Adiabatic Lapse Rates of Planet Atmospheres Harvey F. Blanck Austin Peay State University, Clarksville, TN 37040
I t has recently been suggested (1,2) that derivations of the equation for the dry adiahatic lapse rate and related equations which give the temperature and pressure of planet atmospheres as a function of altitude be included in undergraduate physical chemistry class discussions and textbooks. These derivations have been presented in detail by Earl (2) and may be found along with discussions of atmospheres in many space science and meteorology textbooks (3-9). Using the hydrostatic equation d p = -pgdh, the ideal gas law, and an adiabaticchange of state equation for an ideal gas (I prefer the use of CpdT = Vdp rather than TIT. = [p/p.lRR~),a straightforward derivation yields -dTldh = MglSf,, T = T, - MghlSf,, a n d p = p.[l - MghlSfpT.IcdR provided the heat capacity is treated as being constant. While these derivations are certainly worthwhile additions to physical chemistry material, it seems equally important to discuss the conseouences of these eauations and relate them to actual planet atmospheres where-possible. Notice that the dry adiahatic lapse rate (-dTldh) is independent of pressure and altitude. Thus, the dissimilar pressures of the CO2 dominated atmospheres (95%)of Venus and Mars would not affect their dry adiabatic lapse rates. Venus has a higher acceleration of gravity than Mars (882 cm/sec2 compared to 392 cm/sec2) and also a higher surface temperature (755 K compared to 230 f 60 K). The molar heat caof ('Ol :]I 755 K is pacit). measured n t consrant presaore, 5u.S J mol dea and i15.7 J m d d1.r ill 2:Nl K. lisina these values and the mole weight of C02, the rates calculated for Venus and Mars are 7.8 Klkm and 4.8 Klkm, respectively. For comparison Earth's dry adiabatic lapse rate is approximately 9.7 K h using M = 28.9 glmol, g = 980 cm/sec2, and C, = 3.5 R. -.. If the actual lapse rate is less than the dry adiabatic lapse rate, il warn1 parcel of air will risr n~ttilits reml~eraturewuls to the tempcraturt uf I he curroundinga. Its tempt2raturt~\vill dron a t the d m adiahatic lame if no mixine" occurs and no energy is gained or lost except in doing the work of expansion. When temnerature eauilibrium and hence eaual interior and exterior densities has been achieved, any further vertical displacement would cause a temperature and hence a density change such that the net buoyant force on the parcel would result in a return to its previous equilibrium position.' Warm parcels of air develop a t Earth's surface during the dav. These air warcels then rise and t r a n s ~ o renergy t . . awav t'rmn the surface and thv lupsr rntc d t h e surruuodinrs innate. l'his adiahatir l~urtim creaa>suntil equal t u tht.:~di~l~iltir
r,.
Resented at the 91st meeting of the Tennessee Academy of Science at Austin Peay State University, Clarksville, TN, November 20, 1981. A comment concerning Achimedes' principle seems appropriate. Archimedes' principle is related to the relative number of molecular collisions below compared to above an object in a fluid. Thus a He filled balloon rises because the pressure difference, caused by more molecular collisions below than above, is sufficient to overcome the weight of the balloon. Moelwyn-Hughes ( 12)uses the term "net kinetic force" which keeps the kinetic-moleculartheory more sharply in focus when discussina buovancv. This &uld be the basis for a problem in calculating the height of the base of a cloud given T, and the relative humidity of a warm parcel of air at the surface
'
Calculated Venusian Atmospheric Properties Variable CnB
Average
Gb
Baltzmann'
c.
