c06 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come
THE TEMPERATURE DEPENDENCE OF THE FREE ENERGY 89where we recall that (dG/dT)p=−S.FromG=H−TS, H=G+TS, and
−G−TS=−H,sod(
G
T
)
dT=
−TS−G
T^2
=
−H
T^2
andd(
G
T
)
dT=
−H
T^2
It is convenient to remember thatd(1/T)/dT=− 1 /T^2 ordT/d(1/T)=−T^2 ,
sodTd(
1
T
)
d(
G
T
)
dT=−T^2
(
−
H
T^2
)
=H
ord(
G
T
)
d(
1
T
) =H
This is one form of theGibbs–Helmholtz equation. It is general and applies to
chemical and physical changes. For relatively short temperature intervals,H
may be regarded as a constant. A plot of G/T vs. 1/T givesH. Con-
versely, knowingHenables one to determineG/T, henceGat tempera-
tures other than 298 K. Generally speaking, reactions are more sensitive to tem-
perature changes than to pressure changes of comparable magnitude, hence the
Gibbs–Helmholtz equation is of overarching importance in practical and industrial
chemistry.
We now have methods to determineGat any temperature and pressure from
the tabulatedG^298 values under standard conditions. The usual technical com-
plications arise for real systems. For example,H may not be constant, but it
maybefollowedcloselybyapowerseriesoftheformH=a+bT+cT^2 +···.
These extra terms make the equations look messy, but the principles remain the
same.