Problem 16P

The exponential atmosphere.

(a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.

(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem.) Show, then, that the pressure obeys the differential equation

called the barometric equation.

(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e−mgz/kT. Show also that the density obeys a similar equation.

(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft, 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)

Problem: Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%).

ANSWER:

a)

The basic formula for the vertical pressure variation is,

--------------(1)

Where is the change in height.

In our question, the height is dZ.

So, ----------------(2)

Where is the density of air.

b)

The ideal gas equation is,

-------------(3)

[multiplied mass in both the sides]

----------------------(4)

If we put this result in equation (2),

. (proved)

The barometric equation.

c)

The pressure as a function of height we got as,

We also have equation (4) which gives the relation pressure and density. So, after putting the value we got in equation (4) in the above equation,

So, the density also obeys a similar type of equation apart from a constant multiplied.