Interplay of saturation temperature and pressure

[Urmi Roy]

So we were discussing an interesting situation at school today.

Say we have a room at 1atm pressure, 25C temperature and relative humidity 90%. So from charts we can find what the saturation pressure of water vapor is at 25C. That number multiplied by 0.9 (because of the 90% RH) will give us the actual pressure of water vapor in the room. Let's say its P_vap.

So now the question is, what determines the condition when there will be condensation on the walls of the room.

Someone said that we can find the saturation temperature at P_vap (say itsT_sat*) and if the temperature of the walls is lower than T-sat*, there will be condensation.

I'm not sure this is correct. I think that as long as the walls are at a temperature lower than the saturation temperature at 1 atm pressure (the overall pressure in the room), there will be condensation.
I'd appreciate some help with this!Urmi

 

[Chestermiller]

So we were discussing an interesting situation at school today.

Say we have a room at 1atm pressure, 25C temperature and relative humidity 90%. So from charts we can find what the saturation pressure of water vapor is at 25C. That number multiplied by 0.9 (because of the 90% RH) will give us the actual pressure of water vapor in the room. Let's say its P_vap.

So now the question is, what determines the condition when there will be condensation on the walls of the room.

Someone said that we can find the saturation temperature at P_vap (say itsT_sat*) and if the temperature of the walls is lower than T-sat*, there will be condensation.

I'm not sure this is correct. I think that as long as the walls are at a temperature lower than the saturation temperature at 1 atm pressure (the overall pressure in the room), there will be condensation.
I'd appreciate some help with this!Urmi

Your friend is correct. What you are calling P_vap is what we usually call the partial pressure of the water vapor in the air. Suppose that the wall temperature were lower, and equal to the saturation temperature corresponding to P_vap. Then the relative humidity of the air in contact with the wall would be 100%. We call this the dew point. Water will begin condensing on the wall. You are probably inside a room right now where the temperature is lower than the saturation temperature at 1 atm (100 C). Do you see water condensing on the walls?

Chet

 

[Urmi Roy]

Yeah, that was what was confusing me...since the temperature of the walls in any room we inhabit is less than 100C, according to my initial thoughts, there should be condensation on every wall, which there isn't.
But I'm shocked at how much I lack a fundamental understanding about what's going on here. You're saying that the vapor pressure P_vap has a corresponding saturation temperature (T_sat*). If the walls are at T_sat*, the vapor just above the surface is completely saturated. So if the temperature of the wall is less than that, there will be condensation. Makes sense I think.

I think that perhaps the best way to think about this is through the pressure-temperature chart for a pure substance. Corresponding to any given temperature, if we start off in the liquid phase, and decrease the pressure,we move toward the gas phase and at a certain pressure we're just at the liquid-gas dividing point, which is the saturation state. So now if we bring the substance in contact with something at lower temperature, but maintaining the same pressure, we get straight into the liquid phase.

My confusion till now was about the atmospheric pressure vs the vapor pressure. The key thing here is the vapor pressure, because on the pressure-temperature diagram, we're considering the pressure of the vapor, not air...

 

[Chestermiller]

I think you pretty much have it figured out. The key thing is that, for an ideal gas mixture, the various components behave as if the other species are not even present. So the key parameter for each species is its own partial pressure.

Chet

 

[Urmi Roy]

So I just finished reading up a few sources to make sure I have everything. I have 2 remaining questions that I can't seem to answer on my own. It would be great if I could get some clarification on those points.

1. Let's say we have a bowl of water at 20C...in most problems we consider the region right above it to be completely saturated. But this situation is unlike the one we spoke about last time. Last time we said if the temp. of a surface is equal to the saturation temp corresponding to P_vap, we would have condensation. Here we can't be sure that 20C is that saturation temp...so why is it that the region right above is 100% saturated?

2. There is a certain saturation pressure corresponding to the surrounding temperature. So if the vapor pressure is lower than that saturation value, more water would evaporate from things (like lakes/ plants etc.) But usually the RH of surroundings is stays lower than 100%...then does that mean the water lying around the place is continuously in disequilibrium with the atmosphere and more and more is trying to evaporate?

 

[Chestermiller]

So I just finished reading up a few sources to make sure I have everything. I have 2 remaining questions that I can't seem to answer on my own. It would be great if I could get some clarification on those points.

1. Let's say we have a bowl of water at 20C...in most problems we consider the region right above it to be completely saturated. But this situation is unlike the one we spoke about last time. Last time we said if the temp. of a surface is equal to the saturation temp corresponding to P_vap, we would have condensation. Here we can't be sure that 20C is that saturation temp...so why is it that the region right above is 100% saturated?

