Objectives

To provide a new perspective to students that might assist to infer the effect of humidity in estimating the moist air density; and to present a set of equivalent equations that can be employed in teaching this topic.

Method

In a gas mixture, the physical properties of each constituent contribute to the bulk composition. For instance, according to Dalton’s law, the partial pressure of each compound contributes to the total pressure of the mixture. Based on that, the dry average molecular weight (M_d) of a gas mixture of n compounds can be obtained from:

Where M _i and y _i are the molecular weight and the mol fraction of the constituent i respectively. Moreover, the mole fraction can be obtained from the ratio of the compound with respect to the mixture:

Where: P_i, V_i and n_i are the pressure, volume and moles of compound i, respectively; and P_t, V_t and n_t are the total pressure, volume and moles of the gas mixture. Thus, for the mixture we have ∑ y _i = 1

As for the case of dry air, the mole fraction of oxygen (O₂), nitrogen (N₂) and argon (Ar) are y_O2=0.209, y_N2=0.780 and y_Ar=0.009 respectively (^{Wallace & Hobbs, 2006}). Using the molecular weight of each one, O₂ = 32 g/mol, for N₂ = 28 g /mol and for Ar = 40 g /mol, the dry air molecular weight can be obtained using equation 1:

It can be noticed that the molecular weight of dry air (M_d) is between the molecular weights of oxygen and argon. Since nitrogen is more abundant, the average molecular weight M_d is close to it rather than to the other air components.

From the principle of Avogadro, the wet fraction (f _H) can be defined as the ratio of water molecules and wet air molecules. In addition, we can also use the ratio of water vapor volume and humid air volume, and the ratio of the partial pressure of water vapor and the total pressure:

Where f _H is the wet fraction, n_H2O, V_H2O P_H2O are the moles of water vapor, volume and pressure respectively and n_air, V_air, P_air - are the dry air moles number, volume and pressure respectively. The values of f _H ranges from 0 for dry air and 1 value for pure water vapor.

Using a similar approach, the molecular weight of wet air (M_a), which is a mixture of water (M_w) and dry air (M_d) molecular weights can be estimated as follows:

Where M_w is the water molecular weight of water (18 g/gmol) and M_d is the dry air molecular weight of dry air (28.9 g / gmol).

We define an air parcel as an air volume that has a specific pressure, temperature and fixed composition conditions. Using equation (4) it is possible to calculate and plot the values of M_a for dry air (f _H =0) and for water vapor (f _H =1) Figure 1 presents the possible M_a values. It is observed that when water vapor content increases, the air parcel molecular weight goes from the molecular weight of the dry air (M_d = 29 g/mol) to the molecular weight of the water (M_w = 18 g/mol).

Figure 1 Change in molecular weight with respect to the wet fraction (f_H). 

Nevertheless, as environmental conditions, the general equation of the gases is satisfied and the gas density can be obtained with the following relation:

Where P-pressure (atm), V- volume (L), M - Mass of the compound (g) molecular weight (g / mol), R gas constant (L atm K^-1 mol^-1) and T temperature (K).

Density (ρ) can be obtained from Equation 5:

Where ρ is the gas density (g/L). From the equation 6 it can be observed that the density of a gaseous mixture with the same pressure and temperature conditions is directly proportional to the gas molecular weight. Therefore, in an air parcel that contains water vapor, its density will be lower than that of dry air at the same conditions of T and P. Hence, wet air is lighter than dry air.

Absolute humidity

Absolute humidity is the concentration of water vapor in the air. It refers to the mass of water vapor in a unit of air volume (^{Stull, 2015}), and corresponds to the partial density:

Using equation 6 in 7 the following expression is obtained:

From the above equation, you can identify the partial pressure of the water is P_w = f_HP.

Specific humidity

The specific humidity (q) is the ratio of water vapor mass with respect to the total mass of humid air (^{Jacobson, 1999}):

Substituting water (eq. 8) and wet air (using eq. 4 in eq. 6) densities in eq. 9 the following relation is obtained:

Equation 10 shows an explicit relation between dry and water molecular weights in the specific humidity estimation.

Mass Mixing Ratio

The mass mixing ratio (r) is the water vapor mass (m) with respect to the dry air mass (^{Stull, 2015}):

Substituting equation 6 for dry air and water vapor in equation 11, and simplifying, is obtained:

Comparison between equations

The following equations are commonly used in text books (^{Jacobson, 1999}; ^{Perry et al., 1997}; ^{Stull, 2015}) for computing different quantities using the ratio water-air molecular weight (ε=0.622) in the case of the mass mixing ratio the equation used is:

Where P_w is water vapor partial pressure, P_d is the dry air partial pressure.

The specific humidity is calculated by:

And the moist air molecular weight:

Using the known equations can have an error with respect to the proposed equations of up to 0.21%. For r and q, using ambient water content P_w=10 hPa, P_d=1013 hPa this is mainly because ε has only 3 significant figures even though equations 10 and 14 are equivalent algebraically (see appendix for demonstration). In atmospheric sciences the water content varies from 0 to 5% (^{Wallace & Hobbs, 2006}), however in stack emissions the moist content varies from 10% (in boilers) to 100% (reactor exhaust).