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 column of air, 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.