Page 198 - Essentials of physical chemistry

160 Essentials of Physical Chemistry

Mathematical note: A common student error can occur at this point if the differences of the
reciprocals are rounded, subtracted, and then the final reciprocal is rounded again:

1 1 1
ffi 10, 000
ffi 3 3 ffi 3
1 1 (3:25 10 ) (3:35 10 ) 0:10 10

308:15 298:15
vs.

1 (308:15)(298:15)
¼ 9187:49225,
¼
1 1 10
308:15 298:15

which is easier and more accurate.
Recall that the E* value from the Arrhenius fitas

E* ¼ 90:390 kJ=mol ¼ 21:604 kcal=mol:

Next, we come to a questionable step with this substitution method. What temperature should we
use to calculate DS ? We choose 258C but only because that is a standard for other values.
z
Then at 258Cwe find DS ¼ 19:00862353 J=deg ¼ 4:543170059 cal=deg.
z
We have shown the ten significant figures because the numbers vary over many powers of ten
and we want to be careful in finding DS , which is a small number. This is also one of those tricky
z
calculations where you separate units when you take the natural logarithm, so that you need to make
sure your units are correct before you rearrange the logarithmic equation. Now let us check the DS z
value at 458C:

87, 307:86592
¼ DS 274:4235924
z z
318:15
293:4322159 ¼ DS
so then we find that DS ¼ 19:00862351 J=deg ¼ 4:54317006 cal=deg:
z
While it is very comforting to find the same value for DS at both temperatures as a result of
z
solving two equations in two unknowns, you might expect that if you do not keep all the places on
your eight or ten significant figure calculator you might get a different answer at different
temperatures. You may have noted with annoyance that we have been using the full number of
significant figure values for the physical constants and this is one time that you do have to be
precise. Having said that, the main thing is that DS negative. This means that in this reaction, the
z
change in entropy probably is caused by some sort of solvent ordering around the immanent cation
site. Let us think about that for a moment. The cycloheptane ring is not flat or in a chair form, so
the presence of the added methyl group prevents easy attack by the solvent on the chloro
substituent from one side. The only plausible way the SN1 departure of the Cl can be encouraged

to leave the ring is by some fairly ordered arrangement of solvent molecules, so the negative DS z
value comes from local ordering of solvent near the chloride ion. Essentially, all reactions have a
negative DS value in the Eyring model due to the need for a fairly specific geometry in the
z
activated complex.