Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 13

c)Find the height at which the drag on the rocket is at a maximum.

4.20 Adiabatic Atmosphere (Boston, Maryland)

The lower 10–15 km of t h e atmosphere, the troposphere, is often in a con-

vective steady state with constant entropy, not constant temperature

is independent of the altitude, where

Find the change of temperature in this model with altitude

Estimate in K/km. Consider the average diatomic molecule

of air withmolarmass

4.21 Atmospheric Energy (Rutgers)

The density of the Earth’s atmosphere, varies with height above
the Earth’s surface. Assume that the “thickness” of the atmosphere is
sufficiently small so that it is in a uniform gravitational field of strength

Write an equation to determine the atmospheric pressure given

the function

In a static atmosphere, each parcel of air has an internal energy

and a gravitational potential energy To a very good approxima-

tion, the air in the atmosphere is an ideal gas with constant specific

heat. Using this assumption, the result of part (a), and classical

thermodynamics,show that the total energy in a vertical column of

atmosphere of cross-sectional area A is given by

and that the ratio of energies is

where T is the temperature, is the pressure at the Earth’s surface,

is the molar mass, is the molar specific heat at constant pres-

sure, and is the ratio of specific heats.

Hint: The above results do not depend on the specific way in which

and vary as a function of (e.g.,isothermal,adia-

batic, or something intermediate). They depend only on the fact that

is monotonically decreasing. At some step of the derivation, you

might find it useful to do an integration by parts.

a)

b)

a)

b)