New trends in physics teaching, v.4; The ... - unesdoc - Unesco

New Trends in Physics Teaching IV
WHY THE SKY DOES NOT COLLAPSE: THE HEIGHT OF THE ATMOSPHERE
During the 17th century it was slowly recognized that we live on the bottom of a gigantic sea of
air which extends even above the highest mountains. The question arises why this sea of air does
not collapse and cover the earth in a thin layer with the density of liquid air. This density is of
the same order of magnitude as that of water. A liquid air layer would therefore be only about
10 m high, since a column of water of this height exerts the same pressure as the earth's atmosphere.
The preceding discussion has shown that at a fixed temperature the equilibrium of a thermodynamic
system is not determined by the minimum of the energy E but by the free energy
F = E - TS. Which height H of the earth's atmosphere - we shaIl assume constant density for
simplicity - leads to the minimal free energy at the temperature T = 300 K?
In order to calculate H, we break the volume of a vertical column of air down into N = 2"
sections, each with the 'minimal height' H, = 10 m. The average potential energy of an air
molecule with the mass m in this column is 0.5 X 2" mg H, . This corresponds to an increase in
energy of the molecule AE = (2"-' - 0.5) rng H, when compared to the imaginary liquid air
layer. Similarly the entropy increases with H since we no longer know in which of the N = 2"
sections of the air column the molecule will be found. The loss of information compared to the
(idealized) initial state is thus n bit. The entropy therefore increases by 0.7 nk and the change
in the free energy of the molecule becomes
F(n) = (2"-l - 0.5) rng H, - 0.7 nk T. (Eq. 12)
We now determine the minimum of the free energy as a function of n. This minimum is determined
by F(n) F(n+l), since the derivative of F with respect to n vanishes. From this we obtain
2"-' mgH, = 2"mgH, - 0.7 kT (Eq. 13)
or with 2n H, = H
This result can easily be interpreted. The average kinetic energy of the air molecules at the
temperature T is 1.5 kT. The molecules can therefore rise up to a height H at which their potential
energy rng H is equal to their kinetic energy. Inserting numerical values (k 10-23
J K-l, T x 300 K,
m = 3 X kg, g x 10 m/s2) leads to the correct order of magnitude of the height of earth's
atmosphere, H 14 km [ 131. This corresponds to n X 10 and thus to the well-known fact that
under normal (i.e. atmospheric) conditions the density of gases differs by a factor 21° 1000
from the density of solids or liquids (figure 5).
Our first example has illustrated some of the properties of the entropy and the free energy
with the help of well-known facts. In the following examples we shall use the same mathematical
framework to proceed slowly into more complicated and interesting problems [ 141.
WHEN LIQUIDS LET OFF STEAM: MOLECULES IN SEARCH OF FREEDOM
Every housewife knows that energy is needed for cooking. This energy is used basically for
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