Ideal Gas Law

The temperature, number density, and pressure of a gas are related to each other through the gas law equation:

pressure

=

temperature

x

number density

x

constant

We will use the ideal gas law and the kinetic model representation of a gas to explain the behavior of air. It much simplier to understand this material if one of the three state variable (temperature, pressure, or number density) is held constant, while the other two are allowed to change.

A. Hold number density constant

Suppose we put some air (gas) in a sealed, rigid container. No gas can enter or leave the container, so the number of gas molecules in the container cannot change. The container is rigid so that the size and shape of the container cannot change, in other words the volume of the container cannot change. Therefore, the number density (number of molecules divided by the volume of the container) cannot change.

Now suppose we heat the air in the container. This raises the temperature of the air in the container. At a higher temperature, the average speed of the individual gas molecules increases. Therefore, gas molecules hit the walls of the container harder and more often, increasing the pressure in the container. The reverse happens if you cool the air in the container. In summary, if the number density of a gas is held fixed, increasing the temperature of the gas, increases its pressure and decreasing the temperature of the gas, decreases its pressure.

B. Hold pressure constant

In this case, suppose we put some air (gas) in a sealed, flexible container like a balloon. Gas cannot enter or leave the container, but the size (volume) of the container adjusts so that the air pressure inside the container equals the air pressure outside of the container.

Now suppose we heat the air in the container. The average speed of individual molecules increase, so they hit the walls of the container harder and more often. This forces the container to get larger until the air pressure inside the container equals the air pressure outside the container. Therefore, the number density of the air in the container has decreased because we have the same number of molecules in a larger volume. The reverse happens if you cool the air in the container. In summary, if the pressure of a gas does not change, increasing the temperature of the gas causes the gas to expand (decrease number density) and decreasing the temperature of the gas causes the gas to contract (increase number density).

Relationship between 500 mb height and air temperature

We will now apply B above to explain why the height of the 500 mb pressure surface is related to the temperature of the air in the vertical column from the ground surface to 500 mb. Consider a vertical column of air that extends from the ground surface upward to the top of the atmosphere. Assume that no air is allowed to enter or leave the column. This means that no matter what we do to the temperature of the air in the column, the air pressure at the ground surface will not change (remember that air pressure is determined by the weight of the air above).

If the column of air is heated, it expands upward making the colunm taller. The air pressure at the ground does not change, but the rate at which air pressure decreases with altitude is now slower. The result is that the height of the 500 mb pressure level is now higher. If the column of air is cooled, it contracts, and the 500 mb pressure level becomes lower.

General Rule: Air pressure decreases more slowly with increasing altitude in a warm column of air compared with a colder column of air. This explains why higher 500 mb heights are associated with warmer air and lower 500 mb heights are associated with colder air (see figure).

Thickness: A better estimate for air temperature near the ground

Although we will continue to use the 500 mb height to estimate the pattern of air temperature in the lower troposphere (just above the ground surface where we live), it is not exact. In this section, we will discuss a couple of reasons why the 500 mb height is not completely determined by the air temperature in the lower troposphere. In other words, we will point out why the 500 mb height can sometimes be misleading with regard to air temperature just above the ground.

One issue is that the 500 mb height does depend on the sea level air pressure, i.e., the air pressure at ground level. While the average air pressure at sea level is 1013 mb, the actual sea level pressure at a given location and time varies, typically within the range from 983 - 1043 mb. Thus relatively high sea level pressure tends to raise the 500 mb height since the pressure drop from sea level pressure to 500 mb is greater compared to a case where the sea level pressure is lower. To overcome this dependence on sea level pressure, meteorologists often use a measurement called 1000 mb to 500 mb thickness. This is a measure of the vertical distance between the 1000 mb pressure surface and the 500 mb pressure surface. The 1000 - 500 mb thickness is directly related to the average air temperature between 1000 mb and 500 mb without being influenced by variations in surface pressure (see 1000 - 500 mb thickness figure). Therefore, thickness is a better indicator of how warm or cold the air is in a vertical column above a given location. Note that contour maps of 1000 - 500 mb thickness can be plotted using the University of Wyoming's weather model plotting page.

Another potential problem for interpreting surface air temperature even if using the 1000 - 500 mb thickness is that the 1000 - 500 mb thickness is generally in the range from 4800 to 5700 meters (or about 3 to 3.5 miles). This entire vertical extent need not be uniformly warm or cold, i.e., there can be sublayers of smaller vertical extent that are both relatively warm and relatively cold. For surface temperature, we need to know how warm or cold it is at the bottom of the air column. A better estimate of surface temperature could be obtained by looking at the thickness of a shorter column of air just above the ground surface. For this purpose, meteorologists will look at the 1000 - 850 mb thickness, which is more closely related to the air temperature just above the ground (see 1000 - 850 mb thickness figure).