Notes on Humidity, Vapor Pressure, Partial Pressure, Evaporation, Condensation, and Dew Point

Key concepts

Humidity: the amount of water vapor present in the air.
Relative humidity (RH): a percentage describing how much water vapor is in the air relative to the maximum amount the air can hold at a given temperature. It varies from 0% to 100% (and can exceed 100% in supersaturated situations).
Partial pressure of water, $p{H2O}$ : the actual pressure contributed by water vapor in the air.
Vapor pressure of water $p_{sat}(T)$ : the maximum partial pressure of water vapor that air can hold at temperature $T$ ; it depends on temperature. It increases with temperature.
Evaporation: liquid water → water vapor (in the gas phase).
Condensation: water vapor → liquid water (gas to liquid).
Dew point: the temperature at which the air becomes saturated (i.e., the partial pressure of water equals the vapor pressure, $p{H2O} = p_{sat}(T)$ ). At the dew point, RH = 100% and condensation can begin if the temperature is lowered further.
Saturated, unsaturated, and supersaturated:
- Unsaturated: p{H2O} < p_{sat}(T); RH < 100%; the air can hold more water vapor.
- Saturated: $p{H2O} = p_{sat}(T)$ ; RH = 100%; the rate of evaporation equals the rate of condensation.
- Supersaturated: p{H2O} > p_{sat}(T); RH > 100%; condensation tends to occur to reduce water vapor.

Relative humidity: definition and interpretation

Relative humidity is defined as
$RH = rac{p{H2O}}{p_{sat}(T)} imes 100\%.$
It represents the ratio of how much water vapor is actually in the air to how much water vapor the air can hold at that temperature.
Examples based on given values:
- At 30°C, if $p{H2O} = 18\ ext{Torr}$ and $p_{sat}(30°C) = 31.8\ \text{Torr}$ , then
 $RH = \frac{18}{31.8} \times 100\% \approx 56.6\%.$
- If $p{H2O} = 0\ \text{Torr}$ , then $RH = 0\%.$
- If $p{H2O} = p_{sat}(T)$ , then $RH = 100\%.$

Vapor pressure vs partial pressure

Vapor pressure is a property of water that depends only on temperature: it is the pressure at which water in liquid form is in equilibrium with water in vapor form.
Partial pressure $p{H2O}$ depends on how much water vapor is actually present in the air.
At a fixed temperature, as water evaporates, $p{H2O}$ rises until it approaches (or equals) $p{sat}(T)$ ; if you have more water evaporating than condensing, $p{H_2O}$ climbs toward that limit.

Evaporation, condensation, and equilibrium (using three beakers)

Beaker A (evacuated gas inside, only liquid water):
- Initial state: no water vapor in the air, so $p{H2O}=0$ .
- The temperature is 30°C, so $p_{sat}(30°C) = 31.8\ \text{Torr}.$
- Evaporation occurs (liquid water→gas); some water vapor enters the air, raising $p{H2O}$ (example value given: ≈ 10 Torr).
- Condensation cannot occur yet because there are almost no water molecules in the air, so the rate of condensation is effectively zero.
- Evaporation rate > condensation rate (the arrows show more coming up than going down).
- Result: p{H2O} < p_{sat}(T), so RH < 100% (for this case, $RH\approx \frac{10}{31.8} \times 100\% \approx 31\%$ ).
Beaker B:
- $p{H2O} = 10\ \text{Torr}$ , same temperature (30°C) and same $p_{sat}$ .
- Evaporation still dominates (more water vapor entering the gas phase than condensing).
- $p{H2O}$ continues to rise toward 31.8 Torr.
Beaker C:
- Water vapor has accumulated until $p{H2O} = 31.8\ \text{Torr} = p_{sat}(30°C).$
- Now the rate of evaporation equals the rate of condensation.
- At this point, the air is saturated: $p{H2O} = p_{sat}(T)$ , so $RH = 100\%.$
Summary of the roles:
- Partial pressure measures how much water vapor is actually in the air.
- Vapor pressure is the maximum amount of water vapor the air can hold at a given temperature.
- When p{H2O} < p{sat}, evaporation dominates; when p{H2O} > p{sat}, condensation dominates; when they are equal, you have equilibrium.

The dew point and 100% relative humidity

Dew point is the temperature at which the partial pressure equals the vapor pressure:
$p{H2O} = p{sat}(T{dp}).$
At the dew point, RH = 100%; if you cool further, condensation dominates and you start to see dew, fog, clouds, or precipitation.
Practical takeaway: to measure dew point, gradually cool the air until condensation just begins.
Conceptual link: the dew point is the temperature at which the air becomes fully saturated for that amount of water vapor present.

