8.1.2 Thermodynamics of the air

8.1.2.0.1 Ideal gas

For one molecule with n degrees of freedom (e.g. in the rigid-molecule approximation, n=2 (deg. of rotation) +3 (of transl.) = 5 for O₂ and N₂, and n= 3+3 = 6 for H₂O), the internal energy per molecule is

$$U = {\rm n} \times \frac{1}{2} k T$$

and for ${\cal N}$ molecules of O₂ or N₂,

$$U= \frac{5}{2}{\cal N} k T= \frac{5}{2} n R T$$

where n is the number of moles and R the gas constant.

The Enthalpy is

$$H = U +pV = U + {\cal N} k T\,$$

$$c_v= \frac{dU}{dT}=\frac{5}{2}{\cal N} k ~,~~~ c_p = \frac{dH... ...}{2}{\cal N} k, ~,~~~ \gamma= \frac{c_p}{c_v}= \frac{7}{5}= 1.4$$

For adiabatic expansion, we have the standard relations

$$c_p \mathrm{Log}(\frac{T}{T_0})-R \mathrm{Log}(\frac{p}{p_0}) = 0 ~~{\rm and}~~~ T= Cst\times p^{m}$$

where $m=\frac{R}{c_p}=\frac{\gamma-1}{\gamma}$is the Poisson constant.

8.1.2.0.2 Gas mixture: Dalton's law

A mixture of ideal gases i= 1, 2, ... in a volume V behaves like an ideal gas:

$$p_1 V = {\cal N}_1 k T, p_2 V= {\cal N}_2 k T, ~~...$$ Partial pressures: (8.1)

$$p V = (p_1+p_2)V = ({\cal N}_1 + {\cal N}_2) k T+... = {\cal N} kT$$ Total pressure: (8.2)

Dry air is a mixture of N₂, O₂, ... molecules. It behaves indeed very much like an ideal gas: R_a=c_pa-c_va= 8.3143 J/mol-deg (vs 8.3149 for an ideal gas), $\gamma_a$= 1.404 (vs 1.400 for ideal rigid molecules).

Wet air (without clouds) is a mixture of dry air + H₂O molecules. It is customary to denote by e the partial pressure of water vapor, p_a that of dry air, and p' the wet air pressure. The specific heats of water vapor are not that different from those of ideal gases: c_vw= 25.3 + 2 10^-3 (T-273); $\gamma_w= 1.37$, vs c_v=3R= 24.9 and $\gamma=\frac{4}{3}$ for a rigid asymmetric top.

Then, Dalton's law yields:

$$c'_v=(1-\frac{e}{p'})c_{v_a}+\frac{e}{p'}c_{v_w} \simeq (1+0.2\frac{e}{p'})c_{v_a}$$ (8.3)

The fractional abundance of water vapor and $\frac{e}{p}$ reaching seldom a few percent, the wet air constants are within a small correction term equal to the dry air constants. In particular, introducing the volume density $\rho$, it is customary to write:

$$p'= \frac{R\rho{'}T}{M'}= \frac{R\rho T'}{M_a}$$ (8.4)

where $T'=\frac{M' T}{M_a}= T (1-0.378\frac{e}{p'})^{-1}$, is the virtual temperature.

Then, for the adiabatic expansion of a wet air bubble, one has:

$$T'= Cst \times p^{m'}$$ (8.5)

m' is equal to m_a within few per mil, so, in practice, the adiabatic curves of dry air can be used for wet air (without clouds), provided one replaces T by T'(the difference could reach a few K and could be important near 0$^\circ$C). In the following, we drop the `prime' signs, except for the virtual temperature T'.

  

Figure 8.1: Column densities of important atmosphere components above a given altitude - after Queney (1974)