Numerical Example

The equation we are dealing with states that pressure is equal to density times the gas constant times temperature. Now, if two air parcels have the same pressure, the product of density and temperature must be equal too.

Let's take a specific example. Suppose we're near the surface of the earth, where the air pressure is 1000 mb, or in SI units, 100,000 kg per meter per second squared. The gas constant (you don't know this yet) is roughly 300 square meter degrees per second squared. (When physics units get this complicated, I just make sure I keep everything in SI units and I know it will all come out in the end.) So, suppose an air parcel has a temperature of 300 Kelvins, or 27 Celsius. (Kelvin is the SI unit; if you use Celsius by mistake you'll end up making many remarkable discoveries such as negative densities and the like!) What's the density?

(For clarity, I'll drop the units until the end.)

100,000 = density * 300 * 300

so

100,000 = density * 90,000

so

density = 100,000 / 90,000

so

density = 1.1 kg per cubic meter

Try another case: what if the temperature were only 250 Kelvins, or -23 Celsius?

100,000 = density * 300 * 250

so

density = 100,000 / 75,000

so

density = 1.3 kg per cubic meter

A fifty degree change in temperature in this example corresponds to about a 20% change in density. If pressure doesn't change, then the colder the air, the denser the air.

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