Another important humidity variable is the mixing ratio which is related to the specific humidity through the following relation:
The mixing ratio can be expressed in terms of the vapour pressure by eliminating between eqs.(1) and (2) giving:
At a given pressure , there is an upper limit to the mixing ratio (the saturation mixing ratio ) set by the satuation vapour pressure . This is the partial pressure exerted by water vapour in equilibrium with a plane liquid water surface and depends only on the temperature. The relative humidity is then defined by and is usually expressed as a percentage.
Entropy of moist air
The equation of state provides a diagnostic relation between three variables that characterize the thermodynamic state of a moist air parcel (e.g. , and ). It does not give any information on the changes that will take place if the parcel is radiatively warmed or if a change of phase of the water substance occurs within : for that one needs to appeal to the laws of thermodynamics. The most basic of these is the first law which is a statement of energy conservation. When the working substance is a perfect gas, the first law of thermodynamics takes the form:
where is defined (as a residual) to be the heat energy input/output; is the change in
internal energy (per unit mass) when is the specific heat at constant volume, and is the work done by expansion or contraction where is volume of unit air mass (i.e. the specific volume).
The second law of thermodynamics introduces the fundamental concept of entropy. Much has been written about the interpretation of this law and its implications are often quite difficult to grasp. For our purposes it will be sufficient to regard the entropy as a state variable having the property of being conserved under reversible, adiabatic changes. For general (though reversible) changes in state the entropy is defined by:
and upon substitution into eq.(4), the first law may be written as:
or
where ( ) is the specific heat at constant pressure. If a gas undergoes a reversible process, it passes through a sequence of thermodynamic states sufficiently slowly, and without hysteresis, that the states may be retraced. An example of a non-reversible process is the spontaneous freezing of supercooled droplets that leads to the generation of cirrus fallstreaks : such a process causes an increase in air parcel entropy.
The entropy of a mixture of non-interacting gaseous constituents is equal to the sum of the entropies of each constituent. The entropy of the dry air component of a saturated air parcel satisfies the equation:
which on dividing by T and using the perfect gas equation eq.(1) for gives:
or, on integration:
- to within an arbitrary constant (from here onwards will be the specific heat of dry air). Since
one may write
where , and using an approximate form of eq.(3) (i.e. ) gives: