Question about humidity calculation?

[math1985]

After some pondering and Googling, I realized that the concept of humidity (of air) is not as simple as I was thought in high school. I understand from Wikipedia that there are three kinds of humidity:

• relative humidity (the ratio of the partial pressure of water to the saturated water pressure);
• specific humidity (the mass of water per mass of air); and
• absolute humidity (the mass of water per volume of air).

In particular, I wonder how I can answer the following question:

Outside, the temperature is T_o degrees, and the relative humidity is H_o percent. Inside my room, the temperature is T_i degrees, where T_i > T_o. When I open the room window to the outside, to what percentage will the inside relative humidity H_i tend, assuming I heat the room such that the inside temperature remains constant at T_i?

Can I expect that either the absolute or specific humidity inside and outside will tend to the same values? If yes, why? If not, is there another way to calculate what I want?

[douglasm]

I think the answer is that the absolute partial pressures would equalize.

How relative humidity inside the room compares to outside would depend on what effect the temperature difference has on saturated pressure.
Specific humidity inside the room would, I think, tend towards equaling specific humidity outside. There would be less water density inside than out due to the temperature difference, but the same would be true to exactly the same extent for air in general.
Absolute humidity inside the room would tend to something lower than outside because the higher temperature allows a smaller mass of water per volume to produce the same partial pressure.

[TestTubeGames]

My understanding is that relative humidity is the one we care about most when it comes to comfort. Just looking at the relative humidities in your example, if you are heating up the outside air when it comes inside, then its relative humidity will drop. The air will seem drier. (This is why the air is so dry indoors in winter, not just because the outside air is already fairly dry... but as we heat that air up, its relative humidity drops even more.)

[idobox]

Wouldn't it be the partial pressures that reach equilibrium?

[eSOANEM]

It will be the partial pressures which drive the equilibrium however, because the total air pressures (and all air's constituents' partial pressures) will also equalise, all the different measures of humidity will end up the same inside and outside.

[douglasm]

because the total air pressures (and all air's constituents' partial pressures) will also equalise, all the different measures of humidity will end up the same inside and outside.

That would only be necessarily true if all measures of humidity were defined solely in terms of pressures, and further if all pressures involved are actually present rather than hypotheticals - the saturation water pressure is not a pressure that is actually there, it's more a statement that hypothetically if water's partial pressure were at that point then any additional water vapor would condense into liquid.

In actual fact, none of the three measures of humidity are defined in such terms.

[eSOANEM]

If the partial pressures of all gases present in the air are equal in two regions which are also in thermal equilibrium, how can the density of air (and of each of it's consituents) and therefore both the specific and absolute humidities be different in those two regions?

Certainly, an equal density would imply that if the specific humidities are equal, the absolute humidities are too; and the ideal gas law states that equal pressure and temperature between the two regions implies equal density. This same argument from the ideal gas law can be used for the partial pressure of water vapour implying the mass of water per unit volume must be equal across the two regions.

So I can see good reasons for 1 humidity being equal implying both others are equal but none for the lack of implication.

because the total air pressures (and all air's constituents' partial pressures) will also equalise, all the different measures of humidity will end up the same inside and outside.

That would only be necessarily true if all measures of humidity were defined solely in terms of pressures, and further if all pressures involved are actually present rather than hypotheticals - the saturation water pressure is not a pressure that is actually there, it's more a statement that hypothetically if water's partial pressure were at that point then any additional water vapor would condense into liquid.

In actual fact, none of the three measures of humidity are defined in such terms.

I'm not suggesting they are defined that way, just that their definitions are equivalent to a definition in partial pressures, at least at equilibrium.

[douglasm]

The two regions specified by the OP are allowed to reach pressure equilibrium but not thermal equilibrium. He specified that even after opening the window he would continue heating the room to maintain its higher temperature. The "outside" is presumably a sufficiently large region that heat transfer from the room has a negligible effect on it, so both temperatures remain at their original separate values.

[eSOANEM]

Ah, fair enough. I missed that bit.

In that case, only the partial pressures will equalise but, if I've understood the definition given for relative humidity correctly, this doesn't mean any of the humidities necessarily reach equilibrium because the saturation pressure will be different in the two regions by virtue of the different temperatures and therefore pressures.

[math1985]

I tried testing this experimentally today by opening windows in my house. I used offical weather data for the outside temperature and humidity (10 km from here), so this might have caused an inaccuracy.

I started with outside relative humidity 82% and outside temperature 14 degrees Celcius. The initial inside humidity was 60%, but this should be irrelevant. The inside temperature started at 25 degrees Celcius, and remained 24 degrees Celcius the first hour after I had opened the windows (so remained relatively constant).

Given these temperatures, the saturated pressure outside is 1.6 kPa, and inside 3.2 kPa.

Let us assume that the partial pressures (and thus specific humidity) equalize. Then the partial pressure of water outside is (82/100)*1.6 = 1.312 kPa. As the partial pressures will equalize, the partial pressure of water inside will tend to the same number. Then the relative relative humidity inside would tend to 1.312/3.2*100=41%.

Simplifying, we obtain the formula:
H_i = H_o*sp(T_o)/sp(T_i),
Where sp(T) is the saturated pressure at temperature T.

In reality, after opening a couple of windows, the inside relative humidity dropped to 52% within 15 minutes, and did not change for more than 1% the first 45 minutes after.

What do you think causes the difference between theory and practice? The assumption that the partial pressures equalize? The measurements? My calculations? The fact that other sources in the house contribute to adding humidity inside?

By the way, what would be an argument for the fact that the partial pressures will equalize?

[gmalivuk]

By the way, what would be an argument for the fact that the partial pressures will equalize?

The observation that that's how gas works, for one thing. The most likely configurations of gas molecules in a closed system are all close to evenly distributed, and statistical mechanics will tell you that the sense in which they are evenly distributed is that their partial pressures are the same.