where p(z) is the mass density of air at height z and g is the acceleration due to gravity. From the ideal-gas law, we obtain where Mair is the average molecular weight of air (28.97 g mol-1). Thus which we can rewrite as dp(z) _ Max gp(z) dz RT(z)

where H(z) = RT(z)/M.dirg is a characteristic length scale for decrease of pressure with height.

The temperature in the atmosphere varies by less than a factor of 2, while the pressure changes by six orders of magnitude (see Table A.8). If the temperature can be taken to be approximately constant, just to obtain a simple approximate expression for p(z), then the pressure decrease with height is approximately exponential where H = RT/M[mg is called the scale height.

Since the temperature was assumed to be constant in deriving (1.5), a temperature at which to evaluate H must be selected. A reasonable choice is the mean temperature of the troposphere. Taking a surface temperature of 288 K (Table A.8) and a tropopause temperature of 217 K, the mean tropospheric temperature is 253 K. At 253 K. H = 7.4km.

Number Concentration of Air at Sea Level and as a Function of Altitude The number concentration of air at sea level is tt\\ PoNa MO) = -^r where Na is Avogadro's number (6.022 x 1023 molecules moF1) and po is the standard atmospheric pressure (1.013 x 10s Pa). The surface temperature of the U.S. Standard Atmosphere (Table A.8) is 288 K, so

(6.022 x 1023 molecules mol"1) (1.013 x 105NnT2)

(8.314Nmmol~l K-l)(288K) = 2.55 x 1025 molecules = 2.55 x 1019moleculescm~3

Throughout this book we will need to know the number concentration of air molecules as a function of altitude. We can estimate this using the average scale height H = 7.4 km and

where n¡¿t (0) is the number density at the surface. If we take the mean surface temperature as 288 K, then «a¡r(0) = 2.55 x 1019 molecules cm 3. The table below gives the approximate number concentrations at various altitudes based on the average scale height of 7.4 km and the values from the U.S. Standard Atmosphere;

z (km)

nair (molecules cm 3)


U.S. Standard Atmosphere"


2.55 x 1019

2.55 x 10I!>


1.3 x I01S

1.36 x i019


6.6 x 1018

6.7 x 1018


3.4 x 1018

3.0 x 1018


1.7 x 1018

1.4 x 10,s


8.7 x 1Û'7

6.4 x 10"

"See Tabic A.8.

"See Tabic A.8.

Total Mass, Moies, and Molecules of the Atmosphere The tola! mass of the atmosphere is

p{z)Aedz where Ae == 4nR2e, the toial surface area of the Earth. We can obtain an estimate of the total mass of the atmosphere using (1.5),

= 4nfilPoH

Using Re 6400 km, H Si 7.4 km, and p0 Si 1.23 kg trT3 (Table A.8), we get the rough estimate:

Total mass S 4.7 x 10l8kg An estimate for the total number of moles of air in the atmosphere is total massMiair

Total moles 9* 1.62 X 1020 mol and an estimate of the total number of molecules in the atmosphere is Total molecules ^l.Ox 1044 molecules

An accurate estimate of the total mass of the atmosphere can be obtained by considering the global mean surface pressure (985.50hPa) and the water vapor content of the atmosphere (Trenberth and Smith 2005). The total mean mass of [he atmosphere is

Total mass (accurate) = 5,1480 x I0iskg

The mean mass of water vapor in the atmosphere is estimated as 1.27 x 1016 kg, and the dry air mass of the atmosphere is

Total dry mass (accurate) = 5.1352 x i01Rkg

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