Page 63 - Physical chemistry eng

40 CHAPTER 2 Heat, Work, Internal Energy, Enthalpy, and the First Law of Thermodynamics

the gas. Similarly, in a reversible adiabatic expansion originating at 1 atm,
P adiabatic 6 P isothermal for a given volume of the gas.

EXAMPLE PROBLEM 2.7
A cloud mass moving across the ocean at an altitude of 2000. m encounters a coastal
mountain range. As it rises to a height of 3500. m to pass over the mountains, it
undergoes an adiabatic expansion. The pressure at 2000. m and 3500. m is 0.802 and
0.602 atm, respectively. If the initial temperature of the cloud mass is 288 K, what is
the cloud temperature as it passes over the mountains? Assume that C P,m for air is
–1
28.86 J K mol –1 and that air obeys the ideal gas law. If you are on the mountain,
should you expect rain or snow?

Solution
T f V f
ln ¢ ≤ =-(g - 1)ln ¢ ≤
T i V i
T f P T f P
=-(g - 1)ln ¢ i ≤ =-(g - 1)ln ¢ ≤ - (g - 1)ln ¢ i ≤
T P f T i P f
i
C P,m
a - 1b
(g - 1) P i C P,m - R P i
=- ln a b =- ln a b
g P f C P,m P f
C P,m - R
Energy a -1 28.86 J K mol -1 -1 -1 - 1b
-1
-1
=- 28.86 J K mol - 8.314 J K mol * ln a 0.802 atm b
-1
-1
28.86 J K mol 0.602 atm
-1
-1
28.86 J K mol -1 - 8.314 J K mol -1
(a) (b) (c)
=- 0.0826
FIGURE 2.18
i
f
(a) A system has the translational energy T = 0.9207 T = 265 K
levels shown before undergoing an adia- You can expect snow.
batic compression. (b) After the compres-
sion, the energy levels are shifted upward It is instructive to consider an adiabatic compression or expansion from a micro-
but the occupation probability of the lev- scopic point of view. In an adiabatic compression, the energy levels are all raised, but
els shown on the horizontal axis is
unchanged. (c) Subsequent cooling to the the probability that a given level is accessed is unchanged. This behavior is observed in
original temperature at constant V restores contrast to that shown in Figure 2.5 because in an adiabatic compression, T and there-
the energy to its original value by decreas- fore U increases. For more details, see R. E. Dickerson, Molecular Thermodynamics,
ing the probability of occupying higher W.A. Benjamin, Menlo Park, 1969. If the gas is subsequently cooled at constant V, the
energy states. Each circle corresponds to a energy levels remain unchanged, but the probability that higher energy states are popu-
probability of 0.10. lated decreases as shown in Figure 2.18c.

Vocabulary

cyclic path indicator diagram path
degrees of freedom internal energy path function
enthalpy irreversible quasi-static process
exact differential isobaric reversible
first law of thermodynamics isochoric state function
heat isothermal work
heat capacity