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Chapter 6 (b) If C° is assumed independent of T, then Eq. (5.19) gives H°(T)
Reaction Equilibrium in Ideal Gas P
Mixtures H°(T ) C°(T ) (T T ). Substitution of this equation into (6.37) gives an
1
1
1
P
equation for ln [K°(T )/K°(T )] that involves C°(T ) as well as H°(T ); see
P
1
P
2
P
1
1
Prob. 6.17. Appendix data give C° P,298 2.88 J/mol-K, and substitution in the
4
equation of Prob. 6.17 gives (Prob. 6.17) K° P,600 1.52 10 .
(c) From the NIST-JANAF tables, one finds G° 47.451 kJ/mol, from
600
4
which one finds K° P,600 1.35 10 .
Exercise
Find K° for O (g) ∆ 2O(g) at 25°C, at 1000 K, and at 3000 K using Appendix
2
P
data and the approximation that H° is independent of T. Compare with the
NIST-JANAF-tables values 2.47 10 20 at 1000 K and 0.0128 at 3000 K.
(Answers: 6.4 10 82 , 1.2 10 20 , 0.0027.)
1
Since d(T ) T 2 dT, the van’t Hoff equation (6.36) can be written as
d ln K° ¢H°
P
(6.40)
d11>T2 R
The derivative dy/dx at a point x on a graph of y versus x is equal to the slope of the
0
y-versus-x curve at x (Sec. 1.6). Therefore, Eq. (6.40) tells us that the slope of a graph
0
of ln K° versus 1/T at a particular temperature equals H°/R at that temperature. If
P
H° is essentially constant over the temperature range of the plot, the graph of ln K° P
versus 1/T is a straight line.
If K° is known at several temperatures, use of (6.40) allows H° to be found. This
P
gives another method for finding H°, useful if H° of all the species are not known.
f
G° can be found from K° using G° RT ln K°(T). Knowing G°and H°, we
P
T
T
P
can calculate S° from G° H° T S°. Therefore, measurement of K° over a
P
temperature range allows calculation of G°, H°, and S° of the reaction for tem-
peratures in that range.
If H° is essentially constant over the temperature range, one can use (6.39) to
find H° from only two values of K° at different temperatures. Students therefore
P
sometimes wonder why it is necessary to go to the trouble of plotting ln K° versus 1/T
P
for several K° values and taking the slope. There are several reasons for making a
P
graph. First, H° might change significantly over the temperature interval, and this
will be revealed by nonlinearity of the graph. Even if H° is essentially constant, there
is always some experimental error in the K° values, and the graphed points will show
P
some scatter about a straight line. Using all the data to make a graph and taking the
line that gives the best fit to the points results in a H°value more accurate than one
calculated from only two data points.
The slope and intercept of the best straight line through the points can be found
using the method of least squares (Prob. 6.60), which is readily done on many calcu-
lators. Even if a least-squares calculation is done, it is still useful to make a graph,
since the graph will show if there is any systematic deviation from linearity due to
temperature variation of H° and will show if any point lies way off the best straight
line because of a blunder in measurements or calculations.
Figure 6.6a plots H°, S°, G°, and R ln K° versus T for N (g) 3H (g) ∆
P
2
2
2NH (g). Note that (Sec. 5.5) H°and S°vary only slowly with T, except for low T,
3
where S° goes to zero in accord with the third law. G° increases rapidly and almost
linearly with increasing T; this increase is due to the factor T that multiplies S°in
G° H° T S°. Since H° is negative, ln K° decreases as T increases
P
[Eq. (6.36)]. The rate of decrease of ln K° with respect to T decreases rapidly as T
P
2
2
increases, because of the 1/T factor in d ln K°/dT H°/RT .
P