Page 102 - Essentials of physical chemistry

64 Essentials of Physical Chemistry

Next we use what we have learned about heat capacities and the sign convention for work
done on=by a gas to write

nRT
dU ¼ nC V dT and dw ¼ PdV ¼ dV:
V

Next we use a trick due to the nature of U as a state variable. While heat and work depend on the
way the process is carried out, we should recall that DU ¼ U after U before , so we can imagine a
process that is idealized and as long as it has the same ‘‘before’’ and ‘‘after,’’ we can calculate
DU ¼ U after U before for that process and get the same answer as the real process that has the same
beginning and ending. We can only do this for state variables as for U and H but we will soon learn
there are several state variables that satisfy this ‘‘after-minus-before’’ property. With that informa-
tion we assume that the compression of the air is carried out reversibly so that we can use the idea
that P int ¼ P ext . Then using the first law we can write

nRT dV
,
nC V dT ¼
V
which is a differential equation but we can get rid of the differentials by integrating over the (T,V )
changes in the process. (Drag T from the right numerator to the left denominator, cancel n and then
integrate.)

T ð 2 V ð 2
2 3
dT dV
C V
4 ¼ R 5 :
T V
T 1 V 1 n¼1

ð
dx x 2
x 2
Now recall the integral (which we will use often) ¼ ln and integrate both sides of our
x x x 1

C V T 2 V 2 V 1
equation to obtain ln ¼ ln ¼ ln . Now take the anti-ln of the whole
R T 1 V 1 V 2
C V
C V T 2 R ðÞ V 1
equation and raise the T-ratio to the power to find ¼ or perhaps as
R T 1 V 2
C V C V
V 1 T R ðÞ ¼ V 2 T R ðÞ . Some texts stop here but this is not our favorite form of analysis for a diesel
1 2
engine. We can show an example as an aside to illustrate the important application of adiabatic
nozzle expansion for low-temperature spectroscopy. Much of the history of this technique is
included in the Nobel Address of J. B. Fenn [6].
ADIABATIC NOZZLE EXPANSION SPECTROSCOPY [7]
A recently discovered way to simplify complex electromagnetic spectra of molecules is to sweep a
high-pressure stream of He gas across the sample and carry it to exit into the sample chamber of a
spectrometer at a much lower pressure (nominally 1 atm) whereupon the He and the sample will
drastically cool and this innovation in spectroscopy has resulted in some remarkably sharp spectral
details being resolved in otherwise broad spectral blurs at room temperature. However, the He must
be precooled to below 518K, which is its Joule–Thomson inversion temperature. For simplicity we
can consider N 2 gas to illustrate the same point.