The Direct Relation between Altitude and Boiling Point Boyd L. Earl University of Nevada, Las Vegas, Las Vegas, NV 89154
A recent call from a local citizen set me thinkine about deriving a direct relationship between altitude andvboiling point. I first provided a rough estimate, after which I thought a bit more and came up with a more accurate results. Here I will proceed in the more logical, and economical, reverse direction. I t is common knowledge tbat water's boiling point decreases with increasing altitude, and of course this applies to all substances. Since the physical parameter whose variation is diredlv resoonsible for this is the atmosoheric oressure. one musi maie the connection between bailing doint and altitudevia the oressure. whichisdlrertlvconnected to both. Many physical>hemist& texts presentthe so-called barometric formula for pressure as a function of altitude P(h) = P(0) exp (-MghlRTl
(1)
Decrease In Bolllng Polnt ot Water at Varlous AltHuds, Asumlng & = 373 K at Zero Altitude Altitude (km)
1
2
3
4
5
6
-AT(Mhomeq4 -AT(Mtromeq6
3.26 3.2
6.57 6.5
9.93 9.7
13.36 12.9
16.85 16.2
35.41 32.4
-
-
ic atmosphere, and those invoked in obtaining eq 3: AV V,, RTb/P, and AH is constant. A "back-of-the-envelope" estimate may be obtained by using the linearized small x approximation in (1 - x ) = -x. This yields l/Tb(0)- lIT,,(h)
where h is altitude, M is the average molar mass of air, and other symbols have standard meanings. This is unrealistic, as it assumes an isothermal atmosphere. As I have pointed out in a previous article'. the P ( h ) function for the more realisticadiabaticatrnospLrecan Gederived ina way that is entirely accessible to students of physical chemistry.
Here, C, is the molar constant-pressure beat capacity of air. Given the P(h) function, one needs a pressure-boiling point relationship: the Clapeyron or Clausius-Clapeyron equation. The most direct route to an equation for boiling point as afunction of altitude is touse the integrated formof the Clausius-Clapeyron equation, 1ITdO) - l/T,(h) = (RIAHI lnIP(h)/P(O)I
(3)
or AT,
-
-
-
A T ~ T ~ ( O-MghIAHT(O) )~
-[MgTb(0)2/AHT(O)]h
(5)
Using values2M = 0.02896 kg mol-I and C, = 29.06 J mol-' for dry air, g = 9.81 m s-2, and T ( 0 ) = 300 K (arbitrary choice),MglC,T(O) = 3.26 X m-I, so this linear approximation is good for h 2 3 X 103 m. Since most altitudes of interest are less than about 3000 m, this provides a reasonable estimate of the altitude coefficient of the boiling point, given that one knows the normal hoiling point and the heat vaporization for the substance in question. Using AH = 40.700 J mol-' for water3 yields .
T,,
-
-3.24
X
10@h
(6)
where h is in meters. Results calculated from eqs 4 and 6 are given in the table. I argued in my prior article' that the calculation of the where AH is the molar enthalpy of vaporization, and I have temperature and pressure profiles of the atmosphere is of used Tb to distinguish the boiling point from environmental pedagogic value because it combines results of tbermodytemperature, T. Substituting directly from 2 for In [P(h)/ namics and hydrostatics to yield a result for a problem of P ( 0 ) ]yields eeneral oractical interest. In the same vein. I submit tbat the material in this article could be presented in physicnl cheml'Th(O' 'ITdh) = (Cnlm In I' - M g h l C ~ T ( 0 ) l ( 4 ) istrv in connection with one-com~onentohase eauilibria. I t is a-"natural", since the decreas; of boiling p o i 2 witb altitude is common knowledge, but possibly its physical basis and almost certainly its quantitative explanation would be ~h~ approximations implieit in this equation are an adiabat. new to students. The simplicity of the result and the relative ease witb which it is obtained, based on combination of prior results, are impressive illustrations of the power of tbermo'Earl, 6. L. J. Chem. Educ. 1982,59,826-827. 2~ ~W. J. physics ~ f, the 4ir, ~ 3rd ed.;hD ~N~~~ ~york,~ ~ ~dynamics. : ~ I believe ~ it is also, useful to expose students to the use of approximations, as in obtaining eq 5, and to the 1964. assessment of the range of validity of the resulting equaAlbelty, R. A. Physical Chemistry, 7th ed.; Wiley: New York, 1987. tions.
-
Volume 67 Number 1 January 1990
45
