Exponential Atmosphere

This activity will show how the pressure in a typical atmosphere decreases exponentially with altitude, and will allow students to estimate the atmospheric pressure at different altitudes in Earth's atmosphere.

Time requirement: 30 minutes as an activity for older or mathematically advanced students.



You have learned that pressure in a typical atmosphere decreases with altitude because there is less total mass in the atmosphere (and therefore less weight) above any given point as the height of the point increases. There is actually a mathematical function which closely approximates the relationship between pressure and height in an atmosphere. It is called the exponential function.

There is a very important number which arises in mathematics and the sciences which is called e. This number is said to be irrational because its value has an infinite number of decimal places. Another famous irrational number is pi, which is the ratio of the circumference of a circle to its diameter. The value of pi is approximately 3.142, and the value of e is approximately 2.718. Countless physical processes can be described by the number e raised to a positive or negative exponent. Such processes are said to grow or decay exponentially.

If you know the surface pressure (Psurf) and a quantity called the scale height (H), then you can calculate the atmospheric pressure (P) at any height (h) by:

P = Psurf * e-h/H

Remember that e-h/H means the number e raised to the power (-h/H). Many calculators have a function for raising e to any power. It will be helpful if you have one of these available for making exponential atmosphere calculations.

The equation says that the atmospheric pressure decays exponentially from its value at the mean surface of the body where the height h is equal to 0. When the height is equal to H, the scale height, then the pressure has decreased to a value of e-1 = 0.37 times its value at the mean surface. At two scale heights, the pressure has decreased to a value of e-2 = 0.14 times its value at the surface, and so on.

The surface pressure on Earth is approximately 1 bar, and the scale height of the atmosphere is approximately 7 kilometers. With this information, you can estimate the pressure at any altitude in Earth's atmosphere using the formula from above. For example, at an altitude of 3 kilometers in Earth's atmosphere, the atmospheric pressure is approximately:

P = Psurf * e-h/H
= 1.0 * e-3/7
= 1.0 * e-0.43
= 0.65 bars

Make sure that you understand how the formula works and that you can correctly calculate the atmospheric pressure at an altitude of 3 kilometers in Earth's atmosphere. Once you have done that, then try to calculate the atmospheric pressure at the top of Mount Everest, the highest point on Earth. The altitude there is 8700 meters, and don't forget to convert the altitude to kilometers before making your calculation (1000 meters is equal to 1 kilometer)! Since the pressure in the atmosphere is directly related to the density, why do you suppose that most climbers to the summit of Mount Everest require bottled oxygen to breathe? Also, try to calculate the pressure in Earth's stratosphere at a height of 35 kilometers above the surface. The pressure there is just about the same as it is on the surface of Mars!!!

Last updated: February 05, 1998
Joe Twicken / joe@nova.stanford.edu
Rob Wigand