a. Consider a horizontal slab of air whose thickness 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 denisty of air in terms of pressure, temperature, and the average mass m of the air molecules (air is a mixture of N2 (78% by volume), O2 (21%), and argon (1%)) Show, then, that the pressure obeys the differential equation: dP/dz = -(mg/kT)P 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.