How does air pressure change with temperature?

Air pressure decreases with an increase in temperature.

As the air is warmed, its volume is increased, which means there is less air density in a specific area.  Pressure and density are directly related.  Changes in temperature are the largest factor affecting density during horizontal flight as the temperature may be different at one location than another, when the actual elevation is the same.  As height increases, the temperature decreases, which increases the pressure if everything else was equal.  Air temperature decreases at a rate of 1.98 degrees Celsius per 1,000-foot increase in altitude.  However, the increase in altitude lowers the air pressure at a faster rate than the decrease in temperature raises the pressure.  The net result is that an increase in altitude will decrease pressure even though the temperature is lower.  However, at two points with the same elevation, the lower temperature will have more pressure, resulting in better performance.

Reference(s):

FAA AC-00-6B Aviation Weather pg. 5-4
FAA-H-8083-25B Pilot’s Handbook of Aeronautical Knowledge pg. 4-3, 12-3

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