## Compute Absolute Humidity from Relative Humidity and Temperature (-50°C to 140°C)

Posted: August 1, 2020 in physical chemistry, physics, thermodynamics
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The absolute humidity formula posted in 2012 on this blog has a range of -30°C to 35°C. To expand this range I have developed a new formula to compute absolute humidity from relative humidity and temperature based on a simple but little known polynomial expression (Richards, 1971) for the saturation vapor pressure of water, valid to ±0.1% over the temperature range -50°C to 140°C.

Formula for calculating absolute humidity

In the formula below, temperature (T) is expressed in degrees Celsius, relative humidity (rh) is expressed in %, and e is the Euler number 2.71828 [raised to the power of the contents of the square brackets]:

Absolute Humidity = 1013.25 × e^[13.3185t – 1.9760t^2 – 0.6445t^3 – 0.1299t^4] × rh × 18.01528
(grams/m^3)                                                   100 × 0.083145 × (273.15 + T)

where the parameter t = 1 – 373.15/(273.15 + T)

The above formula simplifies to

Absolute Humidity = 1013.25 × e^[13.3185t – 1.9760t^2 – 0.6445t^3 – 0.1299t^4] × rh × 2.1667
(grams/m^3)                                                                   273.15 + T

To cite this formula please quote: P Mander (2020), carnotcycle.wordpress.com/2020/08/01/compute-absolute-humidity-from-relative-humidity-and-temperature-50c-to-140c

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Study notes

Strategy for computing absolute humidity, defined as water vapor density in grams/m3, from temperature (T) and relative humidity (rh):

1. Water vapor is a gas whose behavior in air approximates that of an ideal gas due to its very low partial pressure.

2. We can apply the ideal gas equation PV = nRT. The gas constant R and the variables T and V are known in this case (T is measured, V = 1 m3), but we need to calculate P before we can solve for n.

3. To obtain a value for P, we can use the polynomial expression of Richards (ref) which generates saturation vapor pressure Psat (hectopascals) as a function of temperature T (Celsius) in terms of a parameter t

Psat = 1013.25 × e^[13.3185t – 1.9760t2 – 0.6445t3 – 0.1299t4]
where t = 1 – 373.15/(273.15 + T)

4. Psat is the vapor pressure when the relative humidity is 100%. To compute the pressure P for any value of relative humidity expressed in %, the expression for Psat is multiplied by the factor rh/100:

P = 1013.25 × e^[13.3185t – 1.9760t2 – 0.6445t3 – 0.1299t4] × rh/100

5. We now know P, V, R, T and can solve for n, which is the amount of water vapor in moles. This value is then multiplied by the molecular weight of water to give the answer in grams.

Absolute humidity (grams/m3) = Psat  ×  rh  ×  mol wt
100 × R × (273.15 + T)

Saturation vapor pressure Psat is expressed in hectopascals hPa
Relative humidity rh is expressed in %
Molecular weight of water mol wt = 18.01528 g mol-1
Gas constant R = 0.083145 m3 hPa K-1 mol-1
Temperature T is expressed in degrees Celsius

6. Summary:
The formula for absolute humidity is derived from the ideal gas equation. It gives a statement of n solely in terms of the variables temperature (T) and relative humidity (rh). Pressure is computed as a function of both these variables; the volume is specified (1 m3) and the gas constant R is known.

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P Mander, July 2020