From the ideal gas law, the density of water, w is given by
w= ( <mw>w fw e)/(R* T)
where <mw>w is the molecular weight of water (18.016 gm/mole),
fw is a correction factor for non-ideal behavior which can be
taken as fw = 1, e is the partial pressure of water, R* is the
universal gas constant, and T is the local temperature.
The total density, , is given by the sum of dry and moist
densities ( i.e., assumes all other species are included in ``dry'').
= d + w = [ <mw>d · (P - fw e)]/(R* T)
+ (<mw>w fw e)/(R* T) =
[<mw>d · (P - 0.37803 fw e)]/(R* T)
where <mw>d is the molecular weight of dry air (28.966),
<mw>w/<mw>d = 0.62197, and (<mw>d - <mw>w)/<mw>d = 0.37803.
The specific humidity is the ratio of the moist to total density and
is dimensionless, but is usually expressed in gm/Kg units.
q = w/ = (0.62197 fw e) / (P - 0.37803 fw e)
w = q ·
and the volumetric mixing ratio, f, (i.e., ratio of
water number density, Nw, to the total number density
Nt) is given by
fw Nw/Nt = <mw>/<mw>w · q
where the average molecular weight, <mw>, is given by
<mw> = <mw>d · (1 - fw) + <mw>w · fw
Mass Mixing Ratio
r = w/d = 0.62197 ·
[(fw e)/(P - fw e)] 0.62197 ·
e/(P - e) q
from US Standard Atmosphere 1976, Table 20, pg. 44,
r(z) ppm by mass
% Relative Humidity
U 100 · r/rs 100 q/qs
where rs r(e = es) and qs q(e = es).
The saturated vapor pressure is approximately given by
es 6.11 · 10[(a T)/(b + T)] mBar
rs 0.62197 · es/(P - es) for es > P
rs 0.62197 for es P
where T is given in °C and a and b are constants.
Dew Point Temperature
The Dew point is the temperature at which the partial pressure of
water reaches the saturation value, that is
e = (w R* T)/(<mw>w fw) = es(Tdp)
The empirical expression for es can be used or a table
lookup can be utilized after e is calculated.