I have a digital weather station with a wireless outdoor sensor. In the photo, the top right quadrant of the display shows temperature and relative humidity for outdoors (6.2°C/94%) and indoors (21.6°C/55%).

I find this indoor-outdoor thing fascinating for some reason and revel in looking at the numbers. But when I do, I always end up asking myself if the air outside has more or less water vapor in it than the air inside. Simple question, which is more than can be said for the answer. Using the ideal gas law, the calculation of absolute humidity from temperature and relative humidity requires an added algorithm that generates saturation vapor pressure as a function of temperature, which complicates things a bit.

*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 base of natural logarithms 2.71828 [raised to the power of the contents of the square brackets]:

Absolute Humidity (grams/m^{3}) = 6.112 x e^[(17.67 x T)/(T+243.5)] x rh x 18.02

(273.15+T) x 100 x 0.08314

which simplifies to

Absolute Humidity (grams/m^{3}) = 6.112 x e^[(17.67 x T)/(T+243.5)] x rh x 2.1674

(273.15+T)

This formula is accurate to within 0.1% over the temperature range –30°C to +35°C

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*Additional notes for students*

Strategy for computing absolute humidity, defined as density in g/m^3 of water vapor, from temperature (T) and relative humidity (rh):

1. Water vapor is a gas whose behavior approximates that of an ideal gas at normally encountered atmospheric temperatures.

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 m^{3}), but we need to calculate P before we can solve for n.

3. To obtain a value for P, we can use the following variant^{[REF, eq.10]} of the Magnus-Tetens formula which generates saturation vapor pressure P_{sat} (hectopascals) as a function of temperature T (Celsius):

P_{sat} = 6.112 x e^[(17.67 x T)/(T+243.5)]

4. P_{sat} is the pressure when the relative humidity is 100%. To compute the pressure P for any value of relative humidity expressed in %, we multiply the expression for P_{sat} by the factor (rh/100):

P = 6.112 x e^[(17.67 x T)/(T+243.5)] x (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 18.02 – the molecular weight of water – to give the answer in grams.

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 m^{3}) and the gas constant R is known.

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UPDATES

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**Formula cited in doctoral dissertation concerning GNSS positioning accuracy**

October 2017: A doctoral dissertation from the University of Connecticut has used my AH formula to quantify real-time kinematic (RTK) positioning with the effect of ground-level AH, which theory and previous research suggests can degrade global navigation satellite system (GNSS) positioning accuracy.

Link: http://opencommons.uconn.edu/cgi/viewcontent.cgi?article=7849&context=dissertations

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**A neat display of RH, T and AH data**

September 2017: This automated data display from a website in Austria is among the best I have seen. Outdoor measurements of RH (%) and T (Celsius) are taken every 10 minutes and fitted to a common 0 -100 scale, which also serves to plot computed AH (g/m^3).

The displayed segment captures the mirror-image movements of RH (blue line) as T (yellow line) rises and falls while AH (red line) remains relatively constant. This neatly visualizes how water vapor density and temperature together determine relative humidity.

Link: http://jdit.at/templog/

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**Formula for computing dew point temperature T _{D} from RH and T**

August 2017: There has been a lot of interest in my formula (P Mander 2012) which computes AH from measured RH and T, since it adds value to the output of RH&T sensors. To further extend this value, I have developed another formula (P Mander 2017) which computes dew point temperature T_{D} from measured RH and T. In this formula the measured temperature T and the computed dew point temperature T_{D} are expressed in degrees Celsius, and the measured relative humidity RH is expressed in %

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If an object whose temperature is at or below T_{D} is present in the local space, the thermodynamic conditions are satisfied for water vapor to condense (or freeze if T_{D} is below 0°C) on the surface of the object.

Further details, including the derivation of the formula and copy-and-paste spreadsheet formulas for computing T_{D} are available on this link:

https://carnotcycle.wordpress.com/2017/08/01/compute-dewpoint-temperature-from-rh-t/

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*Formula cited in two recent academic research papers*

July 2017

Czech Republic: Brno University of Technology, Faculty of Mechanical Engineering

Thesis: The effect of climate conditions on wheel-rail contact adhesion

http://dl.uk.fme.vutbr.cz/zobraz_soubor.php?id=3392

Sweden: Linköping University, Institute for Economic and Industrial Development

Case study: Effect of seasonal ventilation on energy efficiency and indoor air quality

http://www.navic.se/images/Exjobb/rstidsanpassad_ventilation.pdf

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*Formula computes real time AH with DHT22 sensor on single board computer*

June 2017: Single board computers provide low-cost solutions to automation and testing. On element14.com a BeagleBone Black Wireless equipped with a DHT22 RH&T sensor has been used to monitor outdoor and indoor temperature and humidity using my formula to enable AH computations to be processed in real time.

https://www.element14.com/community/roadTestReviews/2398/l/BeagleBoard.org-BBB-Wireless-BBBWL-SC-562

http://element381.rssing.com/chan-13345555/all_p589.html

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*Formula features in Russian Arduino project on YouTube*

April 2017: My formula makes its first live appearance on YouTube. The presentation concerns a humidity/temperature monitoring and management system installed in a cellar affected by mould problems. If you don’t speak Russian don’t worry, the images of the installation give you the gist of what this project is about.

