Note 1.M The 'ideal-gas law'

Air is practically an ideal gas, i.e. it more or less obeys the following law:

pressure = density x temperature x a constant

This is called the equation of state. The constant (called the ideal gas constant R) has a value of 8.314 joules per degree of temperature per mole of gas, i.e. 287 joules per degree per kilogram or 287m2/s2.K in the case of dry air. A joule is a quantity of energy, the amount released in one second when an electrical current of one ampere flows through a resistance of one ohm it is equivalent to about 0.24 calorie. A mole of a substance is the amount in grams equal to the molecular weight M, e.g. 29 grams in the case of air (see Note 1.C).

Thus, the following applies in the case of dry air -

p = r.R.T Pascals

where p is the pressure, T is the temperature (Kelvin) and r is the density (kg/m3). The meaning of the equation is that the pressure exerted by the impacts of the randomly moving molecules of a unit volume of gas on its enclosure is proportional to the number of molecules (ie the density) and their energy (ie the temperature). For example, air with a density of 1.2kg/m3 and a temperature of 280K has a pressure of 964hPa (ie 1.2 x 287 x 280).

Rearrangement of the equation to the following shows that the air's density is proportional to its pressure (as shown in Note 1.L) and inversely proportional to its (Kelvin) temperature -

r = p / R.T kg/m3

A small correction has to be applied if the air contains water vapour, which is lighter than air (Note 1.C). We can still use the ideal gas law, but now with the 'virtual temperature Tv -

p = r.R.Tv Pa

The virtual temperature is slightly higher than the temperature T, thus -

Tv = T.(1+0.61r)

where r is the concentration of water vapour in the air (kg/kg), called the 'mixing ratio' (Chapter 6). Tv is the temperature of dry air with the same density as that of the given unsaturated moist air.

The ideal-gas law shows that the same 0.8% reduction of density, resulting from increasing the air's water-vapour content by 2% (Note 1.C), obtained by warming dry air at 15°C (ie 288K) by 2.3 degrees, since 2.3/288 equals 0.008. The law is sometimes called the Equation of State for gases. It gives either the pressure, temperature or density, once the other two are known.