Humid air and condensation

The first documented study of humid air seems to be the description made by Charles Leroy (1751), a medical doctor in Montpellier (France). He reported in 1751 to the Académie Royale des Sciences that water can be dissolved in air according to air temperature, the higher temperature corresponding to larger dissolution. In support of his claim he described several experiments. The most demounstrative was made with a bottle of air closed at daytime temperature. Once cooled at night, air was unable to hold all water dissolved at higher daytime temperature: The exceeding water lead to well visible condensed droplets inside the bottle.

Air is indeed never completely dry; it always holds in an invisible way some water vapour in different concentrations depending on its temperature. In addition to vapour, humid air can also contain water in visible condensed states: liquid (fog droplets), solid (frosty fog). In the latter cases where vapour and condensed phases coexist, humid air is said supersaturated.

Humid air can thus be considered of being formed of (1) dry air unlikely to condense in the conditions of temperature and pressure considered here, and (2) water vapour likely to condense in liquid or ice. Dry air is mainly composed of Nitrogen (≈ 78%) and Oxygen (≈ 21%). For regular conditions of temperature and pressure found at the earth surface, both gases are far from their critical point coordinates and both fluids can be accepted as ideal gases. Air will be thus considered as a single ideal gas. Water is also far from its critical point coordinates and can be also considered as an ideal gas.

1. Partial pressures: the Dalton Law

The partial pressure of a gas is the pressure that would have the gas if alone in a volume V. As both dry air and water are ideal gases, the total pressure  is equal to the sum of the partial pressures: This is the Dalton law. With p_a (resp p_v) the partial pressure of air (resp., water), one gets p_m = p_a + p_v. This additivity rule corresponds to neglect intermolecular forces among the gases molecules. Pressure being due to impacts of moving gas molecules, the total pressure is simply the addition of impacts of each type of molecules.

2. Equation of state

The equation of state for dry air and water vapour is the equation of ideal gas p_iV = n_iRT where the subscript i stands for air (i=a) or water vapor (i=v); ni = mi /_nMi is the number of moles (i)  of molar mass M_i and mass m_i in a volume V, R = 8.314 J.mole^-1.K^-1 is the molar gas constant. The properties of humid air at given partial pressure of water vapour can be deduced from this simple equation of state.

3. Saturation; condensation

Let us consider the cooling process at constant pressure p_m of a mass of humid air which contains a given mass of water. Mass conservation requires that the total mass and the water vapour mass remain constant during the process. This is therefore also the case for the number of moles of water vapour moles and the corresponding molar fraction n_v / n = p_v / p_m (from the equation of state above). It results that the water vapour pressure remains constant during the cooling process. In the atmosphere, humid air cooling thus occurs at constant water vapour pressure.

During cooling, condensation into liquid can occur (see the Clapeyron phase diagram in Figure 1). Let us consider a mass of humid air initially at point A on isotherm T₁. When temperature decreases at constant pressure p_v, its volume also decreases. The liquid-vapour coexistence curve (the saturation curve) is reached (point B) at some temperature T₂ and liquid drops can appear. Point B is called the dew point and T₂ is the dew point temperature T_d. The vapour pressure is the saturation pressure at temperature T_d, p_s(T_d) = p_v(T₁) where water vapour condensation can start. When air is cooled further, condensation proceeds at constant pressure p_v and temperature T_d. Cooling energy only compensates the release of the condensation latent heat (Read : Pressure, temperature and heat). In C, all water contained in the humid air has condensed. Zone BC is the fog zone where liquid droplets coexist with vapour. Further cooling (until point D) is only concerned with liquid.

The liquid-vapour saturation curve represents in the plane p_v –T the liquid-vapour equilibrium (Figure 2). At given temperature, the maximum pressure above which water vapour changes into liquid water is the saturated vapour pressure p_s. Therefore, in a given mass of humid air, vapour pressure can be such as (i) p_v < p_s: water in humid air is in the vapour state; (ii) p_v = p_s: water in humid air is in both vapour state and liquid state as the phase change is at constant pressure p_s= p_v (T_d) = p_v(T₁). Humid air can be saturated (point B in Figure 1) or supersaturated, when liquid droplets (fog) are present (line BC in Figure 1).

Saturated vapor pressure can then be reached in a given humid air in two ways. (i) Cooling a given mass of humid air: vapor pressure remains constant at p_v but p_s decreases until the equality p_v = p_s(T_d) is fulfilled. (ii) Adding a water mass to a given humid air volume at constant temperature: The vapor pressure increases until it reaches at same temperature p_v = p_s. If more water at constant temperature is added, one obtains the coexistence of saturated vapor pressure and liquid. Humid air is then supersaturated.

From above, one can define the relative humidity RH as the ratio at given temperature of the vapour pressure and saturation vapour pressure, RH = p_v(T)/p_s(T). When cooling a given mass of humid air, RH increases to reach 100% at the saturation line. When adding water at constant temperature, RH also increase to 100% and reaches the saturation line. The relative humidity is a common index to determine how close to saturation is a given humid air. The larger RH, the smaller cooling or added mass is needed to obtain condensation.

Notes

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