of water varies with temperature 
(in Kelvin):Experience tells us to expect that exhaling into sufficiently cold, clear air creates a visible cloud. However, our experience does not guide us in the reverse question: can exhaling into sufficiently hot, cloudy air create a pocket of clear air? (I encourage you to guess before reading further!) It turns out that a subtle aspect of how water behaves in air is responsible for both of these phenomena (or their absence).
First, what is “foggy” or “cloudy” air? The atmosphere is a gas that contains a mixture of different types of molecules, including a variable amount of gaseous water, called water vapor. At any given temperature, there is a saturation pressure, which is the maximum water vapor pressure for that temperature. Any excess water vapor past this maximum condenses into liquid water: this can happen on solid surfaces exposed to the air (which is why water droplets appear on cold surfaces) or around tiny dust particles or other nucleation sites suspended in the air, forming microscopic water droplets we call a cloud. An air mass with more water than the saturation pressure is foggy, whereas an air mass with less water is clear.
To a first approximation, we can ignore all other constituents of air but water. The partial pressure of water vapor is the air pressure times the proportion (volumetric, or molar) of air that is water; this is the pressure that the water vapor would have if the other constituents were removed. For a given temperature and pressure, we can see on a phase diagram of water what phase of water is preferred: but in the absence of other constituents, what is providing this pressure except water vapor?
Phase diagram of water, from Wikipedia. Much more information about water, including details about its phase diagram, can be found here.
So, the way to interpret the “liquid” region of water’s phase diagram is that if the partial pressure of water is so high that the pressure reaches the liquid region, then condensation of water vapor into liquid water is favored. If the partial pressure drops until it reaches the gas region of the diagram, then evaporation of liquid water into water vapor is favored. At the boundary between these regions, which is the saturation pressure, the rate of condensation equals the rate of evaporation. The “relative humidity” is defined as the partial pressure of water divided by the saturation pressure: whenever relative humidity is less than 100%, any standing bodies of water will tend to evaporate over time. The competing processes involved in determining this boundary are the latent heat released by water condensing from gas to liquid, versus the entropically favored expansion of a gas to fill available space.
Generally speaking, air found near the surface of the Earth has moderately high relative humidity. Cooling air causes relative humidity to go up (because the amount of water remains fixed, but saturation pressure goes down) until it reaches 100% (this temperature is the “dew point”) and liquid water condenses out. So as a quick rule of thumb – though not necessarily helpful for the problem we are considering – cooling air causes condensation: for example, cold surfaces gain water droplets by cooling the air near them; rising air in the atmosphere condenses to make clouds; and visible “steam” (actually water droplets) appears above hot beverages when the hot humid air coming off the surface cools.
What makes boiling different from regular evaporation? This is the part where the non-water constituents of the air matter a lot. Consider a pot of water under standard atmospheric conditions. In the interior of the water, the pressure equals the air pressure. However, because of the physical separation of liquid water and the air, the partial pressures within the interior of the water is dominated by water, with small contributions from dissolved gases, while within the air the partial pressures reflect the amount of water vapor or other gases in the air. So evaporation within the interior of the water is only favored when the total pressure of the air is below the saturation pressure of water: when this happens, the water is boiling, and it starts evaporating from its whole volume all at once. If the partial pressure of water in the air is below the saturation pressure, but the total air pressure is above, then evaporation only takes place at the surface, and the water is not boiling.
Another way to understand boiling is that when a bubble of water vapor forms in the interior of the liquid, the pressure within the bubble equals the total air pressure outside; so boiling is only favored if saturation pressure is above total air pressure. The bubble displaces an equal volume of the surrounding air (via displacing some liquid water), but doesn’t immediately gain the entropic benefit of mixing with the surrounding air: the latter only happens if the bubble reaches the surface.
You’ve likely heard that, when exposed to a vacuum, liquid water both boils and freezes. This is true: the low air pressure causes water to boil, cooling the water until it freezes. The process continues with the solid water, which sublimates, cooling it further, and given a perfect vacuum eventually the whole mass will turn to vapor (as can be seen from the phase diagram). However sublimation mostly only happens from the surface, and at low temperatures the saturation pressure of water falls super-exponentially with temperature, so the sublimation can take a very long time.
Now, the resolution of our questions can be found in the phase diagram of water if we know where to look. When you exhale in winter, the warm, humid air in your lungs mixes with the cold, dry air outside. The two initial air masses are clear (there is no liquid water), but under certain conditions their mixture can go above the saturation point (causing a cloud to condense). This is the same process that happens above a hot beverage: hot air touching the beverage can carry a lot of water vapor, creating hot, humid air. When mixed with the ambient air, this can result in saturated air, which we describe as “steam” (although the visible condensate is actually made of liquid droplets).
