It is often explained that warm air holds more water because warmer water is less likely to condense. This and other explanations just seem to be circular arguments. If the warmer air picks up enough moisture it will saturate and then water will still condense at 100% humidity.

Given two cubic feet of air with one warmer than the other and the warmer holding more water, where/how do all those extra water molecules invisibly fit in the warmer cubic foot? What is the deep physics or chemistry reason this happens?

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    HINT - density and temperature relation is inverse. You can assume deviation from ideal gas and then use a virtual temperature in the good ole ideal gas law. The relation does not change however. – gansub Oct 18 at 12:08
  • Higher temp then lower water vapor density. How does this allow more water? – Clark Oct 18 at 12:12
  • I meant lower density of air.So the atmosphere can "hold" more water vapor molecules. – gansub Oct 18 at 12:14
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    @gansub Your brief comments explain the phenomena better than either of the two answers. I wish you'd make an answer so I can upvote it. – horta Oct 18 at 22:55
  • @horta - I looked at BaroclinicCplusCplus and David Hammen's answer. They are both very good. If you have any other question please feel free to ask. – gansub Oct 19 at 6:16

Saying that warm air "holds" more moisture is technically incorrect, but is a common colloquialism. Let's break it down to the technicalities.

Let's consider a glass of water with a vacuum (no air) above it. What will happen? The molecules that are at the top most layer of the water will evaporate. At what rate will the water evaporate? Better yet, what is evaporation?

Evaporation is when the water molecules gain enough kinetic energy (how fast they vibrate) to break the bonds that hold them to one another. Kinetic energy is dependent on temperature. So the molecules vibrate faster, break their bonds, and enter the vacuum as a vapor. Some molecules will stay as a vapor in the vacuum, but others will reenter the liquid. When the molecules enter the liquid as fast as they are leaving, then it is saturated.

If the air is cooled down, then the rate at which molecules leave the liquid slows down. The molecules entering the liquid do not slow down at the same rate, causing the liquid to grow toward it's initial state.

Note that I specifically said it is a vacuum. Instead of a glass of water, picture the water as little drops. The atmosphere can act to warm or cool these drops, and vice-versa.

In the more nitty-gritty aspect of this, the equation that describes the vapor pressure as a function of temperature is called the Clausius-Clapeyeron equation/relation. The American Meteorological Society has one approximate solution, but I prefer this equation: $$e_{sat}(T)=611 Pa \exp[\frac{L_v}{R_v}(273.15^{-1}-T^{-1})]$$, where $L_v$ is the latent heat of vaporization, $R_v$ is the specific gas constant for water vapor, and $T$ is the absolute temperature in Kelvin. Combined with the ideal gas law for water vapor (assuming saturation) $$e_{sat}(T)V=m_vR_vT$$, and given the volume ($V$) we can write an expression for the mass of water vapor $m_v$. The equation comes out to $$m_v=611 Pa \exp[\frac{L_v}{R_v}(273.15^{-1}-T^{-1})]V R_v^{-1}T^{-1}$$

To answer your final question, the molecules are approximated as being infintessimally small, per the ideal gas law. To be more specific, one molecule of water is about 7.08$\times$ 10$^{-19}$ cubic feet (after some math), so the added volume is considered negligible. In short, the molecules are treated as point masses.

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    Thanks very much for the nice explanation. Can you elaborate futher on how this answers my question? – Clark Oct 18 at 14:57
  • You're welcome. I am editting it right now. – BarocliniCplusplus Oct 18 at 14:59

What is the deep physics or chemistry reason this happens?

In one word, entropy.

In a thousand words (i.e., a picture),

Phase diagram of water
Source: Wikipedia article on Phase Diagram

The above is a phase diagram of water. The curve of interest is that between vapor (tan area) and solid or liquid (blue and green areas). For a given temperature (a vertical line in the diagram), it is entropically favorable for water to be solid or liquid when the partial pressure of water vapor is above the sublimation/vaporization curve, but it is entropically favorable for water to be vapor when the partial pressure is below the curve.

Note well: The pressure of interest is the partial pressure of water vapor, not the pressure of the air as a whole. On the other hand, the temperature of the air does dictate the temperature of the water vapor that is present. Also note well: The above diagram is for a flat surface. Small water droplets have a high surface tension, which increases the effective saturation pressure in very clean air. (This is, for example, why snowflakes almost always have a tiny grain of dust at their centers.) Nonetheless, the effective saturation pressure follows the same form: It increases with increasing temperature.

You asked for a deep physics reason for why this happens. Suppose the surface between water ice or liquid water and water vapor is at equilibrium; i.e., the air is the saturated with water vapor. The water vapor molecules have a speed distribution approximately given by the Maxwell–Boltzmann distribution. Some water vapor molecules will be moving slower than others. Those slow-moving molecules that hit the surface are more likely to be absorbed by the surface than are those molecules that are moving faster. The molecules in the ice or liquid water are also vibrating / moving at different speeds. The fast moving solid/liquid water molecules are more likely to escape the surface than are those that are moving slower. Under equilibrium conditions, the rates at which vapor molecules are captured and ice/liquid molecules escape are equal.

Raising the temperature of the solid/liquid and of the vapor means fewer vapor molecules will be moving slow enough to be captured while more solid/liquid will be moving fast enough to escape. This increases the pressure at which equilibrium occurs. Lowering the temperature similarly decreases the pressure at which equilibrium occurs.

The water molecule is just another molecule in the atmosphere - except that it's triatomic, whereas 99.97% of the dry atmosphere is monatomic (Argon) and diatomic (Oxygen and Nitrogen). It's also slightly electrically polarized, and because of this it has a low energy state when in the presence of similar molecules (liquid or solid form), and at a higher temperature.

So "Why doesn't all the water vapor simply fall out of the air?" What's keeping these molecules in the the vapor state is simply the energy of motion. Cool the atmosphere (remove this energy) and the water vapor will be allowed to return to its lower energy state. The same thing will happen with all the other atmospheric constituents - just at different temperatures and pressures.

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