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THE EFFECT OF PRESSURE ON THERMODYNAMIC VARIABLES 149
apples, the total price will depend on the price of each item, and the amounts of each
that we purchase. We could write it as
d(money) = (price of item 1 × number of item 1)
+ (price of item 2 × number of item 2) (4.27)
While more mathematical in form, we could have rewritten Equa- We must use the sym-
tion (4.27) bol ∂ (‘curly d’) in a dif-
ferential when several
∂(money) ∂(money) terms are changing.
d(money) = × N(1) + × N(2) + ··· The term in the first
∂(1) ∂(2)
(4.28) bracket is the rate of
change of one variable
where N is merely the number of item (1) or item (2), and each
when all other variables
bracket represents the price of each item: it is the amount of money are constant.
per item. An equation like Equation (4.28) is called a total differ-
ential.
In a similar way, we say that the value of the Gibbs function
The value of G for
changes in response to changes in pressure and temperature. We
a single, pure mate-
write this as
rial is a function of
G = f(p, T ) (4.29)
both its temperature
and pressure.
and say G is a function of pressure and temperature.
So, what is the change in G for a single, pure substance as the
temperature and pressure are altered? A mathematician would start The small subscripted
answering this question by writing out the total differential of G: p on the first bracket
tells us the differential
must be obtained at
∂G ∂G
dG = dp + dT (4.30) constant pressure. The
∂p ∂T
T p subscripted T indicates
constant temperature.
which should remind us of Equation (4.28). The first term on the
right of Equation (4.30) is the change in G per unit change in
pressure, and the subsequent dp term accounts for the actual change in pressure.
The second bracket on the right-hand side is the change in G per unit change in
temperature, and the final dT term accounts for the actual change in temperature.
Equation (4.30) certainly looks horrible, but in fact it’s simply a statement of the
obvious – and is directly analogous to the prices of apples and sweets we started by
talking about, cf. Equation (4.27).
We derived Equation (4.30) from first principles, using pure mathematics. An alter-
native approach is to prepare a similar equation algebraically. The result of the
algebraic derivation is the Gibbs–Duhem equation:
dG = V dp − S dT (4.31)