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THE EFFECT OF TEMPERATURE ON THERMODYNAMIC VARIABLES 169

Justification Box 4.8

We will use the quantity ‘G ÷ T ’ for the purposes of The function G ÷ T
this derivation. Its differential is obtained by use of the
occurs so often in
product rule. In general terms, for a compound function
thermodynamics that
ab, i.e. a function of the type y = f(a, b):
we call it the Planck
dy db da function.
= a × + b × (4.64)
dx dx dx
so here
All standard signs O
d(G ÷ T) 1 dG 1 have been omitted for
= × + G × − (4.65) clarity
dT T dT T 2
dG
Note that the term is −S, so the equation becomes
dT
d(G ÷ T) S G
=− − (4.66)
dT T T 2
Recalling the now-familiar relationship G = H − TS, we may substitute for the −S
term by saying

G − H
−S = (4.67)
T
d(G ÷ T) G − H 1 G
= × − (4.68)
dT T T T 2
The term in brackets on the right-hand side is then split up; so

d(G ÷ T) G H G
= − − (4.69)
dT T 2 T 2 T 2
On the right-hand side, the first and third terms cancel, yielding

d(G ÷ T) H
=− (4.70)
dT T 2
Writing the equation in this way tells us that if we This derivation as-
know the enthalpy of the system, we also know the sumes that both H
temperature dependence of G ÷ T . Separating the vari- and S are tempera-
ables and defining G 1 as the Gibbs function change at ture invariant – a safe
T 1 and similarly as the value of G 2 at T 2 , yields assumption if the vari-
ation between T 1 and
G 2 /T 2 T 2 1 T 2 is small (say, 40 K

d(G/T ) = H − dT (4.71)
T 2 or less).
G 1 /T 1 T 1