[741]

I'm trying to find an expression for the density of a mixture of air and water vapour at a given RH (relative humidity).

Presumably it is basically a matter of determining the molar proportions of the humid air constituents.

I've been going around in circles somewhat, looking up various definitions.

I read that "Ideally the ratio of partial pressures equals the ratio of the number of molecules". So that looks promising.


If I can find partial pressures for all of dry air's principal constituents and also for H2O, I think that would lead to the answer.

What I'd like to do is create an Excel sheet whereby I can enter temperature and RH and read out the gas density.

[ Note: I think (?) I might need a pressure entry too, to find Molar Volume - except then I wonder whether partial pressure is independent of overall system pressure? ]

[jestearns]

What is the definition of relative humidity with respect to vapor density at a given temperature?

[Enthalpy]

[...] What I'd like to do is create an Excel sheet whereby I can enter temperature and RH and read out the gas density. [...]

Reasonable undertaking.

Algebraic expressions exist that give the equilibrium partial pressure of water vapour in relation with "the" temperature (which one would be an interesting interrogation). Around room pressure and temperature, they work well for water.
https://en.wikipedia.org/wiki/Antoine_equation

Then you can compute the partial pressure or the dry air component (split among its constituents if you wish, but can just treat "dry air" as a pure gas). From both partial pressures, sum the volumes and masses, an deduce a density.

[741]

What is the definition of relative humidity with respect to vapor density at a given temperature?

Wikipedia says
   "ratio of the partial pressure of water vapor to equilibrium vapor pressure of water...depends on temperature and pressure"
Elsewhere I've seen
   RH = 100.(Vapour Density)/(Saturation Density)
Are both of these right & equivalent? I should check really...

[741]

can just treat "dry air" as a pure gas

Thanks.

If the components of air like O₂, N₂ can be considered "as one" to be a pure gas here, then I'm unclear how the vapour is distinguised from the dry air. How are we able to group O₂, N₂ as one, but still treat water as distinct?

[741]

Just thinking it through. Suppose the local atmospheric bench-top pressure is 1 atm = 101,325 Pa and temperature is 20 C.

Then the Antoine equation can approximate the partial pressure of water as

   P (Pascals) = A + B/(C + T)
     
Provided 1 <= T <= 100:
   P (Pascals) = 10.196213 + 1730.63/(233.426 + T)
   
At 20 C I get a low partial pressure for water, just 2330 Pa, scarcely contributing to the 101,325 overall assumed pressure.

*"TO DO" Note - Still have to equate R.H. to the 2330 figure above...

Then the calculation seems to be

Molar ratio (air: vapour) = (101,325-2330):2330 = 42.5 : 1
So, 42.5 parts of air, 1 part of water per 43.5 parts overall

Molar volume of ideal gas = RT/P = 0.0241m3 at 1 Atmosphere.

Molar mass of air   = 0.028969 kg/mol  --> (0.028969/0.024) = 1.207 kg/m3 at 1 Atmosphere
Molar mass of water = 0.018015 kg/mol  --> (0.018015/0.024) = 0.751 kg/m3 at 1 Atmosphere

(1.207*42.5)/43.5 + (0.751*1)/43.5 = 1.197 kg/m3 at 1 Atmosphere

The results I get are reasonable, but even using the "Tetens" equation for saturated vapour pressure, differ significantly from published results.


I'm wondering if the "Clausius–Clapeyron relation" can be used?

---
PS: When I add hyperlinks, how do I add a hyperlink without showing the long 'http' address, i.e. as a proper hyperlink?

So the above is at [/url]https://en.wikipedia.org/wiki/Clausius%E2%80%93Clapeyron_relation[/url], but that looks ugly compared with "Clausius–Clapeyron"
---

[billnotgatez]

Just  informational in case you are interested

When I did a GOOGLE on

Total Molecular Mass of Air

one of the results was
https://www.engineeringtoolbox.com/molecular-mass-air-d_679.html
This was another GOOGLE result on

Clean dry air

https://www.engineeringtoolbox.com/air-composition-d_212.html
Both pages are similar with a few differences.

Also there are many entries in this forum.
You can use the search feature.
for instance try

"clean dry air"

"density of air"

I have done some reading on the components of dry air but yet to delve into water in the air. One thing we know is that the concentration of water vapor is limited unlike Carbon Dioxide. I am watching your posts with great interest.

[Borek]

If the components of air like O₂, N₂ can be considered "as one" to be a pure gas here, then I'm unclear how the vapour is distinguised from the dry air. How are we able to group O₂, N₂ as one, but still treat water as distinct?

You can group components any way you want.

Wikipedia says
   "ratio of the partial pressure of water vapor to equilibrium vapor pressure of water...depends on temperature and pressure"
Elsewhere I've seen
   RH = 100.(Vapour Density)/(Saturation Density)
Are both of these right & equivalent? I should check really...

Yes, these are equivalent. "Equilibrium vapor pressure" means "saturated vapor". You can define the RH using vapor density, partial pressure of water, mass of vapor per cubic meter (and several other quantities, not all though) - they are all linearly dependent on each other and easy to convert, as we are talking about ratio conversion factor cancels out.

Then the Antoine equation can approximate the partial pressure of water as

   P (Pascals) = A + B/(C + T)

Nope, that's not correct. It is not pressure on the left.

[741]

Nope, that's not correct. It is not pressure on the left.

Thanks - what is it though?

[741]

Footnote to earlier post
The above figure, 2330 is the "saturated" or "equilibrium" vapour pressure of water at a 20°C. With RH=60%, the partial pressure of water is 0.6*2330 = 1398.

This link, from NPL, is a useful source for a (presumably) fairly accurate vapour pressure equation, along with tabulated results.
http://www.kayelaby.npl.co.uk/chemistry/3_4/3_4_2.html

Good agreement (at 0% RH) with https://en.wikipedia.org/wiki/Density_of_air is obtained with the following formula

(P_water.RH_F.Density_water + (P_system-P_water.RH_F).Density_air)/P<sub>system

where RHF is relative humidity expressed as a fraction.

Not yet found a source to check the contributions made by the RH fraction.</sub>

[741]

Just found an online calculator (which I can only verify by entering values) here
https://www.omnicalculator.com/physics/air-density I have no way to estimate overall accuracy, it looks OK. It does not allow you to enter 0% RH. I've been after an Excel formula, but this is a possible way to test results for reasonableness.


Here is how I think the calculation works

[Borek]

Nope, that's not correct. It is not pressure on the left.

Thanks - what is it though?

Come on - any web page listing the Antoine equation tells you what is on the LHS (or how to properly calculate the pressure).

[741]

Come on - any web page listing the Antoine equation tells you what is on the LHS (or how to properly calculate the pressure).

I have looked again, and "p is the vapor pressure" according to wikipedia
https://en.wikipedia.org/wiki/Antoine_equation

[Borek]

I have looked again, and "p is the vapor pressure" according to wikipedia
https://en.wikipedia.org/wiki/Antoine_equation

Please list - letter by letter - how the LHS of the equation looks like. Note: I never said p is not pressure - I said "it is not pressure on the left".