Physics · Molecular physics and thermodynamics · Grade 9

Humidity and dew

Why does an ice-cold glass “sweat”, and how does an air conditioner dry a room?

The situation

On a hot summer day you pour ice-cold water from the fridge into a glass — within a minute the outside of the glass “sweats” and droplets start to run down. In winter the window fogs up, and in summer water drips from the pipe under the air conditioner. Where does this water come from? Why does it appear only on the part of the glass filled with water, and why do double-glazed windows mist up less?

Topic

Air humidity. Air conditioner and the dew point

Key ideas

Saturated vapour pressure
Relative humidity
Dew point

Formulas

φ = \frac{e}{e_{s} (T)} \cdot 100 %

Relative humidity: the vapour pressure in the air compared with the saturated vapour pressure at the same temperature.

e_{s} (T) = 6.112 \cdot \exp (\frac{17.62 T}{243.12 + T}) hPa

Magnus formula ( $T$ in °C). The saturated vapour pressure rises steeply with temperature.

T_{d} = \frac{243.12 γ}{17.62 - γ}, γ = \ln φ + \frac{17.62 T}{243.12 + T}

Dew point: if we cool the air to this temperature, the vapour becomes saturated ( $φ$ as a fraction).

ρ = \frac{e}{R_{v} T}

Absolute humidity is the mass of vapour in 1 m³ of air ( $R_{v} = 461.5$ J/(kg·K), $T$ in K).

e = e_{s} (T_{h}) - A p (T - T_{h})

Psychrometer: the wet-bulb thermometer cools because of evaporation. $A \approx 6.6 \cdot 10^{- 4}$ K⁻¹.

Theory

Saturated vapour

Molecules constantly leave the surface of water and return to it. When the vapour pressure reaches a certain value — the saturated vapour pressure $e_{s}$ — the numbers leaving and returning become equal. $e_{s}$ rises steeply with temperature: at 30 °C it is almost 7 times greater than at 0 °C (see the graph). That is why warm air can “hold” more vapour.

Dew point

If we cool the air without changing its vapour content, $e$ stays the same while $e_{s}$ decreases — so the relative humidity rises. At the dew point $T_{d}$ the air becomes saturated. On a surface colder than $T_{d}$ the vapour turns into water: an ice-cold glass, a winter window pane, the cooling radiator of an air conditioner. The glass “sweats” only where cold water touches it — below the water level.

Why does an air conditioner dry the air?

Room air passes through the cooling radiator. If the radiator is colder than the dew point, the vapour condenses on it and is drained outside through a pipe — there is less vapour in the room. If the radiator is warmer than $T_{d}$ , the air is only cooled, not dried. In winter it is the other way round: when cold outdoor air is heated, $e_{s}$ rises and the relative humidity falls — so the air in heated rooms is dry.

The psychrometer and the model

The wet-bulb thermometer loses heat through evaporation and reads lower than the dry-bulb one; the drier the air, the bigger the difference. From the two readings, $φ$ is found in a table. The model is simplified: the room is closed ( $V$ = 54 m³), has no vapour sources (people, cooking) and no ventilation, the room temperature is held by a thermostat, the radiator temperature is constant, and 15 % of the air passes the radiator without touching it. Time is sped up 30 times.

Atmospheric phenomena

When air is cooled without changing its vapour content, the relative humidity rises and the air becomes saturated at the dew point — this is behind many phenomena in the atmosphere. At night the ground and grass lose heat by radiation and cool: if the surface temperature falls below $T_{d}$ , dew forms; if $T_{d} < 0$ °C, hoar frost forms. If all the air near the ground cools through its whole volume, we get fog. Rising air expands and cools (dry air by about 10 °C per kilometre): at the height where it reaches $T_{d}$ a cloud appears, and the droplets grow larger and fall as rain, or as snow in the cold. In the daytime the relative humidity falls because the temperature rises, and at night it rises. A comfortable relative humidity for people is about 40–60 %.

Hygrometers and the lab work

A hygrometer is an instrument for measuring humidity. In a condensation hygrometer a shiny surface is cooled slowly and the temperature $T_{d}$ at which dew appears on it is measured: $φ = e_{s} (T_{d}) / e_{s} (T)$ — the ice-cold glass in the scene works in exactly this way. In a hair hygrometer a degreased hair lengthens in humid air and turns a pointer (it is less accurate); in electronic hygrometers the dielectric of a capacitor absorbs moisture and its capacitance changes. August's psychrometer is two thermometers side by side, with the bulb of one wrapped in wet cloth. Lab work: switch on the “Unknown air” mode (the humidity is hidden), wait until the thermometers settle, note the dry-bulb reading $t$ and the difference $Δ t = t - t_{h}$ , find $φ$ in the psychrometric table (interpolate between rows if necessary) and check your answer. The second method is to find the dew point with the glass and calculate $φ$ from the formula. Water boils when its saturated vapour pressure becomes equal to the external pressure — see the “Melting, boiling and evaporation” experiment.

Did you know?

At 30 °C, 1 m³ of air can hold at most about 30 g of water vapour, but at 0 °C only ≈ 5 g. That is why cold air is “dry” even at 100 % humidity.
Fog and clouds also form because of the dew point: when air cools, the vapour turns into tiny droplets. Morning dew is condensate on grass that has cooled during the night.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Melting, boiling and evaporation — Grade 7
Calorimeter: heat balance — Grade 7
Tandoor and sandal: heat — Grade 7
Gas under a piston — Grade 9
Heat engine: the Otto cycle — Grade 9
Entropy and the arrow of time — Grade 9
Brownian motion and diffusion — Grade 9
Surface tension and capillarity — Grade 9
Crystalline and amorphous solids — Grade 9
Phase transitions — Grade 10