Problems and Solutions on Thermodynamics and Statistical Mechanics

96 Prublema €4 Solutions on Thermodynamics €4 Statistical Mechanics

Solution:
(a) The molecular number density at height h is denoted by n(h). From
the condition of mechanical equilibrium dp = -nmgdh and the equation of
state p = nkT, we find
1 mg
-dp = --dh.
P kT

Thus n(h) = no exp(-mgh/kT). Let Jf n(h)dh/JT n(h)dh = -, then

1

2

The average molecular weight of the atmosphere is 30. We have

8.31 x lo7 x 273

30 x 980

H= x In2 M 8 x lo5 cm = 8 km.

1 mg

P kT

(b) -dp = --dh is still correct and the adiabatic process follows

p('-7)/7T = const

where 7 = 5 w 7/2 (for diatomic molecules). Therefore -~ dT^7 - -
CU T 7-1
-_ mg dh. Integrating we get
kT

T - To = -(7 - l)mg(h - ho)/7k.

Furthermore,

_- dT -

• mg NN -0.1 K/m.
  dh 7k

1099

The atmosphere is often in a convective steady state at constant en-

tropy, not constant temperature. In such equilibrium pV7 is independent

of altitude, where 7 = Cp/Cu. Use the condition of hydrostatic equilibrium

in a uniform gravitational field to find an expression for dT/dz, where z is

the altitude.

(UC, Berkeley)