Calculated using as s function of temperature. aCalculated using C, at an approximately average temperahlre. c C a l ~ ~ l a t eusing d Boitrmann distribution (barometric formuia p = p-e-*WR?
d the atmusphere gradually btcumrs thicker since warmer air u.111 uuntiltue to ri..~while trinwstng this adiabatic reaim. ' h e putenrial remperaturr ( H I , which i; the 1vtnpt.rtnurr any given warcel of air would have if it were comnressed adiaba&all;to 1000 millibars, continues to rise f i r this adiahatic wortion. Convective mixine of the lower wortion of the atmosphere is the major mech&ism for thermal energy transport awav from the surface. Anvone who has ever flown in a small airciaft is well aware of the turbulence associated with this adiabatic reaion on a hot day! Furthermore, the top of this region may sometimes he seenwhile flying, since air p&utants generated at the surface will only he mixed in this region and hence a pollution horizon is often very well defined. If the parcel of air cools snfficientlv, condensation occurs and heat bill he released to the parcel-which abruptly reduces the rate of cooling with an increase in altitude.2 This new rate of cooling is called the moist adiabatic lapse rate. Although the moist adiahatic lapse rate decreases with temperature, it is nearly i~lwayamuch Ieqs than the dry i~dii~lunir lnpse rareand orten les, than ihv lupse r,ire of the surruundinr: air to relatively hidh altiti~des.('ons~qur~ltlg. tups dcl~,uds.uftrn r i s . to great heights. Bv"wav,of contrast. Venus is nermanentlv and heavilv cloud covered which greaiy reduces surface heating and cooling variations. Application of t h e p and T equations as a function of altitude produce rather accurate pressure and temperature wrofiles of Venus' atmoswhere from 0 to 40 km (see the accompanying uhlri. Thr trtltperature of \'enus drcrrasrs from 7% I( ttr a~wr~~xiniatrlv 430 I< and thv rxessure auun,xitni~trl\. . .. drops from approximately 91 atm to approximately 4 atm as the altitude increases from 0 to 40 km (10.11). Much of the atmosphere of Venus is actually adiahatic. The heat capacity value is not especially critical. The E, value at an average tempergure is quite satisfactory. If desired, the values of p , T, and C, can he readily calculated by a computer program using small increments in the altitude from 0 to 40 km. (A hardcopy of this program is available upon request.) Using these derivations and examples in physical chemistry classes helps convince students that perhaps physical chemistry really is relevant especially in this age of space travel and planetary probes. &
ill Blanck, H. F.J. TennasseeAcodemyo/Science,57,41 (19821. (21 E a l , B.L., J.C H E M E D U C . , 1 ~19 ~82 , 1~. ~ ~ (31 Byers, Horace Robsit, "General Meteorology," MeGraw-Hill, New York. 1919, p.
72-95.
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Number 8
August 1983
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(4) Flesgle, Robert G.,snd Businger, Jmst A.,"An Introductionto AtmhspheriePhmia: Academic Press, New York. 1963, p. 4 M 7 . (5) Hsymes, Robert C.. "Introduction to Space Science: John Wilev, New York, 1971, p. 5745. 16) Hw.Seymour L., "Introduction toTheoretical Meteorology," Holt-Dryden. New York, 1969. p. 30-32,80-95. (7) Petternon, Suerre, '"Intmduetion ta Meteomlogy."3rd ed., McGrsw-Hill, New York. 1969, p. 104-115,289-292.
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Journal of Chemical Education
181 Riehl, Herbert. "Introduction to the Atmosphere: 3rd ed.. McGrsw-Hill. NevYork, 1978, P. 54-64. 191 Rogen, R. R.,"AShort Course inCloudPhysies," Znded.,Pergamon Prpss, NewYork, 1979. p. 1 4 8 . (10) Fledrick, Lawrenee W., and Baker, Robert H.,"An Introduction ta Astronomy: 9th ed.. 1981,p. 160. (11) Muhleman. P 0 .Orton, G. S.,and Berge, G L.,Astrophys 5,234,783 11979). (12) Moelwn-Hughes, E. A. "A Short Course of Phyaical Chemistry," American Elseviei. New York, 1987,p. 25.
