Page 56 - Essentials of physical chemistry

18 Essentials of Physical Chemistry

RT c 2a RT c 2a 8a
so ¼ which means that T c ¼ . For those who like Arbie roast
¼
(3b b) 2 27b 3 4b 2 27b 3 27Rb
beef sandwiches, there is a silly way to remember this result as ‘‘If you ate (8a) over (=)27Rb
(Arbie) sandwiches you will reach a critical temperature (T c ) fever.’’ While it is unlikely you
could eat 27 sandwiches, the mnemonic helps you remember the formula. Next we go back to
Equation (1) to find ‘‘a.’’

8a
R
27Rb a 8a a 6 2a 2a a
P c ¼ 2 ¼ 2 2 ¼ 2 ¼ 2 ¼ 2
(3b b) 9b 54b 9b 6 54b 6V 3V
c c
2
so that a ¼ 3V P c .
c
Let us also see what R looks like for the van der Waals gas:
2
8a 83P c V c 8 P c V c
R ¼ ¼ ¼ :
27T c b V c 3 T c
27T c
3

8a 8 P c V c
2
c
27Rb 3 T c
In summary, b ¼ V c =3, a ¼ 3V P c , T c ¼ , and R ¼ . In Figure 1.7 the parameters
are the ‘‘reduced variables’’ which can be defined as

P V T
, , and :
P r ¼ V r ¼ T r ¼
P c V c T c
Therefore, we can insert the definitions of (P r , V r , T r ) and obtain

3 1 8
¼ T r :
P r þ 2 V r
V r 3 3
That is a demonstration of the ‘‘law of corresponding states’’ for the van der Waals equation in terms
of the reduced variables. Other equations of state should also pass the test of freedom from
parameters when expressed in terms of the reduced variables. Learning and=or memorization of
the derivation of the van der Waals critical point parameters will simultaneously improve your math
skills as well as emphasize the importance of the critical point. The formulas for the critical
temperature and critical pressure will be used in later discussion. How fast can you do this
derivation?

SUPERCRITICAL FLUID CHROMATOGRAPHY
What good is the critical point? Recently a new form of chromatography has been developed that
may be of considerable use in forensic applications in that it does not destroy the sample (depending
on the detector) and is capable of resolving some very sticky, gummy materials that normally would
be useless goo! Further the equipment is relatively simple and occupies about the space of a normal
desk. The technique of supercritical fluid (SCF) chromatography is an analytical method for
treatment of materials that are normally difficult to resolve into separate components. While other
materials can be used as a mobile phase, the main idea is to use carbon dioxide at a pressure and
temperature range that is slightly above the critical temperature as an excellent solvent for high-
molecular weight organic materials (Figure 1.8). While detection and identification of various
materials are of main interest, the sample can be recovered after passing though the column as
long as the detector does not use flame ionization.