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 used in study of environmental factors affecting sleep**

April 2018: Researchers in Thailand have made use of my AH formula in a paper published in the Journal of Clinical Sleep Medicine, April 2018.

Link: http://jcsm.aasm.org/ViewAbstract.aspx?pid=31237

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**Formula gets praise on GitHub**

April 2018: My formula has found a new application in HappyBee, a companion set of features to complement ecobee WiFi thermostats. The HappyBee suite includes humidity normalization using a heat recovery ventilator.

Link: https://github.com/ilyabelkin/HappyBee#happybee

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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.

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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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Hi Peter, thank you for your elaborate answer.

Hello Stefan and thank you for the comment. Of course you can multiply those numbers together if you wish. I decided to keep the Bolton approximation visible in the equation to make substitution easy for those who might prefer other saturation vapor pressure approximations, such as

August-Roche-Magnus 6.1094 exp(17.625 x T)/(T + 243.04)

Buck (1981) 6.1121 exp(17.502 x T)/(T + 240.97)

US space agency NASA substituted the August-Roche-Magnus approximation for calibration purposes in its CYGNSS satellite system which monitors hurricane intensity (see Updates – June 2015)

See link below for a comparative review of SVP approximation formulae

https://www.eas.ualberta.ca/jdwilson/EAS372_13/Vomel_CIRES_satvpformulae.html

First, thank you for the information.

In the formula, the numerator is 6.112 x e^[(17.67 x T)/(T+243.5)] x rh x 2.1674. I don’t understand why we cannot simplify the factors 6.112 and 2.1674 further to 13.2471. Where’s my error?

Thanks to Markus for providing coding for the AH formula in Ruby and C.

Thanks Peter.

Hi Milan,

If I understand your question correctly, I think you are asking if there is a maximum pressure that water vapor can exert at a given temperature. The answer is yes: it is called the saturation vapor pressure, and the corresponding temperature is called the dewpoint temperature.

Saturation vapor pressure is the state of maximum relative humidity (100% rh). So using my absolute humidity formula, if you assign a value of 100 to rh, the temperature you choose will automatically be the dewpoint temperature and the formula will return the absolute humidity in g/m^3. This is the mathematical relation you are seeking, but note that it is not a linear function.

To convert g/m^3 into kg/m^3 just divide by 1000.

Hi Peter,

is it correct to state that Absolute Humidity is direct proportional to Dew point ?

Hum[kg/m3] = C * Tdew[Celsius]

if yes what would be exact relation ?

regards,

Milan

Hi Andrew and thank you for the question. This link should be helpful:

https://www.researchgate.net/post/How_does_one_convert_absolute_humidity_expressed_in_g_kg_to_g_m3

Hi Peter,

I’m trying to use this formula to convert large quantities of dry-bulb temperature and RH readings to absolute humidity (g/kg). I have used the formula above to convert the data to absolute humidity in g/m^3, however, have been unsuccessful in converting this information into the required form.

Any help would be greatly appreciated.

Thank you in advance

Thanks thanks a lot Peter. Great job and awesome site!! I’ve been searching for a formula like this to calculate directly the water evaporation in sheets of water, taking the data from weather stations which show only RH and T. And finally I found it, so you saved my a** :)

excellent and easy to follow work. Thanks for sharing the information

Hi Pavel and thank you for the question.

Thermodynamics teaches us that vapor pressure is an inherent property of substances contingent only on temperature. It is independent of atmospheric pressure, whether measured at sea level or any other altitude.

Using my formula, all you need to compute the density of water vapor at the point of measurement is the temperature and relative humidity.

So in a meteostation, where both actual pressure and sea level pressure is measured (altitude is known), we should use what equation construction?

I have fulfiled my promice at http://bit.ly/2dCM1zq

Thank You a lot!

Thank You! I’ll take Yours formula to my home project and report You!

Thank you for the question. The Psat variant I use was developed by Bolton* for meteorological purposes as a best-fit in the -30°C to +35°C temperature range. At +40°C, the accuracy is 0.18%. For accuracy better than 0.005% in the +35°C to +60°C range, you could use the Psat formula developed by Wexler; this equation is numbered (9) in Bolton’s paper.

*a mouse-over link to Bolton’s paper has been included under point 3 of “Additional notes for students”

What would be the accuracy of the results of this equation for temperatures above 35°C, say 35-60?

Thank you so much for this wonderful resource!

Hey Peter, thank you so much for the equations. That you bothered to put equations in so many forms is really appreciable. Great work.

Thanks! :)

Hello lou and thank you for your question. The mathematical constant e is not equal to 10; the approximate value is 2.718

You can read more about e here:

http://en.wikipedia.org/wiki/E_(mathematical_constant)

Hello,

I have tried your formula several time but the result I get is not the right one. I have one question, “in the equation where to I get the value for ‘e'”? I thought e would equal 10 but it does not come out with the correct answer.

(following lou’s question, I added text to clarify the value of ‘e’ – PM)

Thanks for asking Martin. Remember you’re dividing by R and multiplying by the molecular weight of water; the factor is 18.02/8.314 = 2.1674

I still dont get the 2.1674 factor. When you devide P/T by R, there is just R_specific left, which should be 8.3/18.02 = 0.4615 . Can you please provide details on your calculation?

Very interesting derivation of AH formula base on RH and T. Thank you very much!

You’re welcome Robert. Regarding the formula: to simplify computation the factor 10^-2 is amalgamated, along with the molecular weight of water in grams and R in m^3.hPa.K^-1.mol^-1, in the 2.1674 figure.

Thank you for the formula. I’ve got one question though, what happens with the factor 1/100?

At 4. it says rh/100 but at the top of the page where you’ve written the formula, 1/100 is gone.

(following Robert’s comment, I added the uncondensed formula at the top of the page to clarify the math – PM)

Thanks a lot for this, which I can use in my reserch work.

Thanks a lot for this.