The converse question of breathing clear air into a hot fog imagines warm, saturated air in your lungs mixing with very hot, saturated ambient air. Can mixing two saturated air masses result in an unsaturated air mass?
The answer to both of these questions can be found in the fact that the liquid-gas boundary in water’s phase diagram is concave-up. (Note that in the above diagram it looks concave-down due to the log scale of pressure.)
Let us examine this boundary a little more closely. The Clausius-Clapeyron
relation describes how the saturation pressure of water varies with temperature (in Kelvin):
where 
is the specific gas constant of water vapor and
is the latent heat of vaporization of water
(which itself depends on temperature). Thus, crudely speaking, is roughly exponential in temperature. A
high-accuracy approximation for is given by Teten’s equation:
where now is given in Celsius.
Detail in phase diagram of water; the preferred phase is liquid in the saturated region, and gas in the unsaturated region. Example air masses are marked on the diagram, representing cold, clear air in winter; air exhaled from the lungs; and hot, foggy air. The grey region represents ambient conditions where one’s breath will condense on exhalation.
A mixture of two air masses falls somewhere on the line connecting them, with the location depending on the mixing ratio. Note that any mixture of saturated air masses gives a saturated air mass, while some mixtures of unsaturated air masses can be saturated.
When two air masses mix, the resulting temperature and pressure is roughly the average of the two initial masses’ temperatures and pressure, weighted by the relative sizes of the initial masses. Thus, in a plot of temperature and pressure, the final mass should fall on a straight line connecting the initial masses (or more generally, lie in the convex hull if there are more than two initial masses). Shown are hypothetical temperatures and pressures representing cold, wintery air; air as exhaled from the lungs; and hot, foggy air.
For a healthy person, air exhaled from the lungs is saturated, with a temperature slightly below body temperature. When mixed with cold, unsaturated air, we see that the resulting air mass falls on a line that passes through the saturated region of the plot; the location on this line depends on the amount of each initial mass. A typical exhalation has a volume of about 0.5 liters; the amount of ambient air that gets mixed increases with time since exhalation, so the cloud of visible condensate increases in volume over time until so much ambient air is mixed in that it falls back below saturation pressure, and the exhalation disappears. The colder and wetter the ambient air, the more ambient air can be mixed to maintain saturated conditions, and the larger one’s visible breath is. (At cold temperatures the relative humidity of the ambient air matters very little because the ambient air contributes very little water regardless.)
The grey shaded region shows ambient conditions where mixing some ratio of exhaled air and ambient air results in saturated air. Thus it would appear that you should always see your breath at a temperature of 10 C or below, regardless of how dry the air is. However, near the boundary of the grey region, condensation can only occur if a very small amount of ambient air is mixed with your exhalation: to achieve a 10:1 mixing ratio for example, requires only 50 mL of ambient air. Furthermore, the amount of condensate that would (very briefly) form is extremely small, possibly far too little to be visible. In practice, you will only see your breath above 10 C if the air is very humid.
Now we consider the converse question of breathing clear air into a hot fog. First, we emphasize that such conditions do not occur naturally on Earth, and are exceedingly hostile to human survival. Ordinarily it is possible for you to briefly survive high temperatures (like the dot shown in the diagram) through sweating, as the evaporation of water from the skin is endothermic. However, effective cooling through sweating requires that the ambient air be very dry – specifically, dry relative to the water’s saturation pressure at body temperature, not at the ambient temperature.
In our hypothetical, the hot air is not only wet compared to the saturation pressure at body temperature, but even to the much higher saturation pressure at ambient temperature. Instead of evaporation, water would condense quite rapidly onto any exposed skin. Exposure to 60 C water causes serious burns within approximately 5 seconds, but since condensation is exothermic, the condensed water would actually be hotter.
So, if you were to exhale in such conditions, could you breathe clear air into a fog? (We suppose that you stand upside-down to slow the accumulation of hot water in your lungs.) No. Because the saturation pressure is concave-up, there are no combinations of saturated air masses in any mixing ratios that can result in an unsaturated air mass. That is to say, foggy air mixed with foggy air will always make foggy air.
If the saturation pressure were concave-down, the reverse would be true: any clear air masses would always mix to make a clear air mass, and it would be possible to mix foggy air masses to make clear air.
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