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askIITians Faculty 747 Points
7 years ago
To begin to understand the calculation of air density, consider the ideal gas law:

(1) P*V = n*Rg*T

where: P = pressure
V = volume
n = number of moles
Rg = universal gas constant
T = temperature

Density is simply the mass of the molecules of the ideal gas in a certain volume, which may be mathematically expressed as:

(2) D = m / V

where: D = density
m = mass
V = volume

Note that:

m = n * M

where: m = mass
n = number of moles
M = molar mass

And define a specific gas constant for the gas under consideration:

R = Rg / M

where R = specific gas constant
Rg = universal gas constant
M = molar mass

Then, by combining the previous equations, the expression for the density becomes:

(3) [equation 3]

where:D = density, kg/m3
P = pressure, Pascals ( multiply mb by 100 to get Pascals)
R = specific gas constant , J/(kg*degK) = 287.05 for dry air
T = temperature, deg K = deg C + 273.15

As an example, using the ISA standard sea level conditions of P = 101325 Pa and T = 15 deg C, the air density at sea level, may be calculated as:

D = (101325) / (287.05 * (15 + 273.15)) = 1.2250 kg/m3

This example has been derived for the dry air of the standard conditions. However, for real-world situations, it is necessary to understand how the density is affected by the moisture in the air.

Neglecting the small errors due to non-ideal gas compressibility and vapor pressure measurements not made over liquid water (seeref 14), the density of a mixture of dry air molecules and water vapor molecules may be simply written as:

(4a) [equation 4a]

Which, with some substitutions and rearranging (seeref 15), may also be written as:

(4b) [equation 4b]

where: D = density, kg/m3
Pd= pressure of dry air (partial pressure), Pascals
Pv= pressure of water vapor (partial pressure), Pascals
P = Pd+ Pv= total air pressure, Pascals ( multiply mb by 100 to get Pascals)
Rd = gas constant for dry air, J/(kg*degK) = 287.05 = R/Md
Rv = gas constant for water vapor, J/(kg*degK) = 461.495 = R/Mv
R = universal gas constant = 8314.32 (in 1976 Standard Atmosphere)
Md = molecular weight of dry air = 28.964 gm/mol
Mv = molecular weight of water vapor = 18.016 gm/mol
T = temperature, deg K = deg C + 273.15

To use equation 4a or 4b to determine the density of the air, one must know the actual air pressure (which is also called absolute pressure, total air pressure, or station pressure), the water vapor pressure, and the temperature.

It is possible to obtain a rough approximation of the absolute pressure by adjusting an altimeter to read zero altitude and reading the value in the Kollsman window as the actual air pressure. Near the end of this page I'll discuss how to use the altimeter reading to accurately determine the actual pressure. Alternatively, there are many little electronic gadgets that can measure the actual air pressure and the vapor pressure directly, and quite accurately.

The water vapor pressure can easily be determined from the dew point or from the relative humidity, and the ambient temperature can be measured in a well ventilated place out of the direct sunlight.

In the following section, we'll learn to calculate the water vapor pressure.

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