Write the particular equation representing this data expressing air pressure as a function of altitude.

Question

Climbing Mt. Everest is not an easy task! Not only is it a difficult hike, but the Earth’s atmosphere decreases exponentially as you climb above the Earth’s surface, and this makes it harder to breathe. The air pressure at the Earth’s surface (sea level) is approximately 14.7 pounds per square inch (or 14.7 psi). In Denver, Colorado, elevation 5280 feet, the air pressure is approximately 12.15 psi. Write the particular equation representing this data expressing air pressure as a function of altitude. What is the air pressure in Mexico City, elevation 7300 feet? At the top of Mt. Everest, elevation 29,000 feet? (Note: You will need to carry out the decimal values several places to get an accurate equation and air pressures.)

Answer 1

A thermodynamic equation for pressure as a function of altitude, z, is

 

P(z)  =  P₀ [(T₀ - Γ z)/T₀]^p

 

where Γ is the adiabatic lapse rate (~ .0098 °C/m) ,  T₀ is the temperature (in Kelvins) at the surface and P₀ is the pressure at the surface.

 the exponent p is ~ 3.5   and p ~ 0.029 x 9.81/( R Γ)  

     { R is the gas constant ~ 8.3 and 0.29 is the effective molar mass of air in kg}

 

 According to the problem statement,  the ratio of pressure at Denver to that at sea level is 12.15/14.7 = .827 .  To get the formula to work out (with z = 1609 m at Denver),  we need

 

 .827 =[ (T₀ - 1609 x .0098)/T₀ ]^3.5    which implies that [(T₀ - 15.8 )/ T₀] =  .947   So, solving for T₀,  yields   T₀  = 298 K.   This is a reasonable number.

 

At Mexico City  z =  2225 m.   [(298 - 21.8)/298 ]^3.5  =  .767  So the pressure at Mexico City is .767 x 14.7 = 11.3

 

The calculation for Mt. Everest is similar.