Problems and Solutions on Thermodynamics and Statistical Mechanics

98 Problem^8 Sdutiom on Thermcdynamics El Statistical Mechanics

Solution:

Near the ground, we have

dp/dz = -ape.

Dynamic considerations give dpldz = -pg.
Thus a = pog/po, where po is the density of air near the ground. Treating
air as an ideal gas, we have

PO = RTo/Vo = RTopo/M ,

where R is the gas constant, Vo is the volume and M the molecular weight
1
(: ’28 + ‘32 = 29. Thus we have a = Mg/RT.
The slow rising of the gas group can be taken as a quasi-static process.
It has the same p and p as the atmosphere surrounding it. Thus the same
is also true of the temperature T. In the adiabtic process,

Fpl-7 = const ,

with

7 = Cp/Cv = (Cv + R)/Cv = 715.

Differentiating we have

dT _- - (^7) -- - 1 dp
T 7P
On the ground, dT/T = -pdz and dp/p = -adz. We substitute them into
above formula and obtain
7-1 2
p=- a=-a.
7 7
1101
Suppose that the earth’s atmosphere is an ideal gas with molecular
weight p and that the gravitational field near the surface is uniform and
produces an acceleration g.
(a) Show that the pressure p varies as