Atmospheric CO2 – Atmosphere Elements

Density is a measure of the number of particles per volume. Recall that one form of the Beer-Lambert defines attenuation constant as the product of absorption or scattering cross-section of each particle and the number of particles per volume … particle density.

A uniform particle distribution can not be assumed along a vertical atmospheric optical column, particularly in scattering elements. Atmospheric pressure/density is of concern in the determination of number of attenuation/absorption elements within the optical volume.

The particle density varies with temperature, pressure, and altitude. This is true with scattering particles as well as absorption particles.

The optical depth expression may also be expressed in terms of density as:

$d \Theta\;=\;-\Phi_o\,\kappa\,\rho\,dz$

in which describes the opacity of the material and is the material density.

Particle density is not temperature insensitive.
The absorption probability is generally much higher closer to the reflecting surface (assumed a perfectly reflecting Earth).

Ideal Gas Relationship

Density may be determined from the ideal gas relationship PV = nRT

From this, volume is found:

The number of mols of a gas is found from the ratio of the sample mass to the chemical molar mass:

Density is mass/volume

The molar mass of CO2 is 44.01 g/mol. R is the gas constant with a value of 8.31446 J m^-1 K^-1. Pressure P has units of kg m^-1 s^-2 → Pa

Checking units …

$density\;\Rightarrow\;\;\dfrac{\,P\,M\,}{R\,T}\;=\;\dfrac{\;P\left(\frac{kg}{\,m\,s^2\,}\,\right)\,\times\,M\left(\frac{g}{\,mol\,}\right)\;}{R\left(\frac{\frac{\,kg\,m^2\,}{s^2}\,}{mol\,K\,}\,}\,\right)\,\times\,T(K)}\;=\,\dfrac{\,P\,M\,}{R\,T}\left(\dfrac{g}{\,m^3\,}\right)$

The ideal gas law states a relationship between pressure and density:

$P\;=\;\rho\,R\,T$

where P is pressure, $\rho$ is density, R is the gas constant, and T is temperature in Kelvin.

The value for the gas constant is related to the gas of interest such that:

$R_{gas}\;=\;\dfrac{R_c}{\,mW\,}$

where R_c [1] is the universal gas constant and mW is the molecular weight (molar mass) of the gas.

The ideal gas law could also be expressed

$P\;=\;\rho\,T\,\dfrac{R_c}{\,mW\,}$

The gas constant for a dry atmosphere at sea level would be calculated as

where mw_e represents “everything else”: a mixture of all the other gases (except water vapour) in the atmosphere. At such a low percentage, the contribution is considered insignificant.

[1] per NIST: R_c = 8314.3 J K^-1 kg^-1

Atmospheric models

Ideal Gas

The ideal gas expression is:

P is pressure, R is the gas constant, T is temperature, V is volume, and n is quantity. n/V is density in mol/m3. To convert to number of molecules, multiply by Avogadro’s Number:

The nominal concentration of CO2 is 400 ppmv. The molar mass of CO2 is 44.01 g/mol

Using the appropriate values for the physical constants:

The volume of a mol of ideal gas is a physical constant wherein V = 0.22414 m³/mol. The nominal CO2 concentration is given in units of “per volume”, therefore the volume fraction of CO2 is:

It will be convenient to define density in terms of molecules per volume:

Dry Air

The assumptions required to use the ideal gas representation get stretched pretty far in the real world. A basic model could be based on the actual but idealized constituents of the atmosphere.

Using “dry air” parameters

massDensity = 1.2041 kg/m3

molarMass = 28.9644e-3 kg/mol

The molecular density of CO2 is found to be:

Standard Atmosphere

Yet another basic model is based on “standard” parameters. The parameters for the troposphere were defined in 1962; they were reviewed but not changed in 1976 for atmosphere below 32km. The intended altitude for this analysis is 10 km.

The air is clean, dry, and of uniform density. The molecular weight is 28.9664 g/mol

Sea-level values:

            mW_atm = 28.9664 g/mol
            P_atm = 101325 Pa
            T_atm = 288.15 K
            r_atm = 1.2250 kg/m³
            g_atm = 9.80665 m/s²
            R_atm = 8.3143 J mol^-1 K^-1

Comparing basic atmosphere models:

Model	10²¹ molecules/m³	ng/cm³
ideal	10.7417	785.40
dry	9.9915	730.19
stdAtm	10.1878	744.53
AVG	10.307	753.37

Barometric Pressure and Temperature

The previous calculations were for sea-level conditions – recall that density is defined as:

and is dependent on both pressure and temperature – which in turn vary with altitude.

Density is a function of pressure and temperature.

Representative empirical measurements: Spring 30^oN

The MAG line represents the 10 km maximum altitude of interest. This is close to the boundary of the troposphere and stratosphere but far enough that tropospheric conditions remain valid for estimation purposes.

Temperature is essentially a linear function of altitude. From approximations derived from the above charts:

TC_Z = -6.496 mK/m

A more detailed presentation of atmospheric density may be obtained from Density/Altitude information for aircraft. The Density-Altitude value expresses an effective altitude relative to a standard atmosphere. This is a function of temperature, pressure, and to a lesser degree, humidity. A hot and humid atmosphere may cause a density altitude higher than true altitude.

Density/Altitude Expression

An expression relating the three is given:

$DA\;=\;\dfrac{\,T\,}{L}\,\left(\,1\,-\,\dfrac{\,p/P\,}{t/T}\,\right)^{\frac{L\,R}{\,G\,M\,-\,L\,R\,}}$

where:

            P = 101.325     standard sea level pressure kPa
            T = 288.15       standard sea level air temperature K
            G = 9.80665    gravity
            R = 8.31447     gas constant for ideal gas
            M = 0.0289644 molar mass of dry air
            L = 0.0065        lapse rate
            R_dry = 287.058    J/(kg K) gas constant for dry air
            R_H2O =461.495   in J/(kg K) gas constant for water vapor

with p and t representing pressure and spot temperature.

Substituting values, the above expression reduces to

42023.1 [ 1 – 0.145317 (p/t)^0.326034 ]