Problems and Solutions on Thermodynamics and Statistical Mechanics

Thermodynamics 95

in the same way they would be for an adiabatic process, find p(z) and p(z).

(b) Show that for this case no atmosphere exists above a zmaX given

by zmax - - 2 (7) , where R is the universal gas constant per gram.

Solution:

7-1

(SUNY, Buflulo)

(a) When equilibrium is reached, we have

By using the adiabatic relation pp-7 = pOpO7, we obtain,

p7-2(~)dp(~) = --dz SP;.

7PO

With the help of the equation of state p = pRT, we find

and

7/(7-1)

p(2) = po [^1 - -- 7;1;;0]

(b) In the region where no atmosphere exists, p(z,,,) = 0. Thus

z,,, = -^7. - mo.

7-1 9

1098

Consider simple models for the earth’s atmosphere. Neglect winds,

convection, etc, and neglect variation in gravity.

(a) Assume that the atmosphere is isothermal (at 0°C). Calculate an

expression for the distribution of molecules with height. Estimate roughly

the height below which half the molecules lie.

(b) Assume that the atmosphere is perfectly adiabatic. Show that

the temperature then decreases linearly with height. Estimate this rate of

temperature decrease (the so-called adiabatic lapse rate) for the earth.

(GUSPEA)