If you wait long enough, and, if the room is perfectly sealed, then eventually, either all the air in the room will reach 100% RH, or all the water will evaporate before that. However, before any of that happens, at the very interface between the liquid water and the air above it (i.e., the liquid surface), the air will be saturated with water vapor at 20 C. The partial pressure of the water vapor in the air will decrease with distance from the interface, so there will be a gradient near the interface until the partial pressure reaches the bulk partial pressure value throughout most of the room. The gradient in partial pressure is responsible for the diffusion of water vapor through the air away from the interface (i.e., evaporation).

But why is the partial pressure of the water in the air at the interface equal to the equilibrium partial pressure of the liquid at the interface temperature? This is an experimental observation related to fluid mechanics. You are taking a course in fluid mechanics now, so you may have heard of the no-slip boundary condition. If not, look it up. It says that, at the interface between a liquid and a gas, the velocity of the gas exactly matches the velocity of the liquid. That means that, from the frame of reference of the liquid at the interface, the gas immediately above it is stagnant. So the gas right at the interface comes into instantaneous equilibrium with the adjacent liquid.

2. There is a certain saturation pressure corresponding to the surrounding temperature. So if the vapor pressure is lower than that saturation value, more water would evaporate from things (like lakes/ plants etc.) But usually the RH of surroundings is stays lower than 100%...then does that mean the water lying around the place is continuously in disequilibrium with the atmosphere and more and more is trying to evaporate?

Sure. You know that from experience. After it rains, doesn't the ground dry up? As temperatures during the day rise, doesn't the morning dew evaporate? Don't you need to water the garden if it doesn't rain?

Chet

 

[Urmi Roy]

But why is the partial pressure of the water in the air at the interface equal to the equilibrium partial pressure of the liquid at the interface temperature? This is an experimental observation related to fluid mechanics. You are taking a course in fluid mechanics now, so you may have heard of the no-slip boundary condition. If not, look it up. It says that, at the interface between a liquid and a gas, the velocity of the gas exactly matches the velocity of the liquid. That means that, from the frame of reference of the liquid at the interface, the gas immediately above it is stagnant. So the gas right at the interface comes into instantaneous equilibrium with the adjacent liquid.

Wow, this is non-trivial. I would never have thought to make this connection. This is very cool :-)

Sure. You know that from experience. After it rains, doesn't the ground dry up? As temperatures during the day rise, doesn't the morning dew evaporate? Don't you need to water the garden if it doesn't rain?

Chet

Well I have a weird problem with this :-/ I agree with what you said about the every-day examples but if the RH of the surroundings is less than 100% and that causes evaporation from things lying around, then two things can happen: one; the air does becomes saturated at some point, and it starts raining; two- the situation I have trouble with- is that if during the span of the day if the air still didn't manage to get saturated, things will keep getting drier and drier to try and saturate the air...entire lakes could evaporate for this. (I know this sounds silly, but I'm honestly trying to understand what's going on).
So then there are only two situations are possible in reality--either everything's losing water or it's raining. So the environment never reaches equilibrium?

 

[Chestermiller]

Well I have a weird problem with this :-/ I agree with what you said about the every-day examples but if the RH of the surroundings is less than 100% and that causes evaporation from things lying around, then two things can happen: one; the air does becomes saturated at some point, and it starts raining; two- the situation I have trouble with- is that if during the span of the day if the air still didn't manage to get saturated, things will keep getting drier and drier to try and saturate the air...entire lakes could evaporate for this. (I know this sounds silly, but I'm honestly trying to understand what's going on).
So then there are only two situations are possible in reality--either everything's losing water or it's raining. So the environment never reaches equilibrium?

We know that hot air near the surface rises, and, as it rises, it expands and cools. Eventually it cools enough to reach the saturation temperature, and tiny liquid water drops begin to form. We call this clouds. Eventually, the water droplets in the clouds coalesce to large enough size to fall under gravity, and we get rain. So the lakes are replenished long before they dry up (most of the time). So, yes, the environment never reaches equilibrium. We call this meteorology.

Regarding the span of a day: evaporation occurs too slowly for a lake to dry up in the span of a day.

Chet

 

[russ_watters]

No, there are no global equilibriums in these situations, but there are relatively or even completely stable steady states. An evaporating puddle is in a stable steady state until it is gone, so you can consider the situation to be a steady-state, internal (and external if you include the sun) heat transfer/mass flow problem.

Back to your condensation issue, your original issue is not unlike what happens when you over-humidify a house in winter. The stable steady state balances heat flow in from the heater with heat flow out through the walls/windows. And then you have mass flow in from a humidifier, with condensation on the windows (and a growing puddle on the windowsill).

 

[Urmi Roy]

Thanks everyone, especially Chet for all the help and your patience with me!