13.6 Humidity, Evaporation, and Boiling

Humidity and Vapor Pressure

Water vapor and air capacity

Vapor pressure is the pressure exerted by water vapor molecules in a closed system at equilibrium. As temperature increases, water molecules gain kinetic energy and evaporate more readily, which raises the vapor pressure.

How much additional water vapor the air can "hold" depends on the gap between the current vapor pressure and the saturation vapor pressure (the maximum vapor pressure possible at a given temperature).

• When vapor pressure equals saturation vapor pressure, the air is saturated and can't hold more water vapor. That's 100% relative humidity.
• When the gap is large, the air is far from saturation, meaning low relative humidity and plenty of room for more evaporation.

Relative humidity and partial pressure

Relative humidity is the ratio of the actual water vapor pressure to the saturation vapor pressure at that temperature, expressed as a percentage:

$\text{Relative Humidity} = \frac{\text{Actual Vapor Pressure}}{\text{Saturation Vapor Pressure}} \times 100\%$

For example, if the actual vapor pressure is 20 hPa and the saturation vapor pressure is 30 hPa:

$\text{RH} = \frac{20}{30} \times 100\% = 66.7\%$

In a gas mixture like air, each gas contributes its own partial pressure to the total. Water vapor's partial pressure is what we compare to the saturation value. When that partial pressure equals the saturation vapor pressure, relative humidity hits 100% and you've reached the dew point.

Calculations with vapor pressure

Vapor density (mass of water vapor per unit volume of air, in g/m³) can be found using the ideal gas law rearranged:

$\text{Vapor Density} = \frac{P_v \times M_w}{R \times T}$

where:

• $P_v$ = vapor pressure (in Pa)
• $M_w$ = molar mass of water = 18.02 g/mol
• $R$ = gas constant = 8.314 J/(mol·K)
• $T$ = temperature in Kelvin (convert from Celsius: $K = °C + 273.15$)

Absolute humidity uses the same formula as vapor density. It tells you the actual mass of water vapor in a given volume of air.

Specific humidity is the ratio of the mass of water vapor to the total mass of the air parcel. The formula is more involved and typically matters more in meteorology courses, so for an intro physics course, focus on being comfortable with relative humidity and vapor density calculations.

Dew point and humidity relationship

The dew point is the temperature at which air becomes saturated (100% relative humidity) if cooled at constant pressure. Cool the air below its dew point, and water vapor starts condensing into dew, fog, or clouds.

The gap between the current air temperature and the dew point tells you how close the air is to saturation:

• Small gap → high humidity. Air temperature 25°C with a dew point of 20°C feels muggy because the air is nearly saturated.
• Large gap → low humidity. Air temperature 25°C with a dew point of 10°C feels dry because the air can still absorb a lot more moisture before condensation begins.

Evaporation and Boiling

The process of evaporation and factors affecting its rate

Evaporation is the process by which individual liquid water molecules at the surface gain enough kinetic energy to overcome intermolecular forces and escape into the air as vapor. Unlike boiling, evaporation happens only at the liquid's surface and can occur at any temperature.

Four main factors control the rate of evaporation:

1. Temperature — Higher temperatures raise the average kinetic energy of molecules, so more of them can escape. Hot water evaporates faster than cold water.
2. Surface area — A wider surface exposes more molecules to the air. A shallow pan evaporates faster than a tall narrow glass holding the same volume.
3. Air movement — Wind sweeps away vapor molecules near the surface, preventing the air just above the liquid from becoming saturated. That's why clothes dry faster on a breezy day.
4. Humidity — Drier air (lower vapor pressure) means fewer vapor molecules are bouncing back into the liquid, so net evaporation speeds up.

Evaporation requires energy input. The energy absorbed per unit mass during this phase change is called the latent heat of vaporization (for water, about 2,260 kJ/kg at 100°C). This is why evaporation has a cooling effect: the escaping molecules carry energy away from the liquid.

Boiling vs. evaporation

Boiling is the rapid vaporization of a liquid that occurs when its vapor pressure equals the surrounding atmospheric pressure. At that point, vapor bubbles can form throughout the liquid, not just at the surface.

Two key factors change the boiling point:

1. Atmospheric pressure — Lower pressure means a lower boiling point. At high altitude (e.g., Denver at ~1,600 m), water boils below 100°C because atmospheric pressure is lower. This is why cooking times increase at altitude.
2. Dissolved solutes — Adding a solute like salt raises the boiling point through boiling point elevation. Salt water boils at a slightly higher temperature than pure water.

Phase changes and energy

Phase changes like evaporation and boiling involve energy transfer without a change in temperature. The heat of vaporization is the energy required to convert a substance from liquid to gas at constant temperature. For water, this value is large ($L_v \approx 2{,}260 \text{ kJ/kg}$), which is why boiling a pot of water takes significant energy even after it reaches 100°C.

The triple point is the unique temperature and pressure at which a substance can exist as solid, liquid, and gas simultaneously. For water, the triple point occurs at 0.01°C and 611.7 Pa. It's a fixed reference point used in calibrating thermometers.