Page 145 - Essentials of physical chemistry

Gibbs’ Free Energy and Equilibria 107

more complicated method because it produces graphical results similar to the treatment of rate
constants in Chapters 7 and 8, and only depends on DH 0 . The point of this exercise is to
298
reinforce the idea that equilibria are dynamic and are the result of a dynamic trade-off of a
forward reaction and a reverse reaction. We start with the same equation as above,
0 0
DG ¼ RT ln K P ) ln K P ¼ DG =RT . Then the temperature derivative is
298 298
0
0
0
d ln K P ( DG ) 1 d(DG ) þDG 0 1 d(DG )
. But dG ¼ SdT þ VdP, so that

¼
dT RT 2 RT dT RT 2 RT dT
¼

0 0
qG 0 d DG Þ 0
at constant P in a reactor ¼ S . Then at 1 atm ¼ DS , so that
ð
qT dT
P
0
0
0
d ln K 0 DG 0 ( DS ) (DH TDS ) DS 0 DH 0 d ln K 0
P P .
dT ¼ RT 2 RT ¼ RT 2 þ RT ¼ RT 2 ¼ dT
0 0
d ln K DH
0
This equation P ¼ requires only the DH value, although a way is still needed to
dT RT 2
0
obtain K for some initial condition. This equation can be developed further for graphical analysis
P
by integrating the derivative over a range of temperature.
T ð 2 T ð 2 T ð 2
0
0
DH (T)dT DH 0 dT DH 0 1 1 K (T 2 )
0 P
P
d ln K ¼ 2 ffi 2 ¼ ¼ ln 0 :
(RT ) R T R T 2 T 1 K (T 1 )
P
T 1 T 1 T 1
DH 0 1 1
0 0 0
P
P
R T 2 T 1
Thus, ln K (T 2 ) ¼ ln K (T 1 ) , although this equation assumes that DH is con-
stant over a small temperature range.
Example 2
One of the most significant and most highly studied equilibria in the early twentieth century was the
Haber ammonia synthesis for which Fritz Haber was awarded the Nobel Prize in 1918. In spite of
the fact that the atmosphere of Earth is over 70% nitrogen, it is chemically difficult to use that
enormous source of N 2 to produce nitrogen fertilizers because N 2 is quite unreactive. Because of the
Haber synthesis, as much as one-third of the world food supply is a result of increased agricultural
yield due to nitrogen fertilizers, such as NH 4 NO 3 , which can be synthesized from NH 3 on a
commercial scale. Haber might be considered one of the greatest benefactors to humankind but
the original motivation for developing the process was that Germany needed a way to make nitrates
for munitions in WWI. In the early 1900s, nitrates were manufactured by acidifying ‘‘guano’’ (bird
droppings rich in nitrogen compounds) as found in hundred foot layers on islands off the coast of
Chile to obtain nitric acid and then on to nitrates. However, due to various blockades, Germany was
cutoff from obtaining guano. Thus, ammonia synthesis was a strategic process, which lengthened
WWI but later became very beneficial to agricultural yield. Another consideration is that the Nobel
Prize money itself comes from the earnings of the original patent granted to Alfred Nobel for the
invention of stabilized tri-nitro-toluene (TNT), so converting atmospheric N 2 to nitrates has great
significance to humankind whether for war or peace.
0
The 90th Edn. of the CRC Handbook lists values for NH 3 of DH 298 ¼ 45:9 kJ=mol and
DG 0 ¼ 16:4 kJ=mol. Thus, for the equilibrium N 2 þ 3H 2 ƒ 2NH 3 we can calculate
ƒ!
298
0
DG 0 RT ln K ¼ 2( 16:4 kJ=mol) 0 0 ¼ 32:8 kJ and so we can find the value of K 0 .
298 P 298
32, 000 J=mol 0 5 P 2
0
ln K 298 ¼ ¼ 13:2321; K 298 ¼ 5:58 10 ¼ NH 3 3 :
(8:314 J=mol K)(298:15 K) P

P N 2
H 2