Explain relative humidity without adding water

A percentage needs a denominator

A relative humidity report of 50% does not mean that half the air is water. It compares the water-vapor partial pressure with a temperature-dependent saturation pressure. Partial pressure is the contribution water vapor makes to the mixture’s total pressure. The investigation supplies vapor pressure directly, rather than deriving it from a humidity sensor or a wet-bulb observation.

Let e denote water-vapor partial pressure in hPa and e_s(T) denote saturation vapor pressure over liquid water at air temperature T. Relative humidity is RH = 100e/e_s(T), expressed as a percentage. The ratio has no units because both pressures use the same unit. Increasing its denominator at fixed numerator lowers the ratio.

Calculate the reference state

The selected approximation is e_s(T) = 6.112 exp(17.67T/(T + 243.5)), with T in degrees Celsius and pressure in hPa. Here exp is the exponential function, not the vapor-pressure symbol e. Unlike the ideal-gas temperature formula, this empirical expression is written specifically for Celsius input. Its constants incorporate that convention.

At T = 20°C, the exponent is 353.4/263.5, approximately 1.3412. The saturation pressure is about 23.37 hPa. With supplied vapor pressure 20 hPa, RH = 100 × 20/23.37, approximately 85.58%. The same saturation and humidity formulas appear in the NWS calculation guide, although that guide also includes a wet-bulb step not implemented here. NWS humidity and dew-point equations.

On paper, draw two pressure bars using the same scale: supplied e and calculated e_s. Label their ratio separately from their difference. Relative humidity uses division; subtracting the two pressures would produce a pressure deficit with different units and meaning.

Change temperature without changing the mixing ratio

Keep e at 20 hPa and raise temperature to 30°C. Saturation pressure rises, so RH falls to about 47.1%. At 40°C it is about 27.1%. The vapor-pressure bar has not changed; the saturation bar has grown.

The model also fixes total pressure at 1000 hPa. Water-vapor mixing ratio is 1000 × 0.622e/(1000 − e), in grams of vapor per kilogram of dry air. At e = 20 hPa it is about 12.69 g/kg dry air. It stays fixed when only temperature changes, because both e and total pressure remain fixed. The factor 1000 converts a kilogram-per-kilogram ratio to grams per kilogram. NASA description of humidity and mixing-ratio relationships.

This comparison is not heating a sealed container of fixed volume. Under fixed-volume conditions, retaining a fixed amount of vapor while changing temperature would generally change its partial pressure. State which quantities are held fixed before explaining a change in RH.

Distinguish relative and absolute descriptions

A falling RH by itself does not show that vapor was removed. Conversely, raising vapor pressure at fixed temperature raises RH and the modeled mixing ratio. Capture these two different experiments and explain each in terms of numerator and denominator.

The browser does not use the metaphor of air as an empty container that becomes filled with water. Saturation concerns equilibrium vapor pressure over the specified water phase. No droplets, cloud water or rainfall amount is calculated from the percentage alone.

Keep a ledger with columns for T, e, e_s, RH and mixing ratio. Mark supplied controls separately from calculated values. This helps reveal that the many displayed numbers are linked transformations of two chosen inputs rather than independent observations confirming one another.

Transfer to a supplied saturation value

For a paper-only state, suppose e = 12 hPa and e_s = 24 hPa. Calculate RH. If a temperature change makes e_s = 40 hPa while e and total pressure stay fixed, what changes?

Check the relative-humidity transfer

Initial RH is 100 × 12/24 = 50%. The new RH is 100 × 12/40 = 30%. Mixing ratio stays fixed because vapor pressure and total pressure stay fixed. The falling percentage does not establish vapor removal.