A Derivation of
the Virtual Temperature
MJ Mahoney
Created: Oct 26, 2005
Last Revision: Oct 26, 2005
If you consider the ideal gas law, 
, you notice that
the the gas constant in this equation is the gas constant for dry air, Rd.
As written this equation is only valid if the air is dry; that is, the
relative humidity (RH) is zero. If the air is not dry, a different
expression:
(1) 
must be used. Instead of the actual temperature, we define the virtual
temperature (Tv) in a way that allows us to use the gas
constant for dry air. Because moist air is lighter than dry air, and
therefore less dense, the virtual temperature is always greater than
the actual temperature for the same reason that warm air is less dense
than cold air (Charles' Law). To derive an expression for the virtual
temperature we solve equation (1) for the density:
(2) 
The total density (
) is the sum of the
density of dry air (d) and
water vapor (v), which
in turn (using the ideal gas law) can be expressed in terms of the
partial pressures of water vapor (e) and dry air (p-e), and the gas
constants for vapor (Rv) and dry air (Rd). The
gas constants that we are using here are expressed in terms of energy
per unit mass per Kelvin. The Universal
Gas Constant on the other hand is expressed as energy per kmole
per Kelvin (8314.32 J/kmole K). Therefore, our gas constants are
obtained from the Universal Gas
Constant by dividing by the weight of a kmole of vapor or dry
air; that is, the gram molecular weight (gm/mole = kg/kmole). A kmole
of water vapor weighs 18 kg, and a kmole of dry air weight 28.9644 kg.
Because of this, the gas constant for water vapor (Rv = 461.90666 J/kg
K) is much greater than that for dry air (Rd = 287.0531 J/kg K). The
ratio of the molecular weights of dry air to water vapor is given a
special name, "epsilon a":
(3) 
= 0.622004944
Using this, and the second and last expressions in equation (2) we have:
(4) 
Or on inverting:
(5) 
where in the last expression we have replaced the water vapor partial
pressure (e) using the expression for relative humidity (RH):
(6) 
The relative humidity (RH) is the ratio of the vapor pressure (e) at
temperature T to the saturation
vapor pressure (es) at the same temperature expressed as
a percentage. The saturation vapor pressure is only a function of
temperature, and decreases as the temperature is decreased. At constant
pressure, decreasing the temperature will cause the water vapor to
condense out when saturation is reached. This occurs at the dew point or condensation temperature. The
difference between the actual temperature and the dew point temperature
is the dew point depression.
The virtual temperature is a function of pressure (p), temperature (T)
and relative humidity (RH):
(7) 
Saturation Vapor Pressure
The integrated Claussius-Clapeyron equation for atmospheric water vapor
(Curry and Webster, p112), and it's inverted form to determine the dew
point temperature at a given vapor pressure and temperature,
theoretically determine the saturation vapor pressure. However, these
theoretical expressions are not a good fit to experimental data because
Dalton's Law of partial
pressures is not accurate (we're dealing with non ideal gases), the
condensed phase is also under atmospheric pressure, and the condensed
phase is not pure water (it also contains air). A fit to experimental
data is given by (Flatau et al., J. App. Meteorol., 31, 1507-1513, 1992):
es = A(0)
For i = 1 To 6
es = es + A(i) * dT ^ i
Next i
where dT is the temperature in Celcius and
A(0) = 6.1117675
'Saturation vapor pressure (hPa) at 0 Celcius
A(1) = 0.443986062
A(2) = 1.43053301E-02
A(3) = 2.65027242E-04
A(4) = 3.02246994E-06
A(5) = 2.03886313E-08
A(6) = 6.38780966E-11