1.2: Equation of State — Ideal Gas Law
- Page ID
- 46688
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Equation of State
Air pressure is caused by the collisions of rapidly, randomly moving air molecules, so you might expect pressure to increase when there are more molecules in one place (higher density, \(\rho\)) and when they are moving faster (higher temperature, \(T\)). The relationship between pressure, density, and temperature is called the Equation of State.
The gases in the atmosphere have a simple equation of state called the Ideal Gas Law.
For dry air (no water vapor present), the ideal gas law is
\[P=\rho \times R_d \times T \nonumber \]
The ideal gas law assumes that atmospheric gases behave ideally, meaning there are few intermolecular forces and that the molecules are small compared to the space between them. In almost all cases, atmospheric gases can be treated as ideal gases. Again, this is another example of a simplifying assumption we use to approximate relationships in the atmosphere.
Figure \(\PageIndex{1}\) illustrates the effect that air temperature has on its density and pressure. The two air columns above Cities 1 and 2 in (a) have the same temperature, contain the same amount of mass, and have the same volume, so their density is the same. The pressure at the surface is the same in both cities because it is determined by the number of air molecules above it.
In (b), the air temperature above City 1 is lower than the air temperature above City 2. Because of this, the air molecules in the column above City 1 are moving more slowly and take up a smaller volume. Likewise, in City 2, the air molecules are moving faster, and the column's volume is larger. It takes less cold air to achieve the same surface pressure as a larger volume of warm air. The surface pressures are the same, but the temperatures are not.
The surface pressure may be the same in (b), but the pressure aloft is not, leading to air movement visible in (c). At the same altitude above Cities 1 and 2 in (c), there is more air above the same level in City 2 than in City 1. Locally, this creates a higher pressure aloft over the warmer City 2. Because of the difference in pressure aloft, airflow is created that flows from the higher to the lower pressure. This airflow caused by a localized difference in air pressure is what we call wind.
The force arising from a difference in air pressure is called the pressure gradient force (PGF). The word “gradient” here refers to a change over a distance. A pressure gradient force is the force that drives the wind. This concept will be further explored in a later chapter.
The movement of air from City 2 to City 1 creates a falling surface pressure in City 2 and a rising surface pressure in City 1.