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THE EFFECT OF TEMPERATURE ON THERMODYNAMIC VARIABLES 173

OH
CH 3 CH 2 CH 3 O CH 2
O +
H CH 2 H
OH O CH 2
(VII) (VIII) (IX)

Calculate the enthalpy change H O for the reaction if the equilibrium constant for
the reaction halves when the temperature is raised from 300 K to 340 K.

Justification Box 4.10

We start with the Gibbs–Helmholtz equation (Equation (4.62)):

O O
G G 1 1
2 1 O
− = H −
T 2 T 1 T 2 T 1
Each value of G O can be converted to an equilibrium constant via the van’t Hoff
O
O
isotherm G =−RT ln K, Equation (4.55). We say, G =−RT 1 ln K 1 at T 1 ;and
1
O
G =−RT 2 ln K 2 at T 2 .
2
Substituting for each value of G O yields

−RT 2 ln K 2 −RT 1 ln K 1 O 1 1
− = H − (4.79)
T 2 T 1 T 2 T 1
which, after cancelling the T 1 and T 2 terms on the right-hand side, simplifies to

1 1
(−R ln K 2 ) − (−R ln K 1 ) = H O − (4.80)
T 2 T 1
Next, we divide throughout by ‘−R’ to yield

H O 1 1
ln K 2 − ln K 1 =− − (4.81)
R T 2 T 1
and, by use of the laws of logarithms, ln a − ln b = ln(a ÷ b), so the left-hand side of
the equation may be grouped, to generate the van’t Hoff isochore. Note: it is common
O
(but incorrect) to see H O written without its plimsoll sign ‘ ’.

We often talk about
The isochore, Equation (4.81), was derived from the integrated ‘the isochore’ when we
mean the ‘van’t Hoff
form of the Gibbs–Helmholtz equation. It is readily shown that the
isochore’.
van’t Hoff isochore can be rewritten in a slightly different form, as: