obtained. Related historical aspects are also discussed. Keywords Barometric equation · Non-ideality · Positive and negative heights · Historical and technological aspects 1 Introduction In a previous article the barometric formula [1] that gives the pressure dependence with height of an isothermal and ideal gas was discussed. Generalizations of the baro- metric formula for a non-uniform gravitational field and for a vertical temperature gradient were also presented together with a brief historical review. The effect of a gravitational field on fluids and quantum particles as well as a discussion of the van M. N. Berberan-Santos Centro de Química-Física Molecular and IN-Institute of Nanoscience and Nanotechnology, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1049-001 Lisboa, Portugal e-mail: [email protected] E. N. Bodunov Department of Physics, Petersburg State Transport University, 190031 St. Petersburg, Russia e-mail: [email protected] L. Pogliani (B) Dipartimento di Chimica, Università della Calabria, via P. Bucci, 14/C, 87036 Rende, CS, Italy e-mail: [email protected] 123 J Math Chem (2010) 47:990–1004 991 der Waals equation of state was carried out in a series of articles. [2–4] Worth citing about a similar type of problem is a quite recent paper by Dubinova [5] that presents an exact barometric equation for a warm isothermal Fermi gas. In the present paper we will deepen the discussion on the barometric formula with additional remarks on the influence of temperature, gravitational field, Earth rota- tion, and non-equilibrium conditions. A generalization of the barometric formula for negative heights, i.e., for pressure inside shafts and deep tunnels, is also presented. In its simplest form, the barometric formula relates the pressure p(z) of an isother- mal, ideal gas of molecular mass m at some height z to its pressure p(0) at height z = 0, where g0 is the standard acceleration of gravity, k the Boltzmann constant, m the molecular mass, and T the temperature [1,6], mg z z p(z) = p exp − 0 = p exp − . (1) 0 kT 0 H −1 With a scale height H = kT/mg0 = 8.4kmif T = 288 K and m = 29 g mol . The numerator of the quotient in the exponential of Eq. 1 is the potential energy of a molecule, while the denominator, except for a constant numerical factor, is the mean kinetic energy of a molecule. This means that the decreasing pressure with height is the result of a balance between gravity pulling downward and random thermal motion in all directions, upward inclusive. In spite of its approximate nature, namely owing to the assumption of a static iso- thermal atmosphere in thermodynamic equilibrium, and also that g is independent 2 of height, i.e., g(z) = g0 = 9.8m/s ,Eq.1 applies reasonably well to the lower troposphere, i.e., for altitudes up to 6km, the error being less than 5%, and also to the stratosphere up to 20km, where T = 217 K, that is, T = −57 ◦C[7,8]. Certain assumptions that are necessary for the derivation of the simple barometric formula do not apply to the real atmosphere. Equation 1 hides another simplifying assumption, i.e., that the Earth is flat and that it has no finite vertical extent. To get round this problem is to write the hydrostatic equation in spherical polar coordinates and assume that atmospheric variables depend only on the radial coordinate. 2 Discussion of the assumptions 2.1 The atmosphere is not isothermal Temperatures in the real atmosphere range widely, between about 15 ◦C on the surface (average value) to −100 ◦C in the upper mesosphere, not to mention the thermosphere, a highly rarefied layer whose temperatures can exceed 2,000 ◦C. Consider the case of uniform gravitational field but with a vertical temperature gradient. Assuming the fol- lowing linear variation of temperature with height, which is a good approximation for the troposphere (z < 10 km) [7,8], T (z) = T0 − βz. (2) 123 992 J Math Chem (2010) 47:990–1004 Here β is a positive constant in K/m (frequently called temperature lapse rate, often given in K/ km), we have a result already obtained in [1]. / β βz mg0 k p(z) = p0 1 − . (3) T0 This equation represents well the pressure dependence on altitude for the whole tro- 5 ◦ −1 posphere (up to 11km), with p0 = 10 Pa, T0 = 288 K (15 C) and β = 6.5Kkm . If Eq. 1 is used instead, the best empirical fit is obtained with H = 7.8km. The fall of temperature with altitude in the troposphere is due to the fact that air is warmed mainly from the surface of the planet. This fall is, however, smaller than could be expected, because of convection that occurs up to the tropopause [9]. On the other hand, there is a temperature rise in the stratosphere (β =−1.0Kkm−1 from z = 20 to 32 km [8]). This increase is associated with ozone, which concentrates in a layer ca. 20 km thick, and centered at an altitude of ca. 30km. This layer strongly absorbs ultraviolet light from the Sun, and subsequently releases the corresponding energy as heat [9]. A flat temperature minimum at −57 ◦C is observed between 11 and 20 km [7,8], corresponding to a compromise between the cooling and heating profiles. 2.2 The acceleration of gravity is not a constant This constant, in fact, varies with latitude, longitude, and elevation. These variations, however, are quite small and lead to only a small error. In general case for an isothermal atmosphere, ⎛ ⎞ z p(z) mg(u) = exp ⎝− du⎠ . (4) p0 kT 0 Acceleration g depends on altitude z. According to the law of gravitation, and noting that the mass of the atmosphere is very small compared to Earth’s mass, mM mg(z) = G , (5) (R + z)2 where M is the mass of the Earth and R is its radius, while at the Earth’s surface M g = g(R) = G (6) 0 R2 hence 1 g(z) = g0 . (7) (1 + z/R)2 123 J Math Chem (2010) 47:990–1004 993 Inserting Eq. 7 into Eq. 4 we have ⎛ ⎞ z ( ) p z = ⎝−mg0 du ⎠ . exp 2 (8) p0 kT0 (1 + u/R) 0 Integration of this equation gives ( ) p z = −mg0 R − 1 exp 1 + / p0 kT0 1 z R mg z 1 1 z = exp − 0 = exp − . (9) kT0 1 + z/R 1 + z/R H Calculations show that a noticeable difference between data obtained from Eqs. 1 and 9 shows up for z > 0.01R ≈ 64 km(≈ 8%, T0 = 288 K), i.e. well above the stratosphere. Notice that it follows from Eq. 9 that pressure and molecular concentration do not go to zero as the altitude goes to infinity, which is not physically possible. This aspect, already discussed in [1] for the general 3D case, shows that a static and isothermal atmosphere is intrinsically unstable. 2.3 The Earth spins Suppose that the Earth atmosphere is rotating with the same angular velocity ω as the Earth (for the lower atmospheric layers this is appropriate). Due to rotation, the weight of gas is different at the pole and at the equator. At the pole, acceleration of gravity is described by Eq. 6. At the equator, the weight is decreased by the centrifugal force mω2(R + z), i.e., the effective acceleration is smaller than acceleration of gravity and obeys the equation ω2 R ( ) = 1 − ω2 ( + / ) = 1 − ( + / )3 . g z g0 2 R 1 z R g0 2 1 1 z R (1 + z/R) (1 + z/R) g0 (10) −5 2 6 Assuming that ω = 7.27 × 10 rad/s, g0 = 9.8m/s , R = 6.4 × 10 m, we have 2 −3 ω R/g0 ≈ 3.45 × 10 . It is clear from Eq. 10 that the centrifugal force becomes important (change in gravity acceleration ≥ 1%) for ω2 R (1 + z/R)3 ≥ 0.01 ⇒ z/R ≥ 0.43, (11) g0 which leads to a large value (z >2,700km). If it is assumed that the atmosphere rotates with the Earth’s as a whole, i.e., irrespective of height, the upper limit of the atmosphere 123 994 J Math Chem (2010) 47:990–1004 at the equator can be obtained from Eq. 10 by setting the acceleration to zero, ω2 R 1 − (1 + z/R)3 = 0 ⇒ z/R = 5.6, (12) g0 which leads again to a very large value (z = 36,000km). This is of course meaning- less, considering that the density of outer space is attained for altitudes lower than 1,000km. Escape of molecules, atoms and ions from the upper atmosphere occurs in fact by thermal and photochemical mechanisms still in the presence of a significant inward force. Using Eqs. 10 and 4, we obtain for the barometric equation in the case of an iso- thermal atmosphere, ⎛ ⎞ z p(z) mg ω2 R = ⎝− 0 1 − ( + / )3 ⎠ . exp 2 1 1 u R du (13) p0 kT0 (1 + u/R) g0 0 Integration of this equation gives, instead of Eq.
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