Condensation, saturation, and supersaturation explained

If the partial pressure of water is less than the vapor pressure (e.g., $p{H2O} = 0$ or 10 Torr with $p_{sat} = 31.8$ Torr), the air is unsaturated and can hold more water vapor; RH < 100%.
If the partial pressure equals the vapor pressure, the air is saturated; RH = 100%.
If the partial pressure exceeds the vapor pressure, the air is supersaturated; RH > 100% and condensation occurs to bring the system back toward equilibrium.
In general, unsaturated ⇔ RH < 100%, saturated ⇔ RH = 100%, supersaturated ⇔ RH > 100% (for the same temperature).

Supersaturation: how it happens and what it means

Suppose the temperature is 30°C (psat = 31.8 Torr) and the current partial pressure is pH2O = 27 Torr (still below saturation).
If the temperature is rapidly lowered to 20°C (psat(20°C) ≈ 17.5 Torr) while pH2O remains 27 Torr, then p{H2O} > p_{sat}(20°C)
- The air cannot hold all this water vapor at 20°C, so condensation occurs.
- This is a transient supersaturated state: the system moves toward equilibrium by forming liquid water.
Dwell point (dew point) is the temperature where the vapor pressure would equal the actual partial pressure, i.e., where $p{sat}(T{dp}) = p{H2O} = 27\ \text{Torr}$ . This lies somewhere between 20°C and 30°C; a rough estimate might be around 27°C in this illustration, but the exact value depends on the water-vapor saturation curve.
Once the temperature drops below the dew point, condensation proceeds and visible droplets form (fog, dew).

Practical relationships and takeaways

Relationships:
- As temperature increases, it P(sat )increases. For a fixed amount of water vapor, increasing temperature lowers RH.
- As water vapor increases (higher $p{H2O}$ ) at a fixed temperature, RH increases.
- Dew point is the temperature at which the current $p{H2O}$ would equal the new $p_{sat}$ ; condensation begins if you cool past that point.
Real-world relevance:
- Weather patterns: fog, dew, clouds form when air reaches or exceeds its dew point.
- Indoor climate control (HVAC): heating or cooling air changes $p_{sat}$ and thus RH, impacting comfort and comfort-related systems.
- Humidity management affects sweating and cooling; dry air supports faster evaporation of sweat, humid air slows it, making it feel hotter in humid conditions.

Quick reference formulas and definitions

Relative humidity:
$RH = \frac{p{H2O}}{p_{sat}(T)} \times 100\%.$
Vapor pressure vs partial pressure:
- $p_{sat}(T)$ : temperature-dependent maximum partial pressure of water vapor.
- $p{H2O}$ : actual partial pressure of water vapor in air.
Equilibrium (evaporation = condensation):
$p{H2O} = p_{sat}(T).$
Dew point: the temperature $T{dp}$ at which p{H2O} = p{sat}(T_{dp}).}

Practice reflections and quick questions

If at 25°C you measure $p{H2O} = 20\ \text{Torr}$ and $p_{sat}(25°C) = 31.7\ \text{Torr}$ , what is the RH?
- Answer: $RH = \frac{20}{31.7} \times 100\% \approx 63\%.$
What happens if you cool the air from 25°C to 15°C while keeping the same $p{H2O}$ ? Describe the likely condensation behavior.
How would increasing the ambient temperature affect the dew point for a fixed amount of water vapor in the air?

Summary of the key ideas

Humidity is about water vapor in air; relative humidity compares actual water vapor to what the air can hold at that temperature.
Evaporation increases the amount of water vapor in the air until it approaches the vapor pressure; condensation reduces it by turning vapor back into liquid.
The dew point is the critical temperature where the air becomes saturated; below this temperature, condensation occurs.
Saturation, unsaturation, and supersaturation describe the relationship between the actual water vapor pressure and the vapor pressure at the current temperature, with RH providing a percent measure of proximity to saturation.
The concept has broad implications in weather, climate control, human comfort, and various practical environments.

Important glossary recap

p_H2O: actual partial pressure of water vapor in air
p_sat(T): vapor pressure of water at temperature T
RH: relative humidity, a percentage = $\frac{p{H2O}}{p_{sat}(T)} \times 100\%$
Dew point: the temperature where $p{H2O} = p_{sat}(T)$ and condensation begins
Evaporation vs condensation: kinetic-energy-driven exchange between liquid and gas phases depending on the relation between $p{H2O}$ and $p_{sat}(T)$