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150 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Justification Box 4.3
We start with Equation (4.21) for a single phase, and write G = H − TS. Its differen-
tial is
dG = dH − T dS − S dT (4.32)
From Chapter 2, we recall that H = U + pV , the differential of which is
dH = dU + p dV + V dp (4.33)
Equation (4.35) com-
bines the first and Substituting for dH in Equation (4.32) with the expres-
second laws of ther- sion for dH in Equation (4.33), we obtain
modynamics: it derives
from Equation (3.5) dG = (dU + p dV + V dp) − T dS − S dT (4.34)
and says, in effect,
dU = dq + dw.The p dV For a closed system (i.e. one in which no expansion
term relates to expan- work is possible)
sion work and the T dS
term relates to the dU = T dS − p dV (4.35)
adiabatic transfer of
heat energy. Substituting for the dU term in Equation (4.34) with
the expression for dU in Equation (4.35) yields
The Gibbs–Duhem dG = (T dS − p dV) + p dV + V dp − T dS − S dT
equation is also com-
(4.36)
monly (mis-)spelt
‘Gibbs–Duheme ’. The T dS and p dV terms will cancel, leaving the
Gibbs–Duhem equation, Equation (4.31).
We now come to the exciting part. By comparing the total differential of Equa-
tion (4.30) with the Gibbs–Duhem equation in Equation (4.31) we can see a pat-
tern emerge:
∂G ∂G
dG = dp + dT
∂p ∂T
dG = V dp −S dT
So, by direct analogy, comparing one equation with the other, we can say
∂G
V = (4.37)
∂p