Page 201 - Essentials of physical chemistry

More Kinetics and Some Mechanisms 163

but it is better to use the slope from a graph fitted to several points as shown above ( 10,864) or use
two rates far apart in temperature. The best two-point slope would be over the full range of the data
that spans from 08Cto458C here, and we would obtain a slope from those two points as

5
3
ln (2:92 10 ) ln (1:06 10 ) 2:93 10 3 (318:15)(273:15)
¼ ln ¼ 10,856:85:

1 1 1:06 10 5 45
318:15 273:15
We will use the slope of the graph here ( 10,864) since it is available from a least-squares fit:

DH (25 C) ¼ ( R)[(slope) þ T] ¼ ( 8:314 J=mol K)[ 10,864 þ 298:15],

z
so we find
DH ¼þ87:844 kJ=mol ¼þ20:995 kcal=mol,
z

K rate h slope
DS (T) ¼ R ln 1 ,
z
k B T T
4 34
(3:19 10 )(6:6260693 10 ) 10,864
DS (25 C) ¼ (8:314 J=mol K) ln 1 ,

z

(1:3806505 10 23 )(298:15) 298:15
DS (25 C) ¼ 17:208822 J=deg mol ¼ 4:1130072 cal=deg mol:
z

We can calculate DG in two ways. First,
y
4 34
(3:19 10 )(6:6260693 10 )
DG (25 C) ¼ (8:314 J=mol K)(298:15 K) ln

z
(1:3806505 10 23 )(298:15)
or
DG (25 C) ¼þ92:975 kJ=mol:
z

And so finally, DG ¼ DH TDS ¼þ87, 844 (298:15)( 17:208822) ¼þ92:975 kJ=mol.
z
z
z
While it is comforting that we get the same answer for DG in two different ways and that is a
z
check on the calculations for DH and DS , the opportunities for error in these calculations are
z
z
many. Assuming you might be tested on these equations (hint), it would be best to ‘‘practice’’ doing
these calculations several times when you are rested with a clear head.
SUMMARY OF GRAPHICAL METHOD RESULTS AT T ¼ 258C
DH (25 C) ¼þ87:844 kJ=mol ¼þ20:995 kcal=mol,
z

DS (25 C) ¼ 17:208822 J=8 mol ¼ 4:1130072 cal=8 mol, and
z

DG (25 C) ¼þ92:975 kJ=mol ¼þ22:222 kcal=mol:
z

One conclusion from this study of the simple form of Eyring’s absolute reaction rate theory is that
we can obtain DH and DS using two equations in two unknowns over a limited temperature range.
z
z
However, the more general analytical formulas show that the values are definitely dependent on the
temperature. Considering the many ways to make errors in these calculations, it might occur to a