See the YouTube video here:

https://www.youtube.com/watch?v=SO1yugxahpk

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*Formula recommended for use in monitoring comfort levels for exotic pets *

March 2017: A post has appeared on Reddit concerning an Arduino Uno with T&RH sensor and LCD screen, which the poster is using to improve temperature and humidity monitoring of a pet’s habitat – in this particular case a Bearded Dragon (not the one illustrated).

The post has attracted much interested discussion and comment, including a recommendation from one participant to use AH rather than RH, citing my conversion formula. The rationale for the change is so neatly expressed that I would like to quote it:

*“May I recommend absolute humidity instead of relative? Relative humidity only tells you how “full” the air is of moisture, and it’s entirely dependent on temperature; the same amount of moisture will read lower relative humidity at higher temperatures, and vice versa. Whereas absolute humidity is measured in grams of water per cubic meter of air. You can implement this simple conversion formula in your code: (URL for this blogpost)
0-2 is extremely dry, 6-12 is your average indoors, and 30 is like an Amazon rainforest.”*

See the Reddit post here:

https://www.reddit.com/r/arduino/comments/5ysmo5/i_noticed_my_bearded_dragons_habitat_could_use_a/

See the Arduino project here:

https://create.arduino.cc/projecthub/ThothLoki/portable-arduino-temp-humidity-sensor-with-lcd-a750f4

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*Igor uses my formula to keep his cellar dry*

October 2016: I am impressed by this basement humidity control system developed by Igor and reported on Amperka.ru forum.

Inside the short pipe is a fan equipped with a 3D-printed circumferential seal. The fan replaces basement air with outdoor air, and is activated when absolute humidity in the cellar is 0.5 g/m^3 higher than in the street, subject to the condition that the temperature of the outdoor air is lower. This ensures that water in the cellar walls is drawn into the vapor phase and pumped out; the reverse process cannot occur. на русском здесь.

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*Formula cited in doctoral dissertation concerning local weather and infectious disease*

Summer 2016: My AH formula has been cited in a doctoral dissertation entitled *“Seasonality, local weather and infectious disease: effects of heat and humidity on local risk for urinary tract infections and Legionella pneumonia”*

This can be downloaded from http://ir.uiowa.edu/etd/5852

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*Formula powers AH measurements from high-precision RH&T sensor*

April 2016: ** Prof. Antonietta Frani** has made a miniature device for measuring absolute humidity, using my formula to power an Arduino Uno microcontroller board equipped with an SHT75 RH&T sensor which connects to a computer via a USB cable. Systems Integrator

**has developed an interface to transfer the data to Excel spreadsheets with their associated graphical display functions.**

*Roberto Valgolio***– – – –**

*Formula powers online RH←→AH calculator*

March 2016: German website *rechneronline.de* is using my formula to power an online RH/AH conversion calculator.

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*Formula cited in academic research paper*

January 2016: A research article in *Landscape Ecology* (October 2015) exploring microclimatic patterns in urban environments across the United States has used my formula to compute absolute humidity from temperature and relative humidity data.

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*Formula finds use in humidity control unit*

August 2015: Open source software/hardware project Arduino is using my absolute humidity formula in a microcontroller designed to control humidity in basements:

*“The whole idea is to measure the temperature and relative humidity in the basement and on the street, on the basis of temperature and relative humidity to calculate the absolute humidity and make a decision on the inclusion of the exhaust fan in the basement. The theory for the calculation is set forth here – carnotcycle.wordpress.com/2012/08/04/how-to-convert-relative-humidity-to-absolute-humidity.”* на русском здесь.

More photos on this link (text in Russian):http://arduino.ru/forum/proekty/kontrol-vlazhnosti-podvala-arduino-pro-mini

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*AH computation procedure applied in calibration of NASA weather satellite*

June 2015: My general procedure for computing AH from RH and T has been applied in the absolute calibration of NASA’s Cyclone Global Navigation Satellite System (CYGNSS), specifically in relation to the RH data provided by Climate Forecast System Reanalysis (CFSR). The only change to my formula is that P_{sat} is calculated using the August-Roche-Magnus expression rather than the Bolton expression.

The CYGNSS system, comprising a network of eight satellites, is designed to improve hurricane intensity forecasts and was launched on 15 December 2016.

Reference: ddchen.net/publications (Technical report “An Antenna Temperature Model for CYGNSS” June 2015)

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*Formula cited in draft paper on air quality monitoring*

May 2015: Metal oxide (MO) sensors are used for the measurement of air pollutants including nitrogen dioxide, carbon monoxide and ozone. A draft paper concerning the Air Quality Egg (AQE) which cites my formula in relation to MO sensors can be seen on this link:

MONITORING AIR QUALITY IN THE GRAND VALLEY: ASSESSING THE USEFULNESS OF THE AIR QUALITY EGG

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*Formula used by US Department of Energy in Radiological Risk Assessment*

June 2014: In its report on disused uranium mines, Legacy Management at DoE used my formula for computing absolute humidity as one of the meteorological parameters involved in modeling radiological risk assessment.

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